1math::statistics(n) Tcl Math Library math::statistics(n)
2______________________________________________________________________________
6
8 math::statistics - Basic statistical functions and procedures
9
11 package require Tcl 8.4
12 package require math::statistics 1
14 ::math::statistics::mean data
16 ::math::statistics::min data
18 ::math::statistics::max data
20 ::math::statistics::number data
22 ::math::statistics::stdev data
24 ::math::statistics::var data
26 ::math::statistics::pstdev data
28 ::math::statistics::pvar data
30 ::math::statistics::median data
32 ::math::statistics::basic-stats data
34 ::math::statistics::histogram limits values ?weights?
36 ::math::statistics::histogram-alt limits values ?weights?
38 ::math::statistics::corr data1 data2
40 ::math::statistics::interval-mean-stdev data confidence
42 ::math::statistics::t-test-mean data est_mean est_stdev alpha
44 ::math::statistics::test-normal data significance
46 ::math::statistics::lillieforsFit data
48 ::math::statistics::test-Duckworth list1 list2 significance
50 ::math::statistics::test-anova-F alpha args
52 ::math::statistics::test-Tukey-range alpha args
54 ::math::statistics::test-Dunnett alpha control args
56 ::math::statistics::quantiles data confidence
58 ::math::statistics::quantiles limits counts confidence
60 ::math::statistics::autocorr data
62 ::math::statistics::crosscorr data1 data2
64 ::math::statistics::mean-histogram-limits mean stdev number
66 ::math::statistics::minmax-histogram-limits min max number
68 ::math::statistics::linear-model xdata ydata intercept
70 ::math::statistics::linear-residuals xdata ydata intercept
72 ::math::statistics::test-2x2 n11 n21 n12 n22
74 ::math::statistics::print-2x2 n11 n21 n12 n22
76 ::math::statistics::control-xbar data ?nsamples?
78 ::math::statistics::control-Rchart data ?nsamples?
80 ::math::statistics::test-xbar control data
82 ::math::statistics::test-Rchart control data
84 ::math::statistics::test-Kruskal-Wallis confidence args
86 ::math::statistics::analyse-Kruskal-Wallis args
88 ::math::statistics::group-rank args
90 ::math::statistics::test-Wilcoxon sample_a sample_b
92 ::math::statistics::spearman-rank sample_a sample_b
94 ::math::statistics::spearman-rank-extended sample_a sample_b
96 ::math::statistics::kernel-density data opt -option value ...
98 ::math::statistics::tstat dof ?alpha?
100 ::math::statistics::mv-wls wt1 weights_and_values
102 ::math::statistics::mv-ols values
104 ::math::statistics::pdf-normal mean stdev value
106 ::math::statistics::pdf-lognormal mean stdev value
108 ::math::statistics::pdf-exponential mean value
110 ::math::statistics::pdf-uniform xmin xmax value
112 ::math::statistics::pdf-gamma alpha beta value
114 ::math::statistics::pdf-poisson mu k
116 ::math::statistics::pdf-chisquare df value
118 ::math::statistics::pdf-student-t df value
120 ::math::statistics::pdf-gamma a b value
122 ::math::statistics::pdf-beta a b value
124 ::math::statistics::pdf-weibull scale shape value
126 ::math::statistics::pdf-gumbel location scale value
128 ::math::statistics::pdf-pareto scale shape value
130 ::math::statistics::pdf-cauchy location scale value
132 ::math::statistics::cdf-normal mean stdev value
134 ::math::statistics::cdf-lognormal mean stdev value
136 ::math::statistics::cdf-exponential mean value
138 ::math::statistics::cdf-uniform xmin xmax value
140 ::math::statistics::cdf-students-t degrees value
142 ::math::statistics::cdf-gamma alpha beta value
144 ::math::statistics::cdf-poisson mu k
146 ::math::statistics::cdf-beta a b value
148 ::math::statistics::cdf-weibull scale shape value
150 ::math::statistics::cdf-gumbel location scale value
152 ::math::statistics::cdf-pareto scale shape value
154 ::math::statistics::cdf-cauchy location scale value
156 ::math::statistics::cdf-F nf1 nf2 value
158 ::math::statistics::empirical-distribution values
160 ::math::statistics::random-normal mean stdev number
162 ::math::statistics::random-lognormal mean stdev number
164 ::math::statistics::random-exponential mean number
166 ::math::statistics::random-uniform xmin xmax number
168 ::math::statistics::random-gamma alpha beta number
170 ::math::statistics::random-poisson mu number
172 ::math::statistics::random-chisquare df number
174 ::math::statistics::random-student-t df number
176 ::math::statistics::random-beta a b number
178 ::math::statistics::random-weibull scale shape number
180 ::math::statistics::random-gumbel location scale number
182 ::math::statistics::random-pareto scale shape number
184 ::math::statistics::random-cauchy location scale number
186 ::math::statistics::histogram-uniform xmin xmax limits number
188 ::math::statistics::incompleteGamma x p ?tol?
