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StatiCal The Statistician`s Calculator

1. Height Inches Weight Lbs 69 170 62 134 72 210 64 180 you would do this this key sequence gives this result Num NaN 69 Enter 170 Enter 62 Enter 134 Enter n 4 000 72 Enter 210 Enter 64 Enter 180 Test Regr Slope 5 633 Intcp 202 534 RMS Err 21 858 r sq 0 676 r 0 822 p Val 0 178 This means that the straight line Weight 5 633 Height 202 534 fits the data with an overall vertical scatter of about 22 lbs Note that although the correlation was quite high r 0 822 it is not significantly different from zero p 0 178 because of the small sample size Updated March 21 2014 StatiCal User Manual Page 47 of 55 The r 0 key Test the significance of a correlation coefficient This key tests whether an observed correlation coefficient r based on N observations is significantly different from zero It also calculates the confidence interval around the observed r The test is based on the fact that the quantity p r iw 2 is distributed as Student t with N 2 degrees of freedom T To perform this test go to the Num panel and enter r and N onto the stack in that order then switch to the Test panel and tap the r 0 key So for example to test whether an r of 0 5 based on a sample of 30 observations is significantly different from 0 this key sequence gives this result Num 0 5 Enter 30 Test r 0 p Val 0 0049 Hi CL 0 7290 Lo CL 0 1704
2. Need to finish this section Updated March 21 2014 StatiCal User Manual Page 37 of 55 The Tests Panel Statistical Tests StatiCal s final tour de force is its ability to carry out several of the most commonly used statistical hypothesis tests on summarized data counts means and SDs It can also perform several tests t tests ANOVAs correlation tests and linear regression on reasonably small sets of raw data individual values These are done using the keys on the Tests panel fa Fl G 10 23 om You can instantly switch to the Statistical Tests panel at any time by tapping the blue Test key in the left column of any panel The NaN key Entering the Not a Number flag We ve already been introduced to the NaN Not a Number value which up to now has been like an error message that appears when we ask StatiCal to do something it cannot do like take the square root of a negative number But now NaN is actually going to serve a useful purpose and we even have a key that lets us deliberately enter this NaN value onto the stack Why would we ever want to do this Well one of the things StatiCal can do is summarize a set of numbers We can enter say 25 numbers onto the stack and ask StatiCal to summarize them as count mean SD This may be useful in itself but is also very useful for doing subsequent analysis on these numbers like doing a Student t test or ANOVA on two or more such se
3. So for example If the Stack Tapping Swap Then tapping Swap again looks like this gives this result gives this result Sqrt 123 Sqrt 123 Sart 123 Exp 456 Enter 789 Exp 456 Enter 789 Exp 456 Enter 789 Swap is useful if you ve just gotten the result of a big calculation suppose it comes out as the value 5 and you want to do one more operation on it like for example raise 2 to the power of that result You can enter 2 onto the stack but then you can t tap the x y key because that would raise 5 to the power of 2 that is 542 which is equal to 25 rather than raise 2 to the power of 5 that is 2 5 which is equal to 32 Updated March 21 2014 StatiCal User Manual Page 55 of 55 The easiest thing is to enter the 2 then hit the Swap button Then you ll have the 2 and the 5 in the correct positions for doing the x y operation The Roll Up and Roll Dn keys Roll the stack with wrap around The Roll Up key rolls the stack up one position That is every element in the stack moves up by one position The number that was in the top of the stack wraps around to the bottom of the stack The Roll Dn key rolls the stack down one position That is every element in the stack moves down by one position The number that was in the bottom of the stack wraps around to the top of the stack These operations roll only the occupied elements of the stack So If the stack is empty nothing is done when
4. So an r value of 0 5 based on 30 observations is highly significantly different from 0 with a 2 tailed p value of 0 0049 The 95 confidence intervals around the observed r value of 0 5 goes from 0 1704 to 0 729 This could be reported as r 0 5 n 30 CI 0 17 to 0 73 p 0 005 The r r key Compare Two Correlation Coefficients This key tests whether two observed correlation coefficient r1 and r2 based on n1 and n2 observations respectively are significantly different from each other It also calculates confidence limits around the difference between r1 and r2 The test is based on the fact that Fisher Z transform see that section of the manual is approximately normally distributed with a SD of 1 Sqrt n 3 so the difference between two r values r and r2 based on n and nz observations respectively will be normally distributed with a mean of r r2 and a SD of Sqrt 1 n 3 I n2 3 To perform this test go to the Num panel and enter r1 nj r2 and nz onto the stack in that order then switch to the Test panel and tap the r r key So for example to test whether an r of 0 4 based on a sample of 25 observations is significantly different from an r of 0 6 based on a sample of 45 observations this key sequence gives this result Num 0 4 Enter 25 Enter 0 6 Enter 45 Test r r p Val 0 3058 HiCL 0 6558 Lo CL 0 2415 Updated March 21 2014 StatiCal User Manual Page 48
5. The COR produced by this calculator reflects both the effect size and the statistical power of the clinical trial under study Thus trials involving large numbers of patients and or large effect sizes will typically produce undemanding CORs which are easily justified on the basis of current knowledge On the other hand trials involving relatively few patients and or small effect sizes will because they possess relatively little evidential weight produce CORs that demand a substantial amount of prior evidence before the new result can be deemed credible So for example this key sequence gives this result Updated March 21 2014 StatiCal User Manual Page 52 of 55 The Stor Keypad Managing the Calculator s Storage The fifth and final panel of keys may not be very impressive but it s quite important These keys don t do any real calculations a few of them do adds and subtracts instead they deal with the management of StatiCal s various storage locations the stack and the two special memory cells M1 and M2 They can clear these locations out move numbers between them juggle numbers around in the stack and other similar kinds of operations BM E 10 24 pm You can instantly switch to the Storage panel at any time by tapping the blue Stor key in the left column of any panel The memory cells are useful in several kinds of situations Sometimes you may have a number that you know yo
6. StatiCal User Manual Page 12 of 55 Some Conventions Used in This Manual For consistency I ve tried to adhere to certain notational conventions throughout this manual How Examples Are Shown Almost every page of this manual contains specific examples of how the calculator works I try to adhere to a standard format and notational convention for these examples First examples will usually be shown in a table with two or three columns like this All of the examples in this manual will use a consistent notation to indicate your keystrokes For example to enter the number 25 just tap the 2 key and then tap the 5 key then tap the Enter key 2 5 Enter To simplify this notation I ll omit the commas between the individual digits of a single numeric entry 25 Enter To enter the quantity 25 tap the key at the top of the keypad then the 2 key then the 5 key then the Enter key 25 Enter The x y and z designations In the examples and in the naming of many of the keys you ll sometimes see the designations 66 99 66 99 x y and z These stand for numbers you ve just entered onto the stack in that order So when you see something like xy it means that if you first enter a 5 onto the stack and then enter a 2 onto the stack then hit the x y key it will evaluate 5 2 which produces 25 not 2 5 which would produce 32 Updated March 21 2014 StatiCal User Manual Page 13 of 5
7. 2014 StatiCal User Manual Page 26 of 55 The PDFs Panel Probability and Distribution Functions StatiCal as described so far in this manual is in my humble opinion a fairly complete and respectable scientific calculator But here s where it pulls ahead of the others at least if you re a statistician BM 10 21 pm You can switch to the Probability Functions panel at any time by tapping the blue PDFs key in the left column of any panel StatiCal can evaluate the most commonly used statistical probability functions and their inverses e the normal Gaussian distribution Norm the Chi square distribution Chi Sq the Student t distribution Stud t the Fisher F distribution Fish F the Binomial distribution Bin and the Poisson distribution Pois It can evaluate the number of combinations and permutations of n things taken r at a time It can apply the transformations of mean and standard deviation between normal and log normal distributions It can evaluate the arc sine transformation for proportions and the Fisher z transformation for correlation coefficients All of these functions are invoked using keys in the PDFs panel Updated March 21 2014 StatiCal User Manual Page 27 of 55 The Rnge toggle Key Setting the Range of Integration tails of distribution Statisticians generally want probability integrals which are integrals of the probability density functions over some range Stati
8. mean SD N on the stack for each group So if you have case level data you would use the Descr Stats key to summarize them first for each group then tap one of the the tTest 1Grp or 2Grp keys or the AoV key The tTest keys Student t tests There are several Student t tests e one fora single set of numbers testing whether the mean of those numbers is significantly different from zero this is called the single group t test e one for paired samples testing whether the mean of the paired differences is significantly different from zero this is called the dependent samples t test e one for unpaired samples testing whether the difference in the two means is significantly different from zero this is called the independent samples t test The first two types are really equivalent because the first step in doing a dependent samples t test is to calculate the paired differences and then test them with the single group t test The last type for independent samples has two variants e one that assumes that the two samples come from populations with the same SD this is called the equal variance t test or the classic Student t test e one that does not assume that the two population SDs are equal this is called the unequal variance or equal variances not assumed or Welch t test StatiCal has two keys that can handle all of these different types of t tests The 1Grp tTest
9. tap the Enter key for you The Simple Arithmetic Keys x y x y x y X y These keys implement the four basic arithmetic operations between the two most recently entered numbers on the stack In each case the result of the operation replaces the two original operands What happens is this 1 the two original numbers are removed from the stack 2 the arithmetic operation is performed between them and 3 the result is placed onto the stack So for example this key sequence gives this result 5 Enter 2 x y x y 7 5 Enter 2 x y x y 3 5 Enter 2 x y x y 10 5 Enter 2 x y x y 2 5 Notice that at the left of each output field is an indication of what keystroke produced the result displayed in that field Note that this behavior the result of the operation replacing the number or numbers that went into the operation is applied consistently by StatiCal for all of its calculations be they as simple as squaring a number or adding two numbers or as complicated as calculating an ANOVA where many numbers go into the calculation and the calculation produces several results The general rule is that all inputs to the calculation will popped off of the stack and all results will be pushed onto the stack Updated March 21 2014 StatiCal User Manual Page 16 of 55 The Math Panel Advanced Mathematical Calculations The Math Panel implements the mathematical logarithmic exponential
10. you would use the upper row key at the beginning of the number and after the E Don t use this key to subtract two numbers use the x y key at the right of the keypad The E key and Scientific Notation Scientists and engineers often have to work with very large or very small numbers which would be very cumbersome to deal with in the usual notation To enter the quantity one million which is 1000000 or 1x10 you can do it directly 1000000 Enter or using scientific notation 1E6 Enter To enter the quantity one millionth you can do it as 0 000001 Enter or as 1E 6 Enter The largest number StatiCal can deal with is 1 797693 1348623157 E 308 and the smallest nln zero number is 2 2250738585072014 E 308 When is the Enter Key Necessary You must use the Enter key after entering a number only if your next action will be to enter another number So if you want to add 25 and 36 you would do it as follows 25 Enter 36 Updated March 21 2014 StatiCal User Manual Page 15 of 55 You could also have done it as 25 Enter 36 Enter The first Enter is required but the second Enter is optional The first Enter is needed to separate the digits of the number 25 from the digits of the number 36 so that the calculator won t think you were trying to enter the number 2536 The second Enter is not needed because whenever you tap a mathematical operation or function key StatiCal will automatically in effect