190 ::math::statistics::incompleteBeta a b x ?tol?
192 ::math::statistics::estimate-pareto values
194 ::math::statistics::filter varname data expression
196 ::math::statistics::map varname data expression
198 ::math::statistics::samplescount varname list expression
200 ::math::statistics::subdivide
202 ::math::statistics::plot-scale canvas xmin xmax ymin ymax
204 ::math::statistics::plot-xydata canvas xdata ydata tag
206 ::math::statistics::plot-xyline canvas xdata ydata tag
208 ::math::statistics::plot-tdata canvas tdata tag
210 ::math::statistics::plot-tline canvas tdata tag
212 ::math::statistics::plot-histogram canvas counts limits tag
214______________________________________________________________________________
216
218 The math::statistics package contains functions and procedures for
219 basic statistical data analysis, such as:
220 · Descriptive statistical parameters (mean, minimum, maximum,
222 standard deviation)
223 · Estimates of the distribution in the form of histograms and
225 quantiles
226 · Basic testing of hypotheses
228 · Probability and cumulative density functions
230 It is meant to help in developing data analysis applications or doing
232 ad hoc data analysis, it is not in itself a full application, nor is it
233 intended to rival with full (non-)commercial statistical packages.
234 The purpose of this document is to describe the implemented procedures
236 and provide some examples of their usage. As there is ample literature
237 on the algorithms involved, we refer to relevant text books for more
238 explanations. The package contains a fairly large number of public
239 procedures. They can be distinguished in three sets: general proce‐
240 dures, procedures that deal with specific statistical distributions,
241 list procedures to select or transform data and simple plotting proce‐
242 dures (these require Tk). Note: The data that need to be analyzed are
243 always contained in a simple list. Missing values are represented as
244 empty list elements. Note: With version 1.0.1 a mistake in the procs
245 pdf-lognormal, cdf-lognormal and random-lognormal has been corrected.
246 In previous versions the argument for the standard deviation was actu‐
247 ally used as if it was the variance.
248
250 The general statistical procedures are:
251 ::math::statistics::mean data
253 Determine the mean value of the given list of data.
254 list data
256 - List of data
257 ::math::statistics::min data
260 Determine the minimum value of the given list of data.
261 list data
263 - List of data
264 ::math::statistics::max data
267 Determine the maximum value of the given list of data.
268 list data
270 - List of data
271 ::math::statistics::number data
274 Determine the number of non-missing data in the given list
275 list data
277 - List of data
278 ::math::statistics::stdev data
281 Determine the sample standard deviation of the data in the given
282 list
283 list data
285 - List of data
286 ::math::statistics::var data
289 Determine the sample variance of the data in the given list
290 list data
292 - List of data
293 ::math::statistics::pstdev data
296 Determine the population standard deviation of the data in the
297 given list
298 list data
300 - List of data
301 ::math::statistics::pvar data
304 Determine the population variance of the data in the given list
305 list data
307 - List of data
308 ::math::statistics::median data
311 Determine the median of the data in the given list (Note that
312 this requires sorting the data, which may be a costly operation)
313 list data
315 - List of data
316 ::math::statistics::basic-stats data
319 Determine a list of all the descriptive parameters: mean, mini‐
320 mum, maximum, number of data, sample standard deviation, sample
321 variance, population standard deviation and population variance.
322 (This routine is called whenever either or all of the basic sta‐
324 tistical parameters are required. Hence all calculations are
325 done and the relevant values are returned.)
326 list data
328 - List of data
329 ::math::statistics::histogram limits values ?weights?
332 Determine histogram information for the given list of data.
333 Returns a list consisting of the number of values that fall into
334 each interval. (The first interval consists of all values lower
335 than the first limit, the last interval consists of all values
336 greater than the last limit. There is one more interval than
337 there are limits.)
338 Optionally, you can use weights to influence the histogram.
340 list limits
342 - List of upper limits (in ascending order) for the
343 intervals of the histogram.
344 list values
346 - List of data
347 list weights
349 - List of weights, one weight per value
350 ::math::statistics::histogram-alt limits values ?weights?
353 Alternative implementation of the histogram procedure: the open
354 end of the intervals is at the lower bound instead of the upper
355 bound.
356 list limits
358 - List of upper limits (in ascending order) for the
359 intervals of the histogram.