11. 9 4404 and a SD of 7 5069 An indication that you might be dealing with log normally distributed data is that 1 the numbers represent something that cannot be zero or negative and 2 the SD of the numbers is roughly as large as the mean of the numbers Updated March 21 2014 StatiCal User Manual Page 35 of 55 The Arc Sine Key The ArcSine transformation for Proportions Proportions can arise from binomial sampling that is where you divide the number of occurrences of some kind event by the number of opportunities for that event to occur these proportions tend not to be normally distributed For one thing they cannot fall outside of the bounds of 0 0 to 1 0 The Arc Sine key transforms observed proportions into more normally distributed quantities so that they can be used analyses that assume normal distributions You would enter the observed proportion as a fraction between 0 and 1 onto the stack then go to the PDFs panel and tap the Arc Sine key The formula for the arc sine transformation which should actually be called the arc sine square root transformation is Y ArcSine Sqrt p The transformed values will range from 0 if the probability was 0 to pi 2 if the probability was 1 This transformation also tends to make the standard deviations of the samples more well behaved it s called a variance stabilizing transformation The transformed values will have a SD of 1 2 Sqrt n where n
12. Fraction Correct a d t often referred to simply as Accuracy Mis classification Rate 1 Overall Fraction Correct Sensitivity a cl use exact Binomial confidence intervals instead of these Specificity d c2 use exact Binomial confidence intervals instead of these Positive Predictive Value PPV a rl Negative Predictive Value NPV d r2 Difference in Proportions DP a r1 c r2 Absolute Risk Reduction ARR c r2 a r1 which DP Number Needed to Treat NNT 1 absolute value of DP or ARR Relative Risk Reduction RRR ARR c r2 Positive Likelihood Ratio LR Sensitivity 1 Specificity Negative Likelihood Ratio LR 1 Sensitivity Specificity Diagnostic Odds Ratio Sensitivity 1 Sensitivity 1 Specificity Specificity Error Odds Ratio Sensitivity 1 Sensitivity Specificity 1 Specificity Youden s J Sensitivity Specificity 1 Number Needed to Diagnose NND 1 Sensitivity Specificity 1 1 Youden s J Forbes NMI Index Contingency Coefficient Adjusted Contingency Coefficient Phi Coefficient Cramer s Phi and Cohen s w Index for 2x2 table Yule s Q a d b c a d b c OR 1 OR 1 Equitable Threat Score a e a b c e where e rl cl t Entropy H r r1 t log2 rl t r2 t log2 12 t Entropy H c cl t log2 cl t c2 t log2 c2 t Entropy H r c a t log2 a t b t log2 b t c t log2 c t d t log2 d t
13. The 2x2 a key p values for a 2 by 2 cross tab table Cross tabs contingency tables are a very common way to summarize the relationship between two categorical variables These tables can be tested for a significant association between the two variables using a Chi square or Fisher Exact test Also an astounding number of things can be calculated from the four numbers in the table depending on what the row and column variables represent From the point of view of significance testing it doesn t matter which variable is tabulated in rows and which is tabulated as columns nor does it matter which category is the top row or the left column But when calculating parameters like relative risk from the table it does matter So we ll establish a convention that when entering cell counts from a 2x2 table into StatiCal we ll enter them in the order left top right top left bottom then right bottom When testing for causality we ll establish the following cell count naming conventions Outcome Occurs Outcome Does Not Occur Total Causative Factor Present a b a b Causative Factor Absent c d c d Total a c b d a b c d When using StatiCal to evaluate the significance of a 2x2 cross tab you must enter the cell counts for cells a b c and d in that order Then you tap the 2x2 a key to calculate the p values Updated March 21 2014 StatiCal User Manual Page 39 of 55 For example Is there a signifi
14. and trigonometric functions found on most scientific calculators Ga A G 10 14 em 3 141592653589793 1 7724538509055159 7 188082728976037 You can instantly switch to the Mathematical Calculations panel at any time by tapping the blue Math key in the left column of any panel In all of the examples that follow I ll indicate the key that has to be tapped to switch over to the appropriate panel If you re already at that panel you don t have to tap that panel key although there s no harm in doing so The Sqrt key Square Root The Sqrt key takes the square root of the number at the top of the stack the bottom of the three displayed numbers The square root replaces the original number actually the original number is popped off of the stack and its square root is pushed onto the stack The remainder of the stack is unaffected So for example this key sequence gives this result Num 25 Math Sqrt Sqrt 5 Num 0 Math Sqrt Sqrt 0 Num Math Sqrt Sqrt NaN Updated March 21 2014 StatiCal User Manual Page 17 of 55 The square root of zero is zero and the square root of any negative number is the special value NaN Not a Number StatiCal does not work with complex numbers those consisting of a real and an imaginary part If you don t know what that last sentence means don t worry about it just remember that you cannot take the square root of a negative number
15. case to be equal to 1 not 0 as you might expect The above definition doesn t make sense for fractional numbers so the factorial definition can be extended to work for all real numbers positive or negative whole number or fractional There are several ways this can be done the most widely accepted way is to define it as the following integral ni f x e dx This gives exactly the same values as the simpler definition when n is a whole number but also works for fractional n This generalized definition is related to the definition of the gamma function generally represented by I the Greek capital gamma n n 1 or mn n 1 The factorial function gets very large as n gets large 4 24 5 120 10 3 628 800 and 100 9 3326x10 For n larger than 170 n exceeds the limit of the calculator s floating point number system so the calculator returns Infinity as the answer This limit might be different for different types of hardware or operating systems Updated March 21 2014 StatiCal User Manual Page 19 of 55 For negative n the factorial function behaves in a very peculiar fashion n alternates between positive and negative values for negative n and at negative whole numbers it goes to infinity in both the positive and negative directions So for negative whole numbers the calculator returns NaN since n has opposite signs on either side of the whole numb
16. groups and enter them onto the stack for you The 2Grp tTest key places six numbers on the stack e thet value degrees of freedom and p value for the unequal variance Welch t test and e thet value degrees of freedom and p value for the equal variance classic t test The AoV key 1 way Analysis of Variance You can think of the 1 way Analysis of Variance simply called AoV here as an extension of the standard unpaired equal variance Student t test to the situation where there are 3 or more groups being compared The AoV also works when there are only 2 groups in which case it gives exactly the same p value as the unpaired equal variance Student t test So you can think of the t test as a special case of the AoV for only 2 groups being compared Because the AoV is really an extension of the unpaired t test it wants to see for each group the summary statistics mean standard deviation and count for each group on the stack This has to be followed by another entry on the stack telling it how many groups of summary statistics you placed on the stack m1 sd1 nl m2 sd2 n2 mG sdG nG G in that order Then you can tap the AoV key to perform the analysis of variance This will place four numbers on the stack the F value the numerator and denominator degrees of freedom and the p value As with the unpaired t test if you have individual observations for one or more groups you can use the Descr Stats key for these
17. key performs a single sample t test This key wants to see the mean SD and N on the stack in that order If you have individual values you can use the Descr Stats key to calculate mean SD and N and it will put those three numbers onto the stack in the right order Then you tap the 1Grp tTest key to perform the test The 1Grp tTest key is also used for the paired t test you just set up the data differently The mean and SD placed on the stack must be the mean and SD of the paired differences and the N must be the number of pairs Updated March 21 2014 StatiCal User Manual Page 45 of 55 So if you had pairs of numbers x1 yl x2 y2 x3 y3 etc you would do the following e first tap the NaN key to mark the start of your data e then enter x1 then tap Enter then enter y1 then tap x y which will replace x1 and y1 by the difference e then do the same thing for x2 and y2 then for x3 and y3 etc e After entering all your paired differences this way you d tap Descr Stats to calculate the mean SD and N values and put them on the stack e and finally tap the 1Grp tTest key to perform the test The 2Grp tTest key performs two unpaired t tests equal variance and unequal variance This key wants to see two sets of summary statistics on the stack meanl SD1 N1 mean2 SD2 N2 in that order If you have individual observations for one or both groups you can use Descr Stats to calculate the summary statistics for these
18. number fields at the top of the calculator Whatever value that was in that field gets slid up one space to the middle field and whatever value that was in the middle gets slid to the top field The value that was in the top field the third entry in the stack gets pushed out of sight but it s still in the stack in the fourth position Normally you place numbers on the stack and then indicate the operation you want done on those numbers Then the results of the operation are placed on the stack where they are available for further calculations Whenever you perform some operation whether it s as simple as adding two numbers or as complicated as performing an ANOVA on five groups of numbers one rule is always the same All the input to the operation the arguments are removed popped from the stack and all the results of the operation are placed pushed onto the stack Updated March 21 2014 StatiCal User Manual Page 7 of 55 Stack Advantages Hewlett Packard has been making RPN calculators since the 1970 s They have a good web page that explains RPN http h41111 www4 hp com calculators uk en articles rpn html Pl just summarize some of the main benefits of RPN calculators 1 They don t have and don t need an key the result of any calculation is displayed immediately once you say what calculation to perform 2 They don t have and don t need parenthesis keys Complicated arithmeti
19. of 55 So an r value of 0 4 based on 25 observations is not significantly different from an r value of 0 6 based on 45 observations p 0 306 The difference between the two r values is 0 20 with a 95 confidence intervals that goes from 0 24 to 0 66 The Tol Intvs key Tolerance Intervals for Normally distributed data Loosely speaking a tolerance interval for a measured quantity is the interval in which there is some likelihood or of which you feel a some level of confidence that a specified fraction of the population s values lie based on a sample that you measured from this population Tolerance intervals have been widely used in statistical process control The Tol Intvs key will calculate tolerance intervals for any specified population fraction and for any specified level of confidence from the mean and standard deviation of a finite sample under the assumption that the population is normally distributed One sided upper and lower intervals as well as the two sided interval are calculated Reference NIST Sematech Engineering Statistics Handbook Section 7 2 6 3 http www itLnist gov div898 handbook prc section2 prc263 htm The example below is taken from one of the examples in this Handbook Note The very last line on the Handbook page The upper one sided tolerance limit is therefore 97 07 1 8752 2 68 102 096 is wrong The standard deviation in their example is 0 0268 not 2 68 so the answer should