360 list values
362 - List of data
363 list weights
365 - List of weights, one weight per value
366 ::math::statistics::corr data1 data2
369 Determine the correlation coefficient between two sets of data.
370 list data1
372 - First list of data
373 list data2
375 - Second list of data
376 ::math::statistics::interval-mean-stdev data confidence
379 Return the interval containing the mean value and one containing
380 the standard deviation with a certain level of confidence
381 (assuming a normal distribution)
382 list data
384 - List of raw data values (small sample)
385 float confidence
387 - Confidence level (0.95 or 0.99 for instance)
388 ::math::statistics::t-test-mean data est_mean est_stdev alpha
391 Test whether the mean value of a sample is in accordance with
392 the estimated normal distribution with a certain probability.
393 Returns 1 if the test succeeds or 0 if the mean is unlikely to
394 fit the given distribution.
395 list data
397 - List of raw data values (small sample)
398 float est_mean
400 - Estimated mean of the distribution
401 float est_stdev
403 - Estimated stdev of the distribution
404 float alpha
406 - Probability level (0.95 or 0.99 for instance)
407 ::math::statistics::test-normal data significance
410 Test whether the given data follow a normal distribution with a
411 certain level of significance. Returns 1 if the data are nor‐
412 mally distributed within the level of significance, returns 0 if
413 not. The underlying test is the Lilliefors test. Smaller values
414 of the significance mean a stricter testing.
415 list data
417 - List of raw data values
418 float significance
420 - Significance level (one of 0.01, 0.05, 0.10, 0.15 or
421 0.20). For compatibility reasons the values "1-signifi‐
422 cance", 0.80, 0.85, 0.90, 0.95 or 0.99 are also accepted.
423 Compatibility issue: the original implementation and documentation used
425 the term "confidence" and used a value 1-significance (see ticket
426 2812473fff). This has been corrected as of version 0.9.3.
427 ::math::statistics::lillieforsFit data
430 Returns the goodness of fit to a normal distribution according
431 to Lilliefors. The higher the number, the more likely the data
432 are indeed normally distributed. The test requires at least five
433 data points.
434 list data
436 - List of raw data values
437 ::math::statistics::test-Duckworth list1 list2 significance
440 Determine if two data sets have the same median according to the
441 Tukey-Duckworth test. The procedure returns 0 if the medians
442 are unequal, 1 if they are equal, -1 if the test can not be con‐
443 ducted (the smallest value must be in a different set than the
444 greatest value). # # Arguments: # list1 Values in
445 the first data set # list2 Values in the second
446 data set # significance Significance level (either 0.05,
447 0.01 or 0.001) # # Returns: Test whether the given data follow a
448 normal distribution with a certain level of significance.
449 Returns 1 if the data are normally distributed within the level
450 of significance, returns 0 if not. The underlying test is the
451 Lilliefors test. Smaller values of the significance mean a
452 stricter testing.
453 list list1
455 - First list of data
456 list list2
458 - Second list of data
459 float significance
461 - Significance level (either 0.05, 0.01 or 0.001)
462 ::math::statistics::test-anova-F alpha args
465 Determine if two or more groups with normally distributed data
466 have the same means. The procedure returns 0 if the means are
467 likely unequal, 1 if they are. This is a one-way ANOVA test. The
468 groups may also be stored in a nested list: The procedure
469 returns a list of the comparison results for each pair of
470 groups. Each element of this list contains: the index of the
471 first group and that of the second group, whether the means are
472 likely to be different (1) or not (0) and the confidence inter‐
473 val the conclusion is based on. The groups may also be stored in
474 a nested list:
475 test-anova-F 0.05 $A $B $C
478 #
479 # Or equivalently:
480 #
481 test-anova-F 0.05 [list $A $B $C]
482 float alpha
485 - Significance level
486 list args
488 - Two or more groups of data to be checked
489 ::math::statistics::test-Tukey-range alpha args
492 Determine if two or more groups with normally distributed data
493 have the same means, using Tukey's range test. It is complemen‐
494 tary to the ANOVA test. The procedure returns a list of the
495 comparison results for each pair of groups. Each element of this
496 list contains: the index of the first group and that of the sec‐
497 ond group, whether the means are likely to be different (1) or
498 not (0) and the confidence interval the conclusion is based on.
499 The groups may also be stored in a nested list, just as with the
500 ANOVA test.
501 float alpha
503 - Significance level - either 0.05 or 0.01
504 list args
506 - Two or more groups of data to be checked
507 ::math::statistics::test-Dunnett alpha control args
510 Determine if one or more groups with normally distributed data
511 have the same means as the group of control data, using Dun‐
512 nett's test. It is complementary to the ANOVA test. The proce‐
513 dure returns a list of the comparison results for each group
514 with the control group. Each element of this list contains:
515 whether the means are likely to be different (1) or not (0) and
516 the confidence interval the conclusion is based on. The groups
517 may also be stored in a nested list, just as with the ANOVA
518 test.