20. trials produce statistically significant evidence for the efficacy of some new therapy However statistical significance is a notoriously poor indicator of the credibility of a finding that is the extent to which it provides convincing evidence for efficacy as it fails to take full account of the size of the trial or of pre existing insights Bayesian credibility analysis takes the outcome of a given clinical trial as summarized by the published Odds Ratio OR together with its 95 confidence interval CI and uses a Bayesian method to determine the credibility of the finding in the light of what is already known Specifically it shows whether the new finding when combined with existing knowledge can be taken to have credibly demonstrated efficacy at the 95 confidence level Such a finding is said to be credible at the 95 confidence level You can use any confidence level you prefer For a given result to meet this standard the prior evidence for efficacy must exceed a specific level this level is captured by the Critical Odds Ratio COR The assessment of a given clinical trial result can then proceed as follows e If the stated 95 confidence interval excludes an OR of 1 00 corresponding to no effect the result can be deemed statistically significant at the 95 level e If ORs at least as impressive as that indicated by the COR can be justified by existing knowledge the result can also be deemed credible at the 95 level
21. you press the Roll Up or Roll Dn key If there was only one number in the stack nothing is changed it rolls into its own position If there were two numbers in the stack Roll Up and Roll Dn both behave exactly like Swap That is if you entered the two values 11 and 22 in that order onto an empty stack then after pressing either Roll Up or Roll Dn the stack would have the two values 22 and 21 If there are three numbers on the stack those three numbers cycle around That is if you enter the values 11 22 and 33 in that order onto an empty stack then pressing Roll Up would produce 22 33 and 11 while pressing Roll Dn would produce 33 11 and 22 The Roll Up and Roll Dn keys can be a great time saver if you ve entered a lot of numbers on to the stack planning to use the Descr Stats key and then realize that you forgot to mark the beginning of the data with a NaN You don t have to clear out the stack and re enter everything If the stack was empty before you started entering your data just tap the NaN key then go to the Stor panel and tap Roll Dn once and the NaN will wrap around and be at the beginning of the data If there were already other numbers on the stack before you entered your data it s a little more work but still easier than re entering everything Just tap Roll Dn enough times to roll your data down until the first pre existing number that you don t want in the summary is in the bottom field on the scre
22. 21 2014 StatiCal User Manual Page 54 of 55 The Pop key Pop the bottom number from the Stack The Pop key will remove pop the most recently entered number from the stack and slide the other numbers down to fill the gap What happens to the popped number It gets annihilated it vanishes and is never seen or heard from again You will normally use this only when you no longer want the latest number in the stack The Dup key Duplicate the bottom number on the stack Pressing the Dup key will push a duplicate copy of the most recently added number onto the stack pushing all the other numbers to make room for it So if the top of the stack the bottom of the three visible numbers is 123 45 then tapping the Dup key will duplicate this number So for example If the Stack Tapping Dup Then tapping Pop looks like this gives this result gives this result Sqrt 123 Exp 456 Sqrt 123 Exp 456 Enter 789 Exp 456 Enter 789 Dup 789 Enter 789 That 123 entry wasn t destroyed by the Dup it was just slid to the 4m position and beyond the first 3 visible entries Popping the duplicated 789 causes the remaining entries to slide down one position so the 123 comes back into view The Swap key Swap the two bottom numbers on the Stack The Swap key reverses the two top entries in the stack the two bottom fields in the calculator Tapping Swap again will reverse the reversal leaving things as they originally were
23. 5 The Num Panel Entering Numbers and Performing Simple Arithmetic The Num Keypad panel is used to enter numbers into the calculator and to perform the four simple arithmetic operations addition subtraction multiplication and division In effect this panel is your basic 4 banger calculator but with RPN of course B 10 11 em 12345 1 3587E150 eE 5 3 7 gt x lt x L AR You can instantly switch to the Numeric Keypad panel at any time by tapping the blue Num key in the left column of any panel The lt lt key Backspace The lt lt key is used to erase the last keystroke when entering a number You can think of it as a backspace key So the following sequence 1 2 4 lt lt 3 Enter will place the value 123 on the stack You can hit lt lt several times each time will erase the rightmost character from the number being entered Updated March 21 2014 StatiCal User Manual Page 14 of 55 The lt lt key is active only while a number is being entered into the calculator that is while the number appearing in the bottom field is shown in yellow Once you hit Enter the number turns white and is no longer available for editing and the lt lt key will not have any effect The key The key at the top of the keypad is used to enter a negative number or a negative exponent in a scientific notation number So to enter the number 25 or 1 23E 6 or 1 23E 6
24. 9 degrees of freedom is 2 2622 and so we have Conf Limits 5 2 2622 2 sqrt 10 5 1 4307 3 5693 to 6 4307 This can be set up in StatiCal as follows this key sequence gives this result Num 5 Enter 2 Enter 10 Enter Test Conf Intvs Value 5 0000 From the popup dialog select Mean SD n Desc Hi CL 6 4307 Lo CL 3 5693 Confidence intervals can arise in other contexts as well In particular regression programs usually produce a table that shows for each variable appearing in the regression model the value of the regression coefficient or parameter its standard error SE and perhaps a degrees of freedom df for that standard error If the program does not show confidence limits for the coefficient you would have to calculate them from this formula Conf Limits Parameter tcrit SE If the regression table does not show the degrees of freedom it can usually be calculated as the difference between the number of data points and the number of parameters appearing in the model the df points minus parameters rule For example a straight line has two adjustable Updated March 21 2014 StatiCal User Manual Page 50 of 55 parameters a slope and an intercept so the SE for the slope and the SE for the intercept each have degrees of freedom equal to the number of data points minus 2 Sometimes the actual number of data points is not given so you might just want t
25. Cal offers an easy way to select several commonly used ranges for the probability integrals You enter a single number call it x onto the stack and then you tap the Rnge key repeatedly to toggle the yellow range indicator right above the Rnge key between four sets of integration limits For symmetric continuous distributions like the Normal or Student t the range toggles are defined this way Outer the sum of the areas in the two symmetric tails of the distribution that is the integral from infinity to x plus the integral from x to infinity Inner the area between the two limits for a symmetric distribution that is the integral from x to x Left the integral from infinity to x Right the integral from x to infinity For non symmetric continuous distributions like the Chi square or Fisher F it doesn t make much sense to define Outer and Inner the way we did it above for symmetric distributions For one thing these functions take only positive values of x so integrals from infinity to x are meaningless Also for most practical statistical work the left tail is usually not very interesting significance tests almost always utilize only the right tail area So at the present time StatiCal defines the actions of the range toggles this way Outer the integral from x to infinity Inner the integral from 0 to x Left the integral from 0 to x Right the integral from x to infinity Note that f
26. Information shared by descriptors r and c B H r H c H t c A H r c H r C H r c H c Similarity of descriptors r and c S 7 c B A B C Distance between r and c D r c A C A B C Don t confuse these A B and C numbers with the a b and c cell counts they re quite different Updated March 21 2014 StatiCal User Manual Page 41 of 55 In the smoking emphysema example of the previous section we might be interested in the relative risk or perhaps the odds ratio for smoking considered as a risk factor for emphysema We would first enter our cell counts then tap 2x2 a to calculate the p values and more importantly the special info needed for the subsequent calculation of the various parameter values and their confidence intervals then tap the 2x2 b key The program will bring up a list of all the things it can calculate from a 2x2 table Tap the entry you want for this example say the Relative Risk and it will display that parameter and its confidence limits Then tap 2x2 b again and this time select the Odds Ratio Here s what you can expect this key sequence gives this result Num 22 Enter 10 Enter 12 Enter 16 Test 2x2 a ChiSq 0 0435 Yates 0 0787 Fisher 0 0673 2x2 b then select Relative Risk from the popup list Rel Rsk 1 604 HiCL 2 719 Lo CL 0 956 2x2 b then select Odds Ratio from the popup list Odds Ratio 2 933 HiCL 9 766 Lo CL 0 901 So
27. Rad toggle is Deg The Circ Hyp modifier key Circular or Hyperbolic functions You might remember from high school that the ordinary trig functions we know and love so well are sometimes called the circular trig functions and that there are also a set of strange beasts called the hyperbolic functions that have similar names as the circular functions StatiCal can calculate the hyperbolic functions by means of the Cire Hyp modifier key If you press this key before pressing the Sin Cos or Tan key you ll get the hyperbolic sine cosine or tangent function respectively Like the Inv key the Cire Hyp key is a modifier key not a toggle key That is it s a one time key as soon as you calculate a hyperbolic function the yellow Hyp indicator changes back to Circ You have to hit Hyp every time you want a hyperbolic function You can get the inverse hyperbolic functions by pressing both the Inv modifier and the Hyp modifier in either order before pressing the Sin Cos or Tan key I m not aware of hyperbolic functions being used with degree arguments so when you use the Hyp key the current status of the Deg Rad toggle is ignored If people actually do use the hyperbolic functions with degree arguments let me know and I ll change StatiCal s behavior accordingly So for example this key sequence __ gives this result Num 1 Math Hyp Cos CosH 1 5431 Updated March 21
28. StatiCal User Manual Page 1 of 55 StatiCal The Statistician s Calculator release 1 User Manual By John C Pezzullo PhD Updated March 21 2014 StatiCal User Manual Page 2 of 55 Table of Contents Table OF COMLSINS vis insscsnssnnieins nanddsiiewebeedtionuaen minon neninn ananin diverts uwunedunnsudedinnaumnadenciudetinenatare 2 TER O OU CHOI asasucitiscsstcnciauntcuetaaesentinnabenstebiewdcavanadicdvaeciisaudeavasuvdceu sdeavessubucte lt v navacdacnbenseatan 4 LISTON Y AAE TE 5 General Operation of the CalCulator eececcceeeseseeeeeeeeeeeeeeeeseneeeeeeeeeeeeeeeenesseeneneeees 6 Reverse Polish Notation RPN enina rinier a AEE R EAE AEA ERa 6 ANNE SUACIC nt a n n a a a a a a a u R 6 Stack Advantages arpia ina ea i E EE A E E E RRE EERTE R R EET E ee 7 Exiting the StatiCal app Quitting vs Suspending seeesesseesesresseserrsresstseresressesnresressessresresseese 7 K ys and Panels espesie ee e Se ca et nets Red a EE iae esa etd EEEE e a SA ea 8 Result Fr ldSisne ninnan eia iae a E AAR EAT AE AEE R E E ORAE 9 Field Labels rire ata nda a A E A E E coos 9 setting the Display POrial lt 3 c 428i iae na a ei ee E 9 Toggle and Modifier Values ccceict 2s cienias an eda Gael ee ees 10 Getting w Loy 0 PR eee on e han RTRs See ane e Ee ee e ra me mete EE E 11 Some Conventions Used in This Manual nnsnnnnsnnnnnnnrnnnnrnnnnrnnnnnnnnnrnnnnnnnnnnnnnnnnnnn nnn 12 How Examples Are SNOW sieszonerscccpeatulac
29. The square root of Infinity is Infinity and the square root of Infinity is NaN The square root of NaN is NaN The x12 key Squaring a Number The x 2 key simply squares the number at the top of the stack the bottom of the three displayed numbers That is it multiplies that number by itself The result replaces the original number on the top of the stack the rest of the stack is unaffected The number to be squared can be any number positive or negative whole or fractional The square will always be positive negative times negative is positive If the number to be squared is greater than about 1 34E 154 the square will exceed the largest implemented floating point number and the result will be the special value Infinity The square of Infinity is Infinity and the square of Infinity is also Infinity The square of NaN is NaN Squares and square roots undo each other That is squaring 5 produces 25 and taking the square root of 25 produces 5 again Similarly the square root of 16 is 4 and squaring 4 produces 16 again This is always true but the results might be affected by round off error in the calculator s floating point arithmetic So for example this key sequence __ gives this result because Num 1E200 Math x 2 x 2 Infinity 1E400 is beyond the SiatiCars floating point range Num 1E 200 Math x 2 x 2 0 1E 400 is beyond the StatiCals floating point range The x y k
30. alue of the chi square integral will replace them on the stack That is the two bottom fields in the calculator will be replaced by a single field and the remaining stack contents will be slid down If you want the inverse of the chi square probability value you would enter the probability value then the degrees of freedom in that order You can also toggle between the range indicators if you want Then tap the Inv key and the Chi Sq key and the value of chi square will replace the values of the probability and degrees of freedom on the stack So for example examples to be provided this key sequence gives this result The Stud t key the Student t Distribution This key gives the probability integrals and inverses for the Student t family of distributions You must enter the value of t and also the degrees of freedom n in that order You can also toggle between the four range indicators if you want Then tap the Stud t key The values of t and n will be popped off of the stack and the value of the t integral will replace them on the stack That is the two bottom fields in the calculator will be replaced by a single field and the remaining stack contents will be slid down one position to fill the vacant slot Updated March 21 2014 StatiCal User Manual Page 30 of 55 If you want the inverse of the t probability value you would enter the probability value then the degrees of freedom n in that order Yo
31. ateactastatlecbs Anareh ousted shantieat E S aS eaS eks 12 The x y and PCOS SIAL OnStar igs elas E E EEEE A ad E E EEE E E AE EARE 12 The Num Panel Entering Numbers and Performing Simple Arithmetic 13 ithe lt key Backspace aneen r a a A E RA A Ea 13 WS A E E EE E E E E N ht eORnOe 14 The Ek y and Scientific Notati Nssssioneresiisgt irino e igi asa e s aE 14 Whenis th Enter Key Necessary 3 2 03 cisssscdaasi sastcais trdietsoacanavessdents eia se a RE E ents 14 The Simple Arithmetic Keys x y X y X Y X Y cssceceesecctseccesseeconteccesseccesneeseeesecenteccensensontess 15 The Math Panel Advanced Mathematical Calculations ccssssessseessseeeeeenees 16 he Sort key Square Roots en acne cc na e E A A A R Aaa EA HRANA EAR aia 16 he x 2 key Squaring a Numbet resien n a eee eae 17 The x y key Raising a Number to a Power iscs sscccevsssecasassnsseansesdceva seeduasevsdsccsvacedan dacdedsaesencedvess 17 The x key Factorials serioen n E a a a a E E Pale coche tame cas 18 The Ln key Natural Logarithm of the Factorial of a number eeeseseeeesesesesreesrrereerresreseesereses 19 The Gam key Gamma FUNCIONA rran ce aa a inaia ENR ahha ee dence Gla 20 The Ln Gam key Natural Logarithm of the Gamma Function ssseeseseeseesesereeeesersresreeseeserseresee 21 The key Reversing the Sinica ea da eae ene 21 The xly key Modulo remainder after division eeeeeeseeseeseeesesresseseresresses
32. be 97 12 not 102 096 The calculation of normal based tolerance intervals requires the input of five numbers The count mean and SD of the original sample the desired population fraction and the level of certainty which is put into the CL field So using the example in the Sematech Handbook if I measure a sample consisting of 43 items and get a mean value of 97 07 and a standard deviation of 0 0268 then I can be 99 certain that 90 of the population lies e within the interval from 97 0106 to 97 1294 a 2 sided tolerance interval or e below the value 97 1203 an upper 1 sided tolerance limit or e above the value 97 0197 a lower 1 sided tolerance limit Here s how you d determine those tolerance limits with StatiCal this key sequence gives this result Num 97 07 Enter 0 0268 Enter 43 Enter 90 Enter 99 Test CL Tol Intvs H1 TL 97 1203 then select 1 sided from the popup menu L1 TL 97 0197 H2 TL 97 1294 L2 TL 97 0106 Note that Tol Intvls puts four result numbers on the stack the two 1 sided limits and the pair of 2 sided limits Only the bottom three of them will show L1 H2 and L2 but you can slide the view window up to see the H1 value by tapping on the L1 field label Updated March 21 2014 StatiCal User Manual Page 49 of 55 Like many of the tests on this key panel Tol Intvls can work with summary data mean standard deviation and observation count or with i
33. c formulas can be evaluated in a very simple and intuitive way once you get the hang of RPN and you can see the intermediate results of complicated expressions as you evaluate them 3 You don t have to memorize any operator priority rules like multiplication having a higher priority than addition 4 RPN calculators are very consistent Simple algebraic calculators are actually partly stealth RPN calculators without telling you that leading to conceptual inconsistencies For example to subtract a number you first hit the subtract key then enter the number to be subtracted But to evaluate square roots on a simple calculator you first enter the number to be square rooted then hit the square root key Not very consistent is it 5 RPN calculators are very transparent in their evaluation of complicated expressions showing you the results of intermediate sub expressions that go into the big expression Algebraic calculators those that let you enter complicated algebraic expressions to be evaluated actually maintain a stack to hold intermediate results arising from parenthesized sub expressions or operator hierarchy rules but they hide it from you 5 The RPN Stack framework smoothly extends to handle statistical tests and functions that require several arguments and return several results Exiting the StatiCal app Quitting vs Suspending As you already know the standard Android user interface provides four s
34. calculations on your data When you start up StatiCal these fields are empty because the stack is initially empty and you might not even be aware that there are three fields there Field Labels Each field has a label in small type to the left of the field that describes how that entry in the stack got there whether it was a number you entered directly from the numeric keypad or whether it was the result of some operation on data you had already entered Setting the Display Format When you first launch StatiCal it will show the first three stack positions Since nothing is yet on the stack these three positions will show the value 0 Numbers can be displayed in two different formats Fixed point and Floating point Updated March 21 2014 StatiCal User Manual Page 10 of 55 Fixed point numbers will always show the same number of digits after the decimal point You can control how many of these digits you want to see You might want no fractional decimal digits at all values would be rounded to nearest whole number or you might want 2 decimal digits good for money calculations or you might want 6 decimal digits for high precision work But very large and very small numbers don t display well in fixed point format Very large numbers might spill over onto two or more lines in a very unsightly and hard to read form And very small numbers will have a lot of leading zeros after the decimal point Scientists and engineer
35. can calculate from a 2x2 cross tab The 2x2 b key Other Quantities from a 2 by 2 cross tab table As mentioned in the previous section an astounding number of things can be calculated from the four numbers in a 2x2 cross tab table depending on what the row and column variables represent e risk factors for unfavorable outcomes odds ratio relative risk difference in proportions absolute and relative reduction in risk number needed to treat e measures of the effectiveness of a diagnostic criterion for some condition sensitivity specificity pos amp neg predictive values pos amp neg likelihood ratios diagnostic and error odds ratios e measures of inter rater reliability correct or consistent mis classification rate kappa Forbes NMI e other measures of association contingency coefficient Cramer s phi coefficient Yule s Q Updated March 21 2014 StatiCal User Manual Page 40 of 55 First the four cell counts a b c and d from the following table layout have to be entered into StatiCal Outcome Positive Outcome Negative Total Causative Factor Present a b a b Causative Factor Absent c d c d Total a c b d a b c d and the 2x2 a button has to be tapped to produce the three p values Then any of the following parameters and their confidence intervals can be calculated by tapping the 2x2 b key Odds Ratio OR a b c d Relative Risk RR a r1 c r2 Kappa Overall
36. cant association between smoking and emphysema If we were to study 60 people we might get the following results Developed Emphysema No Emphysema Total History of Smoking 22 10 32 No Prior Smoking 12 16 28 Total 34 26 60 These are totally made up numbers strictly for the sake of an example Entering these four cell counts we don t enter the row totals or column totals in the appropriate order gives us this this key sequence gives this result Num 22 Enter 10 Enter 12 Enter 16 Test 2x2 a ChiSq 0 0435 Yates 0 0787 Fisher 0 0673 The simple uncorrected Chi square test gives a significant p value 0 0435 for 2x2 tables but this is not the best test for a 2x2 cross tab We should use the Yates corrected chi square test instead which gives a non significant p value 0 0787 This may be disappointing but it s closer to the exact p value provided by the Fisher Exact test 0 0673 In addition to providing these three p values the 2x2 a key also does something behind the scenes that is very important It save certain other computational results in a special hidden area not on the stack or in the M1 or M2 memory cells These results are used by the 2x2 b key to calculate the values and confidence intervals for various quantities derived from the 2x2 table like odds ratios kappa etc See the following section for a description of all the things you
37. cause calculating 300 in StatiCal or just about any calculator will give Infinity or a floating point overflow error and the log of infinity is still infinity Somehow you have to be able to calculate Ln 300 without having to calculate 300 first So StatiCal provides a Ln key that calculates the natural log of x by a special procedure that does not have to calculate x first So it will work for huge numbers For example this key sequence gives this result Num 300 Math Ln Ln 1 414 9058 Num 300 Math Ln Exp 89 700 Num 298 Math Ln Num Math Exp As another example the natural log of 10 000 is easily calculated using StatiCal s Ln key as 82 108 9278368 1436 which agrees with the number provided by Wolfram Alpha The Gam key Gamma Function The Gamma function is related to the factorial function by the expression Gamma x x 1 or x Gamma x 1 So if we can calculate either one of these two functions gamma or factorial we can easily calculate the other so I only needed to provide either factorial or gamma But since people like me are very prone to forgetting which one is the 1 and which one is the 1 relationship StatiCal provides both functions So for example this key sequence _ gives this result Num 4 Math x x 24 Num 5 Math Gam Gam 24 Num 2 5 Math Gam Gam 0 9453 Updated March 21 2014 Sta
38. completely free and it will remain so as long as I m around It has no ads banners or other commercial add ons There are no trial or limited edition or Pro versions there s only the Full version e StatiCal requests no special permissions to run it does not modify the device s SD Card storage does not access phone information does not access the network etc e StatiCal currently does not run on the iPhone iPad or any other iOS app Updated March 21 2014 StatiCal User Manual Page 5 of 55 History I m a semi retired statistician but I still do statistical consulting work Many years ago I had a Palm PDA personal digital assistant I bought a program called CASL that let me develop my own apps they weren t called apps back then for the Palm operating system One of these apps was a scientific calculator that could also calculate p values for the Normal Chi square Student t and Fisher F distribution and their inverses This proved to be very handy and I used the Palm app as my main calculator until smart phones supplanted PDAs Unfortunately CASL was never updated to work with the newer smart phones or tablets and it s now a dead product But I really missed my old calculator and I searched around for an Android app that could do statistical calculations within the context of a general scientific calculator Not finding any pre existing apps fitting that description I then looked around
39. e cosecant is the reciprocal of the sine So in common algebraic usage Sin x means the arcsine of x but Sin x j means the reciprocal of the sine of x that is the cosecant of x and Sin x means the sine of the reciprocal of x Confusing Well fortunately when using StatiCal you don t have to worry about this I don t use the notation I use the Inv notation Here s where the Inv key comes into play The Inv key might appear to be a toggle key just like the Deg Rad key and indeed if you press Inv repeatedly the yellow Inv indicator above it will go on and off This is convenient in case you accidentally hit Inv when you didn t want to just hit it again to turn it off But it s important to note that while the Deg Rad toggle stays as you set it until you change it the Inv key is a one time thing as soon as you evaluate the inverse trig function the Inv indicator goes off and the next time you want an inverse trig function you have to hit Inv again just prior to the trig function key So I prefer to call the Inv key a modifier rather than a toggle In other words toggles are persistent but modifiers are one time So for example Updated March 21 2014 StatiCal User Manual Page 25 of 55 this key sequence gives this result comment Num 0 5 Math Inv Sin ASin 0 5236 if the Deg Rad toggle is Rad Num 0 5 Math Deg Rad Inv Sin ASinD 30 if the Deg