519 Note: some care is required if there is only one group to com‐
521 pare the control with:
522 test-Dunnett-F 0.05 $control [list $A]
525 Otherwise the group A is split up into groups of one element -
528 this is due to an ambiguity.
529 float alpha
531 - Significance level - either 0.05 or 0.01
532 list args
534 - One or more groups of data to be checked
535 ::math::statistics::quantiles data confidence
538 Return the quantiles for a given set of data
539 list data
542 - List of raw data values
543 float confidence
546 - Confidence level (0.95 or 0.99 for instance) or a list
547 of confidence levels.
548 ::math::statistics::quantiles limits counts confidence
552 Return the quantiles based on histogram information (alternative
553 to the call with two arguments)
554 list limits
556 - List of upper limits from histogram
557 list counts
559 - List of counts for for each interval in histogram
560 float confidence
562 - Confidence level (0.95 or 0.99 for instance) or a list
563 of confidence levels.
564 ::math::statistics::autocorr data
567 Return the autocorrelation function as a list of values (assum‐
568 ing equidistance between samples, about 1/2 of the number of raw
569 data)
570 The correlation is determined in such a way that the first value
572 is always 1 and all others are equal to or smaller than 1. The
573 number of values involved will diminish as the "time" (the index
574 in the list of returned values) increases
575 list data
577 - Raw data for which the autocorrelation must be deter‐
578 mined
579 ::math::statistics::crosscorr data1 data2
582 Return the cross-correlation function as a list of values
583 (assuming equidistance between samples, about 1/2 of the number
584 of raw data)
585 The correlation is determined in such a way that the values can
587 never exceed 1 in magnitude. The number of values involved will
588 diminish as the "time" (the index in the list of returned val‐
589 ues) increases.
590 list data1
592 - First list of data
593 list data2
595 - Second list of data
596 ::math::statistics::mean-histogram-limits mean stdev number
599 Determine reasonable limits based on mean and standard deviation
600 for a histogram Convenience function - the result is suitable
601 for the histogram function.
602 float mean
604 - Mean of the data
605 float stdev
607 - Standard deviation
608 int number
610 - Number of limits to generate (defaults to 8)
611 ::math::statistics::minmax-histogram-limits min max number
614 Determine reasonable limits based on a minimum and maximum for a
615 histogram
616 Convenience function - the result is suitable for the histogram
618 function.
619 float min
621 - Expected minimum
622 float max
624 - Expected maximum
625 int number
627 - Number of limits to generate (defaults to 8)
628 ::math::statistics::linear-model xdata ydata intercept
631 Determine the coefficients for a linear regression between two
632 series of data (the model: Y = A + B*X). Returns a list of
633 parameters describing the fit
634 list xdata
636 - List of independent data
637 list ydata
639 - List of dependent data to be fitted
640 boolean intercept
642 - (Optional) compute the intercept (1, default) or fit to
643 a line through the origin (0)
644 The result consists of the following list:
646 · (Estimate of) Intercept A
648 · (Estimate of) Slope B
650 · Standard deviation of Y relative to fit
652 · Correlation coefficient R2
654 · Number of degrees of freedom df
656 · Standard error of the intercept A
658 · Significance level of A
660 · Standard error of the slope B
662 · Significance level of B
664 ::math::statistics::linear-residuals xdata ydata intercept
667 Determine the difference between actual data and predicted from
668 the linear model.
669 Returns a list of the differences between the actual data and
671 the predicted values.
672 list xdata
674 - List of independent data
675 list ydata
677 - List of dependent data to be fitted
678 boolean intercept
680 - (Optional) compute the intercept (1, default) or fit to
681 a line through the origin (0)
682 ::math::statistics::test-2x2 n11 n21 n12 n22
685 Determine if two set of samples, each from a binomial distribu‐
686 tion, differ significantly or not (implying a different parame‐
687 ter).
688 Returns the "chi-square" value, which can be used to the deter‐
690 mine the significance.
691 int n11
693 - Number of outcomes with the first value from the first
694 sample.
695 int n21
697 - Number of outcomes with the first value from the second
698 sample.
699 int n12
701 - Number of outcomes with the second value from the first
702 sample.
703 int n22
705 - Number of outcomes with the second value from the sec‐
706 ond sample.
707 ::math::statistics::print-2x2 n11 n21 n12 n22
710 Determine if two set of samples, each from a binomial distribu‐
711 tion, differ significantly or not (implying a different parame‐
712 ter).
713 Returns a short report, useful in an interactive session.
715 int n11
717 - Number of outcomes with the first value from the first
718 sample.
719 int n21
721 - Number of outcomes with the first value from the second
722 sample.
723 int n12
725 - Number of outcomes with the second value from the first
726 sample.
727 int n22
729 - Number of outcomes with the second value from the sec‐
730 ond sample.
731 ::math::statistics::control-xbar data ?nsamples?
734 Determine the control limits for an xbar chart. The number of
735 data in each subsample defaults to 4. At least 20 subsamples are
736 required.
737 Returns the mean, the lower limit, the upper limit and the num‐
739 ber of data per subsample.
740 list data
742 - List of observed data
743 int nsamples
745 - Number of data per subsample
746 ::math::statistics::control-Rchart data ?nsamples?
749 Determine the control limits for an R chart. The number of data
750 in each subsample (nsamples) defaults to 4. At least 20 subsam‐
751 ples are required.
752 Returns the mean range, the lower limit, the upper limit and the
754 number of data per subsample.
755 list data
757 - List of observed data
758 int nsamples
760 - Number of data per subsample
761 ::math::statistics::test-xbar control data
764 Determine if the data exceed the control limits for the xbar
765 chart.
766 Returns a list of subsamples (their indices) that indeed violate
768 the limits.
769 list control
771 - Control limits as returned by the "control-xbar" proce‐
772 dure
773 list data
775 - List of observed data
776 ::math::statistics::test-Rchart control data
779 Determine if the data exceed the control limits for the R chart.
780 Returns a list of subsamples (their indices) that indeed violate
782 the limits.
783 list control
785 - Control limits as returned by the "control-Rchart" pro‐
786 cedure
787 list data
789 - List of observed data
790 ::math::statistics::test-Kruskal-Wallis confidence args
793 Check if the population medians of two or more groups are equal
794 with a given confidence level, using the Kruskal-Wallis test.
795 float confidence
797 - Confidence level to be used (0-1)
798 list args
800 - Two or more lists of data
801 ::math::statistics::analyse-Kruskal-Wallis args
804 Compute the statistical parameters for the Kruskal-Wallis test.
805 Returns the Kruskal-Wallis statistic and the probability that
806 that value would occur assuming the medians of the populations
807 are equal.
808 list args
810 - Two or more lists of data
811 ::math::statistics::group-rank args
814 Rank the groups of data with respect to the complete set.
815 Returns a list consisting of the group ID, the value and the
816 rank (possibly a rational number, in case of ties) for each data
817 item.
818 list args
820 - Two or more lists of data
821 ::math::statistics::test-Wilcoxon sample_a sample_b
824 Compute the Wilcoxon test statistic to determine if two samples
825 have the same median or not. (The statistic can be regarded as
826 standard normal, if the sample sizes are both larger than 10.
827 Returns the value of this statistic.
828 list sample_a
830 - List of data comprising the first sample
831 list sample_b
833 - List of data comprising the second sample
834 ::math::statistics::spearman-rank sample_a sample_b
837 Return the Spearman rank correlation as an alternative to the
838 ordinary (Pearson's) correlation coefficient. The two samples
839 should have the same number of data.
840 list sample_a
842 - First list of data
843 list sample_b
845 - Second list of data
846 ::math::statistics::spearman-rank-extended sample_a sample_b
849 Return the Spearman rank correlation as an alternative to the
850 ordinary (Pearson's) correlation coefficient as well as addi‐
851 tional data. The two samples should have the same number of
852 data. The procedure returns the correlation coefficient, the
853 number of data pairs used and the z-score, an approximately
854 standard normal statistic, indicating the significance of the
855 correlation.
856 list sample_a
858 - First list of data
859 list sample_b
861 - Second list of data
862 ::math::statistics::kernel-density data opt -option value ...
864 ] Return the density function based on kernel density estima‐
865 tion. The procedure is controlled by a small set of options,
866 each of which is given a reasonable default.
867 The return value consists of three lists: the centres of the
869 bins, the associated probability density and a list of computa‐
870 tional parameters (begin and end of the interval, mean and stan‐
871 dard deviation and the used bandwidth). The computational param‐
872 eters can be used for further analysis.
873 list data
875 - The data to be examined
876 list args
878 - Option-value pairs:
879 -weights weights
881 Per data point the weight (default: 1 for all
882 data)
883 -bandwidth value
885 Bandwidth to be used for the estimation (default:
886 determined from standard deviation)
887 -number value
889 Number of bins to be returned (default: 100)
890 -interval {begin end}
892 Begin and end of the interval for which the den‐
893 sity is returned (default: mean +/- 3*standard
894 deviation)
895 -kernel function
897 Kernel to be used (One of: gaussian, cosine,
898 epanechnikov, uniform, triangular, biweight,
899 logistic; default: gaussian)
900
902 Besides the linear regression with a single independent variable, the
903 statistics package provides two procedures for doing ordinary least
904 squares (OLS) and weighted least squares (WLS) linear regression with
905 several variables. They were written by Eric Kemp-Benedict.