40. eeseeeeeeseseeeeesersreerresreseesereses 39 The MxN Xtab key Chi square analysis of any size cross tab table oneens 42 The Ord Xtab key Analysis of a cross tab of ordinal categories eseeeeeeseseeeseseeerieeresressrreresee 42 The Dscr Stats key Basic Descriptive Statistics of a set of NUmMbers eeseeseererreeresreerrrerees 43 The tlest keys Student 1680S irera a EE EE E RE ee ea arene 44 The AoV key l way Analysis of Variance i2 225 si ssetesiceeigsideciaesmncetaci eee dasteacd heed eats 45 The Regr key Straight line regression and correlation analysis of x y data 0 eee 45 The r 0 key Test the significance of a correlation coefficient 0 0 eee eeeeesseeceeeceeeeeeeeeeeeeenaeens 47 The r r key Compare Two Correlation Coefficients 2 0 0 ceecesscecsseceseceeeeeeaeeceaeceeeseeeeeneeenaeen 47 The Tol Intvs key Tolerance Intervals for Normally distributed data eee eee eeeeeeneeeneeees 48 The Conf Intvs key Confidence Intervals for Normal Data 0 eee eeeceeeeeeseeceseeeseeeeeeeeneeenaeees 49 The Bay Cred key Bayesian Credibility Analysis from Odds Ratios ceeeeeseeeeeeeeeeeneeenseees 51 The Stor Keypad Managing the Calculator s Storage ccccccseeeeeeeeeeeeeeeeeeeeeeeeeeees 52 Th Clr keys Seta Memory Cell to Zeros 1 esicssdeus eia peronea s eei aes oseane SENEN 53 The Sto keys Store a number into a separate Memory Cell ssssssesesesseseeseseresrresersrrsrresreseeseresee 53 The Rec keys Recall a n
41. egrees and other times in radians most scientific calculators also have some way to toggle between degree and radian mode And since many scientific calculators provide hyperbolic sines cosines and tangents as well as the more familiar circular trig functions they also might provide a way to toggle between the these two families of functions Without this toggling scheme scientific calculators would have to devote 24 keys to the circular and hyperbolic functions and their inverses in degree and radian mode But with toggling only 6 keys are required the Sin Cos and Tan keys an Inverse toggle a Degrees Radians toggle and Circular Hyperbolic toggle Updated March 21 2014 StatiCal User Manual Page 11 of 55 Getting Help There are currently two sources of help for using StatiCal One is the StatiCal Manual you re reading now Since free PDF reader apps are readily available for most smart phones and tablets you might want to store this PDF file on your device so that it will always be handy The other is a set of help messages that are built into SratiCal If you long press any key hold it down for about a second a box will pop up containing a concise description of what that key does and how to use it that is what you have to place on the stack before you tap the key what will be on the stack after you tap the key and how the various toggles and modifiers affect the operation of that key Updated March 21 2014
42. en That will be one tap for every number you do want in the summary Then go to the Num panel and tap NaN Then go back to the Stor panel and tap Roll Up that same number of times so that your last desired number is in the bottom field You will have inserted a NaN between the wanted and unwanted data and you can now tap the Descr Stats key Updated March 21 2014
43. er and one cannot say whether it s or at the whole number So for example this key sequence gives this result Num 1E200 Math x x Infinity The Ln key Natural Logarithm of the Factorial of a number We sometimes want to use factorials of very large numbers as intermediate expressions in a calculation where the final result of the calculation might not be very large For example 300 divided by 298 happens to equal 299 298 which is equal to the relatively small value 89 700 But if you try to calculate the value of 300 298 in the obvious way calculate 300 calculate 298 then divide it won t work because the intermediate values 300 and 298 will both be calculated as Infinity and Infinity Infinity is undefined and produces NaN For example this key sequence gives this result Num 300 Math x x y NaN Num 298 Math x Num x y In such cases it is convenient to work in logarithms or as mathematicians like to say in the logarithmic domain For example the rules of logarithms tell us that Ln x y Ln x Ln y SO Ln 300 298 Ln 300 Ln 298 Updated March 21 2014 StatiCal User Manual Page 20 of 55 therefore 300 298 Exp Ln 300 Ln 298 Now Ln 300 happens to equal 1414 90585 but you can t arrive at this value by entering 300 calculating the factorial then taking the log be
44. ey Raising a Number to a Power The x y key raises any base x to any power y that is it generates x often written as x y or x y in various computer languages Enter the base number x then the power number y then press the x y key The result x will replace x and y on the stack So to raise 4 to the third power this key sequence gives this result Num 4 Enter 3 Math xy x4y 64 Updated March 21 2014 StatiCal User Manual Page 18 of 55 The power can be positive or negative number integer or fractional and the base can be any positive number integer or fractional for any value of the power So we have this key sequence gives this result Num 1 7 Enter 3 1 Math xy x4y 0 19302 But the base can be negative if and only if the power is an even integer So we have this key sequence gives this result Num 1 7 Enter 2 Math x y x4y 2 89 If the base is negative and the power is anything other than an even integer the calculator will return the value NaN because in those cases the answer is not a real number it is a complex number with real and imaginary parts and StatiCal does not work with complex numbers The x key Factorials For a positive whole number n the factorial of n written as n is the product of all the whole numbers from to n inclusive So 4 is equal to 1 2 3 4 or 24 Zero factorial 0 is defined as a special
45. finity The reciprocal of Infinity is 0 The reciprocal of Infinity is 0 which is for all practical purposes treated the same as 0 Updated March 21 2014 StatiCal User Manual Page 22 of 55 The reciprocal of NaN is NaN So for example this key sequence gives this result Num 5 Math 1 x 1 x 0 2 The pi key Entering the value of pi The pi key pushes the precise value of pi 3 141592653589793 onto the stack The other numbers on the stack push up to accommodate the new value So for example this key sequence gives this result Math pi pi 3 1416 The Exp key Exponentiating The Exp key raises e to the power of the number at the top of the stack e is the number 2 71828 Exp of Qis 1 Exp of 1 is e or 2 718281828459045 Exp of Infinity is Infinity Exp of Infinity is 0 Exp of NaN is NaN So for example this key sequence gives this result The Ln key Natural Logarithms The Ln key replaces the number at the top of the stack with its natural base e logarithm The Ln of Infinity is Infinity The Ln of 0 is Infinity Updated March 21 2014 StatiCal User Manual Page 23 of 55 The Ln of any negative number is NaN StatiCal does not deal with complex numbers So for example this key sequence gives this result Num 100 Math Ln Ln 4 606 The Log key Common Logarithms The Log key replace
46. for an easy to use software environment to develop Android apps for the smart phone I was using After several false starts I discovered Basic for Android B4A which was just what I needed It had all the math and programming features needed to create a compact app and used a programming language that was close enough to what I was already familiar with so that I could actually learn it So over the course of a month or so I developed StatiCal This manual took another two months to write I was also able to port the application over to the Blackberry Playbook tablet StatiCal does everything my old Palm calculator did and a lot more I wanted to put into it every statistical calculation that was in any way amenable to a hand held device Obviously this would not include multiple regression on data sets with hundreds of variables and thousands of cases but I wanted it to handle anything where the data could be conveniently keyed into a hand held device So Icreated StatiCal for myself But now that it s a reasonably stable and reliable working app I m sharing it with the world This is in keeping with the philosophy that has guided another of my pet projects the StatPages org web site which provides free statistical software via online web based calculators If you like StatiCal or if you think of other features that would be useful to have let me know If you don t like it at least you have the consolation of knowing
47. groups to calculate the summary statistics and enter them onto the stack for you The Regr key Straight line regression and correlation analysis of x y data e The Regr key performs a linear straight line least squares regression and correlation analysis on a set of x y data Updated March 21 2014 StatiCal User Manual Page 46 of 55 Data is entered as pairs of numbers representing the x and y values for each data point You must first enter a NaN flag to mark the start of the data on the stack Then you must place the paired data onto the stack in the order x1 yl x2 y2 x3 y3 etc Finally you tap the Regr key to perform the analysis The Regr key calculates the following results and places them on the stack e the number of data points number of x y pairs of values e the slope of the fitted line e the intercept of the fitted line e the RMS Error which is loosely speaking the average scatter of the points from the fitted line actually the standard deviation of the vertical distances of the points from the fitted line e the square of the Pearson correlation coefficient 12 e the Pearson correlation coefficient r the p value indicating the significance of the regression also indicating the significance of the slope of the line root mean square error of the fit So for example to calculate the correlation coefficient between the height and weight of four subjects
48. is the denominator of the fraction that produced the proportion When preceded by the Inv toggle the Arc Sine key will perform the inverse transformation turning a number between 0 and pi 2 into a probability between 0 and 1 p Sin y So for example this key sequence gives this result Num 0 5 PDFs Arc Sin ArcSine 0 7854 Num 1 5 PDFs Inv Arc Sine Prop 0 9950 The Fish z key The Fisher z transformation for correlation coefficients Correlation coefficients do not have a normally distributed sampling distribution For one thing they cannot have values outside of the range 1 to 1 But R A Fisher discovered a transformation which he called the z transformation that turned them into more nearly normally distributed numbers Z 0 5 Ln 1 r C1l r The transformed z numbers can lie anywhere between infinity when r 1 and infinity when r 1 Updated March 21 2014 StatiCal User Manual Page 36 of 55 This transformation also has the interesting property that the standard deviation of the z numbers will always tend to be 1 sqrt N 3 where N is the number of data points x y pairs from which the r value was calculated So this is another variance stabilizing transformation So for example this key sequence gives this result Num 0 6 PDFs Fish Z FishZ 0 6931 Num 0 99 PDFs Fish Z FishZ 2 6467 Num 5 PDFs Inv Fish Z r 0 9999092
49. lar Chi square test previous section couldn t detect the gradual trends in the two rows because the Chi square test doesn t know left from right from middle columns or top from bottom from middle rows But the ordinal test is on the lookout for these progressive directional trends The Dscr Stats key Basic Descriptive Statistics of a set of Numbers The most common way to concisely summarize a set of numbers is by quoting the count mean and standard deviation SD of the numbers The Descr key calculates these three summary measures from a set of numbers that you ve entered onto the stack StatiCal s stack currently can hold up to 100 numbers if your sample is larger than that use some other program like Excel to summarize them Updated March 21 2014 StatiCal User Manual Page 44 of 55 Before entering your numbers you must first place a special flag called NaN for not a number onto the stack The Num keypad has a NaN key for just this purpose So for example to calculate summary statistics on the six numbers 18 22 20 28 19 and 30 you would do the following this key sequence gives this result Num NaN 18 Enter 22 Enter 20 Enter Mean 23 4000 28 Enter 19 Enter 30 Enter Test Descr Stats SD 5 3666 N 5 0000 While this is nice in itself the main utility of the Descr Stats key is as a precursor to using the tTest or AoV key These keys want to see summary data