906 In addition to these two, it provides a procedure (tstat) for calculat‐
908 ing the value of the t-statistic for the specified number of degrees of
909 freedom that is required to demonstrate a given level of significance.
910 Note: These procedures depend on the math::linearalgebra package.
912 Description of the procedures
914 ::math::statistics::tstat dof ?alpha?
916 Returns the value of the t-distribution t* satisfying
917 P(t*) = 1 - alpha/2
920 P(-t*) = alpha/2
921 for the number of degrees of freedom dof.
924 Given a sample of normally-distributed data x, with an estimate
926 xbar for the mean and sbar for the standard deviation, the alpha
927 confidence interval for the estimate of the mean can be calcu‐
928 lated as
929 ( xbar - t* sbar , xbar + t* sbar)
932 The return values from this procedure can be compared to an
935 estimated t-statistic to determine whether the estimated value
936 of a parameter is significantly different from zero at the given
937 confidence level.
938 int dof
940 Number of degrees of freedom
941 float alpha
943 Confidence level of the t-distribution. Defaults to 0.05.
944 ::math::statistics::mv-wls wt1 weights_and_values
947 Carries out a weighted least squares linear regression for the
948 data points provided, with weights assigned to each point.
949 The linear model is of the form
951 y = b0 + b1 * x1 + b2 * x2 ... + bN * xN + error
954 and each point satisfies
957 yi = b0 + b1 * xi1 + b2 * xi2 + ... + bN * xiN + Residual_i
960 The procedure returns a list with the following elements:
963 · The r-squared statistic
965 · The adjusted r-squared statistic
967 · A list containing the estimated coefficients b1, ... bN,
969 b0 (The constant b0 comes last in the list.)
970 · A list containing the standard errors of the coefficients
972 · A list containing the 95% confidence bounds of the coef‐
974 ficients, with each set of bounds returned as a list with
975 two values
976 Arguments:
978 list weights_and_values
980 A list consisting of: the weight for the first observa‐
981 tion, the data for the first observation (as a sublist),
982 the weight for the second observation (as a sublist) and
983 so on. The sublists of data are organised as lists of the
984 value of the dependent variable y and the independent
985 variables x1, x2 to xN.
986 ::math::statistics::mv-ols values
989 Carries out an ordinary least squares linear regression for the
990 data points provided.
991 This procedure simply calls ::mvlinreg::wls with the weights set
993 to 1.0, and returns the same information.
994 Example of the use:
996 # Store the value of the unicode value for the "+/-" character
999 set pm "\u00B1"
1000 # Provide some data
1002 set data {{ -.67 14.18 60.03 -7.5 }
1003 { 36.97 15.52 34.24 14.61 }
1004 {-29.57 21.85 83.36 -7. }
1005 {-16.9 11.79 51.67 -6.56 }
1006 { 14.09 16.24 36.97 -12.84}
1007 { 31.52 20.93 45.99 -25.4 }
1008 { 24.05 20.69 50.27 17.27}
1009 { 22.23 16.91 45.07 -4.3 }
1010 { 40.79 20.49 38.92 -.73 }
1011 {-10.35 17.24 58.77 18.78}}
1012 # Call the ols routine
1014 set results [::math::statistics::mv-ols $data]
1015 # Pretty-print the results
1017 puts "R-squared: [lindex $results 0]"
1018 puts "Adj R-squared: [lindex $results 1]"
1019 puts "Coefficients $pm s.e. -- \[95% confidence interval\]:"
1020 foreach val [lindex $results 2] se [lindex $results 3] bounds [lindex $results 4] {
1021 set lb [lindex $bounds 0]
1022 set ub [lindex $bounds 1]
1023 puts " $val $pm $se -- \[$lb to $ub\]"
1024 }
1025
1028 In the literature a large number of probability distributions can be
1029 found. The statistics package supports:
1030 · The normal or Gaussian distribution as well as the log-normal
1032 distribution
1033 · The uniform distribution - equal probability for all data within
1035 a given interval
1036 · The exponential distribution - useful as a model for certain
1038 extreme-value distributions.
1039 · The gamma distribution - based on the incomplete Gamma integral
1041 · The beta distribution
1043 · The chi-square distribution
1045 · The student's T distribution
1047 · The Poisson distribution
1049 · The Pareto distribution
1051 · The Gumbel distribution
1053 · The Weibull distribution
1055 · The Cauchy distribution
1057 · The F distribution (only the cumulative density function)
1059 · PM - binomial.
1061 In principle for each distribution one has procedures for:
1063 · The probability density (pdf-*)
1065 · The cumulative density (cdf-*)
1067 · Quantiles for the given distribution (quantiles-*)
1069 · Histograms for the given distribution (histogram-*)
1071 · List of random values with the given distribution (random-*)
1073 The following procedures have been implemented:
1075 ::math::statistics::pdf-normal mean stdev value
1077 Return the probability of a given value for a normal distribu‐
1078 tion with given mean and standard deviation.
1079 float mean
1081 - Mean value of the distribution
1082 float stdev
1084 - Standard deviation of the distribution
1085 float value
1087 - Value for which the probability is required
1088 ::math::statistics::pdf-lognormal mean stdev value
1091 Return the probability of a given value for a log-normal distri‐
1092 bution with given mean and standard deviation.
1093 float mean
1095 - Mean value of the distribution
1096 float stdev
1098 - Standard deviation of the distribution
1099 float value
1101 - Value for which the probability is required
1102 ::math::statistics::pdf-exponential mean value
1105 Return the probability of a given value for an exponential dis‐
1106 tribution with given mean.
1107 float mean
1109 - Mean value of the distribution
1110 float value
1112 - Value for which the probability is required
1113 ::math::statistics::pdf-uniform xmin xmax value
1116 Return the probability of a given value for a uniform distribu‐
1117 tion with given extremes.
1118 float xmin
1120 - Minimum value of the distribution
1121 float xmin
1123 - Maximum value of the distribution
1124 float value
1126 - Value for which the probability is required
1127 ::math::statistics::pdf-gamma alpha beta value
1130 Return the probability of a given value for a Gamma distribution
1131 with given shape and rate parameters
1132 float alpha
1134 - Shape parameter
1135 float beta
1137 - Rate parameter
1138 float value
1140 - Value for which the probability is required
1141 ::math::statistics::pdf-poisson mu k
1144 Return the probability of a given number of occurrences in the
1145 same interval (k) for a Poisson distribution with given mean
1146 (mu)
1147 float mu
1149 - Mean number of occurrences
1150 int k - Number of occurences
1152 ::math::statistics::pdf-chisquare df value
1155 Return the probability of a given value for a chi square distri‐
1156 bution with given degrees of freedom
1157 float df
1159 - Degrees of freedom
1160 float value
1162 - Value for which the probability is required
1163 ::math::statistics::pdf-student-t df value
1166 Return the probability of a given value for a Student's t dis‐
1167 tribution with given degrees of freedom
1168 float df
1170 - Degrees of freedom
1171 float value
1173 - Value for which the probability is required
1174 ::math::statistics::pdf-gamma a b value
1177 Return the probability of a given value for a Gamma distribution
1178 with given shape and rate parameters
1179 float a
1181 - Shape parameter
1182 float b
1184 - Rate parameter
1185 float value
1187 - Value for which the probability is required
1188 ::math::statistics::pdf-beta a b value
1191 Return the probability of a given value for a Beta distribution
1192 with given shape parameters
1193 float a
1195 - First shape parameter
1196 float b
1198 - Second shape parameter
1199 float value
1201 - Value for which the probability is required
1202 ::math::statistics::pdf-weibull scale shape value
1205 Return the probability of a given value for a Weibull distribu‐
1206 tion with given scale and shape parameters
1207 float location
1209 - Scale parameter
1210 float scale
1212 - Shape parameter
1213 float value
1215 - Value for which the probability is required
1216 ::math::statistics::pdf-gumbel location scale value
1219 Return the probability of a given value for a Gumbel distribu‐
1220 tion with given location and shape parameters
1221 float location
1223 - Location parameter
1224 float scale
1226 - Shape parameter
1227 float value
1229 - Value for which the probability is required
1230 ::math::statistics::pdf-pareto scale shape value
1233 Return the probability of a given value for a Pareto distribu‐
1234 tion with given scale and shape parameters
1235 float scale
1237 - Scale parameter
1238 float shape
1240 - Shape parameter
1241 float value
1243 - Value for which the probability is required
1244 ::math::statistics::pdf-cauchy location scale value
1247 Return the probability of a given value for a Cauchy distribu‐
1248 tion with given location and shape parameters. Note that the
1249 Cauchy distribution has no finite higher-order moments.
1250 float location
1252 - Location parameter
1253 float scale
1255 - Shape parameter
1256 float value
1258 - Value for which the probability is required
1259 ::math::statistics::cdf-normal mean stdev value
1262 Return the cumulative probability of a given value for a normal
1263 distribution with given mean and standard deviation, that is the
1264 probability for values up to the given one.
1265 float mean
1267 - Mean value of the distribution
1268 float stdev
1270 - Standard deviation of the distribution
1271 float value
1273 - Value for which the probability is required
1274 ::math::statistics::cdf-lognormal mean stdev value
1277 Return the cumulative probability of a given value for a log-
1278 normal distribution with given mean and standard deviation, that
1279 is the probability for values up to the given one.