50. lows formula to be provided The Bin key evaluates the sum of one or more terms of the binomial distribution Before you tap Bin you must enter values for N and x and either one or two additional numbers depending on how the Rnge toggle has been set that indicate the range of x values in the sum you want The Rnge toggle works this way Outer You enter two numbers after N and x call the two numbers Lo and Hi The sum will include terms from 0 to Lo inclusive and from Hi to N inclusive Updated March 21 2014 StatiCal User Manual Page 31 of 55 Inner You enter two numbers after N and x call the two numbers Lo and Hi The sum will include terms from Lo to Hi inclusive Left You enter one number after N and x call the number Hi The sum will include terms from 0 to Hi inclusive Right You enter one number after N and x call the number Lo The sum will include terms from Lo to N inclusive So for example examples to be provided this key sequence gives this result The Inv key modifies the action of the Bin key but in a different way from how Inv modifies the four continuous probability functions Strictly speaking Inv doesn t produce the inverse of the binomial distribution That is it doesn t tell you what range of terms in the discrete binomial distribution sum up to a particular probability Rather it returns exact Clopper Pearson confidence limits around an observed pr
51. me things to note about invoking the 2x2 b calculaton 1 You must run 2x2 a on your data first You cannot simply enter the four cell counts and then tap 2x2 b Remember a then b 2 You can run as many 2x2 b taps as you want after you ve run the 2x2 a You don t have to re enter the cell counts for the 2x2 b taps Each time you run 2x2 b you select one of the parameters to be evaluated How the method works Confidence intervals for the estimated parameters are computed by a very general method based on constant chi square boundaries described in Statistical Methods for Rates and Proportions 2 Ed Section 5 6 by Joseph L Fleiss Pub John Wiley amp Sons New York 1981 This method is also described in Numerical Recipes in C and Ed Section 15 6 by William H Press et al Pub Cambridge University Press Cambridge UK 1992 This method calculates two limiting tables that represent how far away from your observed table on either side the truth could be and still have your observations not be too unlikely with too unlikely being quantified by the confidence level you want Then when you plug the numbers from these limiting tables into the formula for whatever parameter you want they yield the confidence limits for that parameter You can get the contents of those two limiting tables by selecting the last entry in the popup list Limiting Tables You can use these tables if you re interested in
52. n one position to fill the gap The M2 Sto key does exactly the same thing but with the M2 memory cell instead of the M1 cell The Rec keys Recall a number from a Memory Cell and put it onto the stack The M1 Rec key takes the number currently sitting in the M1 memory cell and pushes it onto the stack pushing the other stack contents up one position The number also remains intact in the M1 cell so it can be recalled again and again if you want The M2 Rec key does exactly the same thing but with the M2 memory cell instead of the M1 cell The M keys Add a number from the stack onto a Memory Cell The M1 key takes the most recently added number on the stack and adds it to the contents of the M1 memory cell putting the sum into M1 and popping the number off of the stack The M2 key does exactly the same thing but with the M2 memory cell instead of the M1 cell The M keys Add a number from the stack onto a Memory Cell The M1 key takes the most recently added number on the stack and subtracting it from the contents of the M1 memory cell putting the new value into M1 and popping the number off of the stack The M2 key does exactly the same thing but with the M2 memory cell instead of the M1 cell The AC key Clear out All Values from Stack and Memory Cell The AC key works like the All Clear keys on most calculators everything gets set to zero all 100 elements of the stack and both memory cells Updated March
53. nCr key Combinations binomial coefficient The nCr key gives the number of combinations of n objects taken r at a time the number of different ways of drawing a sample of r objects from a collection of n distinct objects where the sequence in which the objects are drawn doesn t matter The definition of nCr is n r n r nCr is sometimes read aloud as from n choose r Updated March 21 2014 StatiCal User Manual Page 33 of 55 nCr is also referred to as the binomial coefficient For certain combinations of n and r the factorials appearing in the above expression might exceed the device s floating point limits but the final value of nCr might not be excessively large due to partial cancellation of the large numerator and denominator StatiCal uses logarithms when necessary to obtain nCr values for these extreme cases So for example this key sequence gives this result Num 20 Enter 6 PDFs nCr 20C6 38 760 Num 1000 Enter 957 PDFs nCr 1000C957 6 622295559 E75 The nPr key Permutations The nPr key gives the number of permutations of n objects taken r at a time the number of different ways of drawing a sample of r objects from a collection of n distinct objects where the sequence in which the objects are drawn does matter The definition of nPr is n n r For certain combinations of n and r the factorials appearing in the above expression might exceed the de
54. ndividual observed values x1 x2 x3 xn If have individual observations you can enter these numbers onto the stack and then use the Descr Stats key to calculate the mean sd and n The Descr Stats key will place these three summary values on the stack in the correct order for the Tol Intvls calculation then all you have to add is the population fraction and the level of certainty if it s different from what s showing above the CL key By the way there is an online web based tolerance intervals calculator at http StatPages org tolintvl html The Conf Intvs key Confidence Intervals for Normal Data For normally distributed data confidence limits are generally calculated as a value plus and minus the standard error of that value multiplied by a number which is the critical value of Student t for the appropriate confidence level and degrees of freedom The simplest case is for a symmetric 2 sided 95 confidence interval around the observed mean M of a sample of N numbers with a standard deviation of SD Conf Limits M tcrit SD sqrt N where tcrit is the critical 2 tailed Student t value for p 0 05 with N 1 degrees of freedom The quantity SD sqrt N is equal to the standard error of the mean So for example a sample of 10 numbers with a mean of 5 and a SD of 2 would have a symmetrical 2 sided 95 confidence interval going from 3 5693 to 6 4307 This is because the critical 2 sided Student t value for p 0 05 and
55. o Hi inclusive Left You enter one number after M call the number Hi The sum will include terms from 0 to Hi inclusive Right You enter one number after M call the number Lo The sum will include terms from Lo to N inclusive So for example examples to be provided this key sequence gives this result The Inv key modifies the action of the Pois key but in a different way from how Inv modifies the four continuous probability functions Strictly speaking Inv doesn t produce the inverse of the Poisson distribution That is it doesn t tell you what range of terms in the discrete Poisson distribution sum up to a particular probability Rather it returns exact Garfield confidence limits around an event count defined as N The confidence level is determined by the value shown in yellow above the CL key So for example if you observe the occurrence of 10 events N 10 per unit of time or space your best guess of the true event rate is 10 Then assuming that the CL has been set to show 95 CL this key sequence gives this result Num 10 PDFs Inv Pois Hi CL 18 3904 Prop 10 0000 Lo CL 4 7954 To get confidence limits for other confidence levels you would set the CL value at some time before tapping the Pois key this key sequence gives this result Num 10 Enter 90 PDFs CL Inv Pois Hi CL 16 9622 Prop 10 0000 Lo CL 5 4254 The
56. o assume that it is very large The critical Student t value then is replaced by the critical normal z value There is one more complication For some kinds of regression like logistic Poisson or Cox Proportional Hazards CPH regression the most useful results are obtained by raising e to the power of the regression coefficients Exponentiating logistic regression coefficients gives odds ratios exponentiating Poisson regression coefficients gives relative event rate ratios and exponentiating CPH regression coefficients gives hazard ratios For these situations the confidence limits around the regression coefficients would be calculated as described above then the coefficient and both confidence limits would be exponentiated to get the confidence limits around the odds or rate or hazard ratio For such situations choose the option that ends in Ln For example this key sequence gives this result Updated March 21 2014 StatiCal User Manual Page 51 of 55 The Bay Cred key Bayesian Credibility Analysis from Odds Ratios The Bay Cred key allows the credibility of a clinical trial finding to be assessed in the light of current knowledge It is based on the groundbreaking work of Robert Matthews Matthews R A J 2001 Methods for assessing the credibility of clinical trial outcomes Drug Information Journal vol 35 4 1469 1478 Available online at ttp bit ly Credibility AnalysisPaper Many clinical
57. o be ordinal So let s take a look at a different example that happens to have the same cell count numbers as the example of the previous section on MxN Xtab but representing something entirely different Suppose the columns didn t represent dessert preferences but rather the attitude of the subject toward action adventure movies like Cowboys vs Aliens The cross tab would now look like this with the same numbers but different column labels Dislike Neutral Like Male 15 20 23 Female 13 12 5 Notice that the numbers in the top row steadily increase from left to right while the numbers in the bottom row steadily decrease from left to right Males tend to lean more toward action adventure movies than women do This gradual trend across columns was irrelevant in the previous example because the sequencing of the columns was totally arbitrary But now with an ordinal column variable this steady trend means something and we want an analysis that is on the lookout for such a trend and can take advantage of it You would test this ordinal cross tab this way this key sequence gives this result Num 15 Enter 20 Enter 26 Enter 13 Enter 12 Enter 5 Enter Conc 355 0000 2 Enter 3 Test Ord Xtab Disc 835 0000 pVal 0 0241 So this table indicates a significant association between the gender and preference for action adventure movies p 0 0241 The regu
58. oportion defined as x N The confidence level is determined by the value shown in yellow above the CL key So for example if you flip a coin 10 times and get 7 heads N 10 x 7 your observed proportion of heads is 0 7 P 0 7 Then assuming that the CL has been set to show 95 CL this key sequence gives this result Num 10 Enter 7 PDFs Inv Bin HiCL 0 9333 Prop 0 7000 Lo CL 0 3475 To get confidence limits for other confidence levels you would set the CL value at some time before tapping the Bin key this key sequence gives this result Num 10 Enter 7 Enter 90 PDFs CL Inv Bin HiCL 0 9127 Prop 0 7000 Lo CL 0 3934 The Pois key The Poisson Distribution and Confidence Intervals The Poisson distribution is defined as follows formula to be provided The Pois key evaluates the sum of one or more terms of the Poisson distribution Before you tap Pois you must enter a value for M and either one or two additional numbers depending on how the Rnge toggle has been set that indicate the range of x values in the sum you want The Rnge toggles work this way Updated March 21 2014 StatiCal User Manual Page 32 of 55 Outer You enter two numbers after M call the two numbers Lo and Hi The sum will include terms from 0 to Lo inclusive and from Hi to N inclusive Inner You enter two numbers after M call the two numbers Lo and Hi The sum will include terms from Lo t
59. or the non symmetric distributions Outer is the same as Right either one gives the area of the right tail of the distribution and Inner is the same as Left either one gives the area in the body of the distribution starting from x 0 and excluding the right tail And I d point out that Inner Outer for the same value of x always equals 1 0 and Left Right for the same value of x also always equals 1 0 For the discrete distributions Binomial and Poisson the probability integrals aren t really integrals in the calculus sense they re the sums of the discrete terms of the distribution so the range designations are handled in a slightly different way See the sections in Bin and Pois for an explanation of how the Rnge toggle is interpreted for these functions Updated March 21 2014 StatiCal User Manual Page 28 of 55 The Inv Key Inverse of Probability Functions and Confidence Intervals The PDFs panel has an Inv modifier key just like the Math panel has but in the PDF s panel it has two different interpretations For continuous PDFs it refers to the inverse of a continuous distribution that is an Inv probability function calculates the value of the integration limit for which the integral equals the value that has been entered onto the stack The range of the integral is determined from the Rnge toggle in the same way as for the probability integrals So for example this key sequence gives this resul
60. ramming 100 different keys into a phone s window would result in tiny closely spaced keys This might be workable for someone with tiny fingers but not for my man hands So I took a different approach StatiCal s keyboard layout looks very simple and uncluttered There are only 25 keys on the screen in a fairly spacious 5 by 5 layout How do I get 100 keys into a 25 key screen Simple I define 5 different keypad screens called panels each containing 25 keys the PDFs functions panel BM G 10 21 pm the Num Keypad Panel BM G 10 11 ow the Math functions panel DM G 10 14 12345 3 141592653589793 1 3587E150 1 7724538509055159 7 188082728976037 ajaj EEEE S E x k aE l Ria CECE apo0g pogga BES A K B i Updated March 21 2014 StatiCal User Manual Page 9 of 55 the statistical Test panel the Stor manipulations panel BM G 10 23 em BME 10 24 em You ll notice that the five left most keys with the blue letters are the same on all five panels they re the panel switching keys and tapping any of these keys instantly switches you from one panel to another Result Fields StatiCal has at the top of the screen three fields where the results of data entry and calculations are displayed These three fields show you the top three entries in the stack these will be your most recently entered data and or the results of the most recent
61. s the number at the top of the stack with its common base 10 logarithm So for example this key sequence gives this result Num 100 Math Log Log 2 0000 The Trigonometric Function keys Sin Cos Tan The three trigonometric keys can perform 18 different kinds of calculations depending on how three toggles have been set prior to tapping the trigonometric key In the simplest case the way StatiCal starts up before you do any special toggling these three keys cause the number at the top of the stack to be replaced with the trigonometric sine cosine or tangent of the number The number is considered to be the angle expressed in radians If you want the cotangent secant or cosecant you have to remember your basic trigonometry these are just the reciprocals of the tangent cosine or sine respectively So just press the 1 x key if necessary e For cotangent press Tan then 1 x e for secant press Cos then 1 x e for cosecant press Sin then 1 x The Deg Rad toggle key Degrees or Radians Now things get interesting What if you have the angles in degrees rather than radians That is what if you want to calculate the Sin of 30 degrees Notice that there s a Deg Rad button on the top row with Rad in tiny yellow letters above it This is a toggle one of several that StatiCal uses The yellow Rad means that the argument to the trig function will be interpreted as radians If yo
62. s to Left Num 1 96 PDFs Rnge Nrml Nrml 0 025 Another tap on Rnge toggles to Right Updated March 21 2014 StatiCal User Manual Page 29 of 55 As the above example illustrates the sum of the Inner Outer integrals for any particular value of x always equals 1 0 and the sum of the Left Right integrals for any particular value of x always equals 1 0 Note that Left means left integral that is the area of everything to the left of x It is not the left tail area unless x is given as a negative number Also if you start up StatiCal in its default configuration and carry out the following four example in succession without doing other stuff in between you get this key sequence gives this result comment Num 0 05 PDFs Inv Nrml InvNrml 1 96 Rnge defaults to Outer Num 0 95 PDFs Rnge Inv Nrml InvNrml 1 96 Single tap on Rnge toggles to Inner StatiCal computes the normal distribution integrals using a formula developed by Graeme West based on work by J Hart The Chi Sq key the Chi square Distribution This key gives the probability integrals and inverses for the chi square family of distributions You must enter the value of chi square and also the degrees of freedom in that order You can also toggle between the four range indicators if you want Then tap the Chi Sq key The values of chi square and degrees of freedom will be popped off of the stack and the v
63. s use what s called scientific notation to easily represent very large or very small numbers in the format X XXXXXXXXXXXXXXXE XX Scientific notation will show as many significant digits as necessary up to 16 with an exponent part if necessary You can control the appearance of the numbers by tapping on the number fields e To switch to scientific notation display tap the middle number e To switch from scientific notation to fixed point display tap the upper or lower number e To increase the number of decimal places in fixed point display tap the upper number e To decrease the number of decimal places in fixed point display tap the lower number Toggle and Modifier Values To keep the number of required keys down to a manageable level most scientific calculators have several toggle keys to modify the way certain other keys work And StatiCal is no exception For example many scientific calculators have three trigonometric function keys Sin Cos and Tan They could have added three more keys for the inverse trig functions ArcSin ArcCos and ArcTan but it s easier to just have a single Arc key so that if you tap the Arc key and then tap the Sin key you ll get the ArcSin Some functions have a shift or alternate or function or some other kind of general toggle key and then print the name of the alternate function in the space above the key Since we sometimes want trig functions to work in d
64. some other quantity that s not in the list Just get the tables and manually plug them into the formula for the parameter you re interested in Updated March 21 2014 StatiCal User Manual Page 42 of 55 The MXN Xtab key Chi square analysis of any size cross tab table The Fisher Exact test gets incredibly difficult for cross tab tables larger than 2x2 and I won t attempt to perform that test on those tables But cross tab tables of any size can be analyzed quickly and easily by the Chi square test To use the MXN Xtab key you first enter the cell counts of the table Enter them in row order that is enter all the numbers in the first row then all the numbers in the second row etc Don t enter them column wise or you ll get the wrong answer After entering all the cell counts you have to enter two more numbers to tell StatiCal how many rows and how many columns the table has So for example Suppose you want to perform a chi square test on the following 2 by 3 table showing the dessert preferences of a sample of males and females Pie Cake Jello Male 15 20 23 Female 13 12 5 You would do it this way this key sequence gives this result Num 15 Enter 20 Enter 23 Enter 13 Enter 12 Enter 5 Enter Chi Sq 5 3456 2 Enter 3 Test MxN Xtab df 2 0000 p Val 0 0690 So this table indicates that there is no significant association between the gender and
65. sresressersresreeseeseeseresee 21 The lackey Reciprocal of a number Siea a e E E 21 The pi key Entering the value of pi sc9civeiiie cece 22 The Exp key Exponentiatin geenien a a stan e g iiai io is 22 Th Laskey Natural Logarithms eici e e ilar ee E nee 22 The Lag key Common LogathmMs sson apeta oi Taa REAT E n aaa aiia hie 23 The Trigonometric Function keys Sin Cos Tan eeesessseeseesreesssressessresressessresresserseesreeseeseeserssee 23 The Deg Rad toggle key Degrees or Radians essessseseesessiesresressessresressessresresserstesreeseeseeserssee 23 The Inv modifier key Inverse Trigonometric Functions eseseseeseseseereeserssresreesersresreeseeseeseesee 24 The Circ Hyp modifier key Circular or Hyperbolic functions essseseeseeerseseeeresersreereesresersereses 25 Updated March 21 2014 StatiCal User Manual Page 3 of 55 The PDFs Panel Probability and Distribution Functions ccccceeseeeeeeeeeeeeeeeeees 26 The Rnge toggle Key Setting the Range of Integration tails of distribution eee 27 The Inv Key Inverse of Probability Functions and Confidence Intervals 00 0 0 ee eeeeeeeeeeeeeeeee 28 The CL Key Setting the Confidence Level for confidence intervals 0 ese eeeeeeeeeeneeeneeees 28 The Norm key the Normal Distribution 2 20 5 ocs seneendd eu Nota teenie ee sien 28 The Chi Sq key the Chi square Distr bution ie ateepeeke cee Ei aea eee ede 29 The Stud t key the Student t Dis
66. t comment Num 0 05 PDFs Inv Norm InvNorm 1 96 Assuming Rnge is set to Outer For discrete PDFs the Inv key generates exact confidence intervals by the method of Clopper amp Pearson for the binomial and by the analogous method of Garwood for the Poisson See the sections in Bin and Pois for an explanation of how the Inv modifier is interpreted for these functions The CL Key Setting the Confidence Level for confidence intervals By default StatiCal will evaluate 95 confidence limits but you can change this by using the CL key You place a number on the stack indicating the confidence level expressed in percent that you want then hit the CL key The confidence level that you set will remain in effect until you change it or until StatiCal is re started The Norm key the Normal Distribution This key gives probability integrals and their inverses for the standard normal Gaussian distribution The value of the calculation replaces the value of the input on the stack So for example So if you start up StatiCal in its default configuration and carry out the following four example in succession without doing other stuff in between you get this key sequence gives this result comment Num 1 96 PDFs Nrml Nrml 0 05 Rnge defaults to Outer Num 1 96 PDFs Rnge Nrml Nrml 0 95 Single tap on Rnge toggles to Inner Num 1 96 PDFs Rnge Nrml Nrml 0 975 Another tap on Rnge toggle
67. tandard buttons either actual buttons or on screen icons Menu Home Back and Search Two of these buttons can be used to exit StatiCal and return to the screen from which you had launched it either one of your home screens or the all apps window These two keys work in different ways The Home button suspends the StatiCal app and saves its current status so that the next time you launch StatiCal it will come up in exactly the same state it was in when you left it The stack and memory cells will still have their earlier values it will show the panel you were using the toggles and modifiers will be as you left them etc Use the Home button when you want to temporarily leave StatiCal to do something else but plan to return to StatiCal to continue what you were working on The Back button terminates the StatiCal app and doesn t save your work The next time you launch StatiCal you ll see the standard start up configuration the stack and memory cells will be zeroed out the Num keypad panel will be showing etc Updated March 21 2014 StatiCal User Manual Page 8 of 55 Keys and Panels Scientific calculators can have lots of keys because there are lots of scientific functions you might want to evaluate Because I wanted StatiCal to be a full function scientific calculator and to also have statistical functions and tests I realized that StatiCal would require 100 or more keys Since smart phones normally use on screen keyboards c
68. that you didn t sink any money into it Updated March 21 2014 StatiCal User Manual Page 6 of 55 General Operation of the Calculator To understand how StatiCal works you have to be aware of two things Reverse Polish Notation RPN First and foremost StatiCal is a reverse Polish notation or RPN calculator That means that you enter numbers and operations differently from how you enter them into simple algebraic calculators Usually to add two numbers on a simple algebraic calculator you might hit the keys this way 5 3 and you d see the answer 8 But an RPN calculator doesn t have an key and it has an Enter key that you don t find on simple calculators For the above calculation you hit the keys this way 5 Enter 3 and you d see the answer 8 as before Notice that with an RPN calculator you first give it a number then tell it what you want it to do with that number The Stack The second thing to be aware of is that StatiCal uses a stack to hold the results of calculations StatiCal s stack is a storage mechanism that in the current implementation of StatiCal can hold up to 100 numerical quantities Each quantity can be a positive or negative number with up to 16 significant digits Numbers can range in magnitude from about 10 to 10 Numbers get pushed onto the stack when you enter them or do calculations on them A pushed number appears at the bottom of the three
69. the dessert preference p 0 069 The Ord Xtab key Analysis of a cross tab of ordinal categories The Chi square and Fisher Exact tests assume that the rows and columns represent nominal categorical variables that is categories whose values don t fall into any natural logical sequence like the following e Cauc AfrAm Asian Other e Pie Cake Jello The sequence is unimportant and switching the rows around or switching the columns around doesn t affect the p value from either the Chi square or Fisher Exact test But many categorical variables are ordinal that is they do fall into a logical sequence e Neonate Infant Child Adolescent Adult Geriatric Extra Small Small Medium Large Extra Large Emaciated Normal Overweight Obese Morbidly Obese Absent Slight Moderate Loaded Strongly Disagree Disagree No Opinion Agree Strongly Agree Cured Improved No Change Worse Died Updated March 21 2014 StatiCal User Manual Page 43 of 55 The Chi square and Fisher Exact tests are by their very nature not on the lookout for gradual consistent trends across a range of ordinal categories and so are not particularly efficient at detecting such trends They deal very well with nominal categories but not with ordinal categories Special tests have been designed for analyzing cross tabulations of ordinal categorical variables By the way you can consider any dichotomous categorical variable like Gender t
70. tiCal User Manual Page 21 of 55 The Ln Gam key Natural Logarithm of the Gamma Function The Ln Gam key is provided for the same reason that the Ln key is provided to be able to evaluate expressions involving gamma of very large numbers in the logarithmic domain So for example this key sequence gives this result Num 1E50 Math Gam Gam Infinity Num 1E50 Math Ln Gam Ln Gam1 14129 E52 The key Reversing the Sign The key reverses the sign of the number at the top of the stack a positive number becomes negative and a negative number becomes positive So for example this key sequence gives this result Num 25 Enter Math 25 Num 25 Enter Math 25 The x y key Modulo remainder after division The simple definition of the modulo function xly spoken as x modulo y is the remainder when x is divided by y So 13 modulo 5 is 3 because 13 5 2 with a remainder of 3 Similarly 5 4 modulo 1 38 is 1 26 It s a little tricky when one or both of the two numbers are negative 13 modulo 5 is 3 as you might have expected 13 modulo 5 is 3 not 3 as you might or might not have expected and 13 modulo 5 is 3 as you might or might not have expected Anything modulo 0 is NaN The 1 x key Reciprocal of a number The 1 x key simply replaces the number at the top of the stack by its reciprocal The reciprocal of 0 is In
71. tribution ssessesseeseesesssesresseserssressessresresstssresreeseesresreeseeseeseresee 29 The Fish F key The Fisher F Distribution sssssssesssesssesesseeesseesseessesseresseeessresseesseeeseeessseesseese 30 The Bin key The Binomial Distribution and Confidence Intervals sseeeseneeeneeeereeresreerreereee 30 The Pois key The Poisson Distribution and Confidence Intervals 0 eee eeeeeseceeeeeeeeeeneeenneee 31 The nCr key Combinations binomial coefficient 0 0 eee eeeeesceesseceeeceseeeeaeecaeceseeseeeesneeenaeens 32 The nPi key Permutations iesse arene aa A E EO CR TA 33 The LgN Nrm and key Transforming between Normal and Log Normal Distributions 33 The Arc Sine Key The ArcSine transformation for Proportions esssssesrseseeeeesrrsreeeresressesereses 35 The Fish z key The Fisher z transformation for correlation coefficients eeeeeceeeeeeereeeeeeee 35 The Tests Panel Statistical Tests s ssssscsesssessssssesesssessesnsnsssneesesnsnssnsnensesnsons 37 The NaN key Entering the Not a Number flag cee ceeeeeescecsseceseeeseeeeseeceaeceseeseeeesneeenaeees 37 The CL key Setting the Confidence Level s snnnssesssesssseessseesseesseesseeesseeesseesseesseeeseressseesseeso 38 The 2x2 a key p values for a 2 by 2 cross tab table sseeeeesseseeeesseesresressessresressersresrreseesreseresee 38 The 2x2 b key Other Quantities from a 2 by 2 cross tab table es
72. ts of numbers We can enter up to 100 numbers onto the stack that s its current capacity and we need to tell StatiCal how far down the Updated March 21 2014 StatiCal User Manual Page 38 of 55 stack to look for the numbers You might have other numbers sitting on the stack perhaps from earlier calculations which you want to save and you don t want these other numbers to be included in the descriptive summarization So for that reason there s a NaN key in the Num keypad to push this special flag onto the stack We can hit NaN before we start entering our actual data values so it will serve as a marker What if you ve painstaking entered 50 numbers on the stack and then realize to your horror that you forgot to hit the NaN key first Don t panic just read the The CL key Setting the Confidence Level Several of the keys in the Tests panel calculate confidence limits of one type or another By default these will be 95 confidence intervals but you can set any confidence level you d like by entering that number onto the stack as a percentage not as a fraction and then tapping the CL key So for example if you d like 90 confidence limits you d enter 90 from the Num keypad then go to the Tests keypad and tap CL You ll see the indicator 90 CL in small yellow type above the CL button This new confidence level will stay in effect until you change it or until you terminate the running of StatiCal
73. u re going to have to use over and over again during a set of calculations and you don t want to have to enter it every time it may have a lot of digits Or you may have the result of some calculations that you know you ll have to use at several places in subsequent calculations It s very convenient to be able to stash this number someplace safe at least safer than on the stack where there s always a lot of action going on There are keys on the Stor panel that let you move numbers between the stack and the memory cells There are also keys to clear out zero out the contents of a memory cell And there are keys that let you use these memory cells as accumulators to add or subtract a set of numbers There are also a few keys that are useful in dealing with the stack itself like popping numbers off the stack or swapping the positions of two numbers Updated March 21 2014 StatiCal User Manual Page 53 of 55 The Clir keys Set a Memory Cell to Zero The M1 Clr cell sets the M1 memory cell to zero The stack is not altered The M2 Clr key does exactly the same thing but with the M2 memory cell instead of the M1 cell The Sto keys Store a number into a separate Memory Cell The M1 Sto cell pops the most recently added number off of the stack and puts it into the M1 memory cell replacing anything that might have been in M1 at the time The number is removed from the stack and the other numbers on the stack slide dow
74. u can also toggle between the range indicators using the Rnge key if you want Then tap the Inv key and the Stud t key and the value of t will replace the values of the probability and n on the stack So for example this key sequence gives this result The Fish F key The Fisher F Distribution This key gives the probability integrals and inverses for the Fisher F family of distributions You must enter the value of F and also the degrees of freedom for the numerator n1 and denominator n2 in that order You can also toggle between the four range indicators if you want Then tap the Fish F key The values of F nl and n2 will be popped off of the stack and the value of the F integral will replace them on the stack That is the three fields in the calculator will be replaced by a single field and the remaining stack contents will be slid down two positions to fill the vacant slots If you want the inverse of the F probability value you would enter the probability value then the values of n1 and n2 in that order You can also toggle between the range indicators if you want Then tap the Inv key and the Fish F key and the value of F will replace the values of the probability n1 and n2 on the stack So for example examples to be provided this key sequence gives this result The Bin key The Binomial Distribution and Confidence Intervals The binomial distribution is defined as fol
75. u press the Deg Rad button once the yellow Rad turns into a yellow Deg which means that from now on StatiCal will be evaluating the trig functions for angles in degrees Updated March 21 2014 StatiCal User Manual Page 24 of 55 The Deg Rad toggle flips back and forth every time you press that key and stays the same until you press it again So for example this key sequence gives this comment result Num 30 Math Deg SinD 0 5 Tapping the Deg Rad toggle once will change the toggle from its Rad Sin default Radians setting to the Degrees setting Num 45 Math Tan TanD The Deg Rad toggle will remain at Degrees until you toggle it to something else The Inv modifier key Inverse Trigonometric Functions Now what about getting the inverse trigonometric functions What if you want to know the angle whose sine is 0 5 The inverse trigonometric functions give you the angle whose sine or cosine or tangent is the value you have specified There are several conventions for how the inverse functions are named when speaking or writing them in a programming language The are usually spoken as inverse sine or arcsine and written as InvSin ArcSin or Asin or Sin among several other ways That last way Sin can be confusing because a superscript 1 usually means raised to the power of 1 which means the reciprocal and the arcsine is definitely not the reciprocal of the sine th
76. umber from a Memory Cell and put it onto the stack eee 53 The M keys Add a number from the stack onto a Memory Cell ee eee eeseceeeeeeeeeeeeenaeen 53 The M keys Add a number from the stack onto a Memory Cell cece eeseeeseeeeeeeneeenneees 53 The AC key Clear out All Values from Stack and Memory Cello ee eee eeseceseeeeeeeeeeenaeens 53 The Pop key Pop the bottom number from the Stack esseessesseseesesssesressessresressersresreesreseeseresee 54 The Dup key Duplicate the bottom number on the stack sesesesseseeeseesesseseresressersresressresseseesee 54 The Swap key Swap the two bottom numbers on the Stack eeeseseseesessesesesresersreseresressrsereses 54 The Roll Up and Roll Dn keys Roll the stack with wrap around eee eeeeeeeeceeeeeeeeeeeeeeaeees 55 Updated March 21 2014 StatiCal User Manual Page 4 of 55 Introduction This manual describes the app StatiCal The Statistician s Calculator StatiCal is a full function scientific calculator that also performs a variety of statistical calculations Rather than listing them all here I d just ask you to take a look at the Table of Contents for this manual to get an idea of StatiCal s capabilities A few things about StatiCal as an smartphone tablet app e StatiCal should run on all reasonably current versions of the device s operating system e StatiCal is a lean and mean app the installed app is about 1 2 MB e StatiCal is
77. vice s floating point limits but the final value of nPr might not be excessively large due to partial cancellation of the large numerator and denominator StatiCal uses logarithms when necessary to obtain nPr values for these extreme cases So for example this key sequence gives this result Num 20 Enter 6 PDFs nPr 20C6 27 907 200 Num 1000 Enter 957 PDFs nPr 1000P957 Infinity Num 100 Enter 95 PDFs nPr 100P95 7 77718 E155 Note that permutations tend to be a lot bigger than combinations in fact r times larger The LgN Nrm and key Transforming between Normal and Log Normal Distributions Many things in nature tend to have distributions that are very strongly right skewed The log normal distribution very often describes such data nicely Though the numbers may be very skewed their logarithms are often nicely normally distributed Mathematically e ifa set of numbers X are log normally distributed then their logarithms x Ln X are normally distributed Updated March 21 2014 StatiCal User Manual Page 34 of 55 And conversely e ifa set of numbers x are normally distributed then their exponentials X Exp x are log normally distributed If we denote the mean and standard deviation of the X as M and S and the mean and standard deviation of the x as m and s For infinitely large populations there are exact relationships between M S and m s formulas to be pro
78. vided Log Normal Normal to Normal to Log Normal Mean m M Standard Deviation s S The LgN Norm key transforms the M and S of a lognormal distribution into the m and s of the normal distribution that would result from log transforming the numbers You don t actually work with the X or x numbers you need only the mean and standard deviation When preceded by the Inv toggle the LgN Norm key will work in the opposite direction transforming the m and s of a normally distributed set of numbers into the M and S of the log normal distribution that would result from exponentiating each of the x numbers Regardless of which direction you want to go you would first enter the mean onto the stack then the standard deviation So for example this key sequence gives this result Num 10 Enter 8 PDFs LgN Norm mbLn 2 0552 sdLn 0 7033 Num 2 Enter 0 7 PDFs Inv LgN Norm mL N 9 4404 sd L N 7 5069 So if an infinitely large collection of log normally distributed numbers had a mean of 10 and a SD of 8 then if you calculated the natural log of each number those logs would be normally distributed with a mean of 2 0552 and a SD of 0 7033 And if an infinitely large collection of normally distributed numbers had a mean of 2 and a SD of 0 7 then if you exponentiated each number raised e to the power of each number those numbers would be log normally distributed with a mean of

StatiCal The Statistician`s Calculator

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