1280 float mean
1282 - Mean value of the distribution
1283 float stdev
1285 - Standard deviation of the distribution
1286 float value
1288 - Value for which the probability is required
1289 ::math::statistics::cdf-exponential mean value
1292 Return the cumulative probability of a given value for an expo‐
1293 nential distribution with given mean.
1294 float mean
1296 - Mean value of the distribution
1297 float value
1299 - Value for which the probability is required
1300 ::math::statistics::cdf-uniform xmin xmax value
1303 Return the cumulative probability of a given value for a uniform
1304 distribution with given extremes.
1305 float xmin
1307 - Minimum value of the distribution
1308 float xmin
1310 - Maximum value of the distribution
1311 float value
1313 - Value for which the probability is required
1314 ::math::statistics::cdf-students-t degrees value
1317 Return the cumulative probability of a given value for a Stu‐
1318 dent's t distribution with given number of degrees.
1319 int degrees
1321 - Number of degrees of freedom
1322 float value
1324 - Value for which the probability is required
1325 ::math::statistics::cdf-gamma alpha beta value
1328 Return the cumulative probability of a given value for a Gamma
1329 distribution with given shape and rate parameters.
1330 float alpha
1332 - Shape parameter
1333 float beta
1335 - Rate parameter
1336 float value
1338 - Value for which the cumulative probability is required
1339 ::math::statistics::cdf-poisson mu k
1342 Return the cumulative probability of a given number of occur‐
1343 rences in the same interval (k) for a Poisson distribution with
1344 given mean (mu).
1345 float mu
1347 - Mean number of occurrences
1348 int k - Number of occurences
1350 ::math::statistics::cdf-beta a b value
1353 Return the cumulative probability of a given value for a Beta
1354 distribution with given shape parameters
1355 float a
1357 - First shape parameter
1358 float b
1360 - Second shape parameter
1361 float value
1363 - Value for which the probability is required
1364 ::math::statistics::cdf-weibull scale shape value
1367 Return the cumulative probability of a given value for a Weibull
1368 distribution with given scale and shape parameters.
1369 float scale
1371 - Scale parameter
1372 float shape
1374 - Shape parameter
1375 float value
1377 - Value for which the probability is required
1378 ::math::statistics::cdf-gumbel location scale value
1381 Return the cumulative probability of a given value for a Gumbel
1382 distribution with given location and scale parameters.
1383 float location
1385 - Location parameter
1386 float scale
1388 - Scale parameter
1389 float value
1391 - Value for which the probability is required
1392 ::math::statistics::cdf-pareto scale shape value
1395 Return the cumulative probability of a given value for a Pareto
1396 distribution with given scale and shape parameters
1397 float scale
1399 - Scale parameter
1400 float shape
1402 - Shape parameter
1403 float value
1405 - Value for which the probability is required
1406 ::math::statistics::cdf-cauchy location scale value
1409 Return the cumulative probability of a given value for a Cauchy
1410 distribution with given location and scale parameters.
1411 float location
1413 - Location parameter
1414 float scale
1416 - Scale parameter
1417 float value
1419 - Value for which the probability is required
1420 ::math::statistics::cdf-F nf1 nf2 value
1423 Return the cumulative probability of a given value for an F dis‐
1424 tribution with nf1 and nf2 degrees of freedom.
1425 float nf1
1427 - Degrees of freedom for the numerator
1428 float nf2
1430 - Degrees of freedom for the denominator
1431 float value
1433 - Value for which the probability is required
1434 ::math::statistics::empirical-distribution values
1437 Return a list of values and their empirical probability. The
1438 values are sorted in increasing order. (The implementation fol‐
1439 lows the description at the corresponding Wikipedia page)
1440 list values
1442 - List of data to be examined
1443 ::math::statistics::random-normal mean stdev number
1446 Return a list of "number" random values satisfying a normal dis‐
1447 tribution with given mean and standard deviation.
1448 float mean
1450 - Mean value of the distribution
1451 float stdev
1453 - Standard deviation of the distribution
1454 int number
1456 - Number of values to be returned
1457 ::math::statistics::random-lognormal mean stdev number
1460 Return a list of "number" random values satisfying a log-normal
1461 distribution with given mean and standard deviation.
1462 float mean
1464 - Mean value of the distribution
1465 float stdev
1467 - Standard deviation of the distribution
1468 int number
1470 - Number of values to be returned
1471 ::math::statistics::random-exponential mean number
1474 Return a list of "number" random values satisfying an exponen‐
1475 tial distribution with given mean.
1476 float mean
1478 - Mean value of the distribution
1479 int number
1481 - Number of values to be returned
1482 ::math::statistics::random-uniform xmin xmax number
1485 Return a list of "number" random values satisfying a uniform
1486 distribution with given extremes.
1487 float xmin
1489 - Minimum value of the distribution
1490 float xmax
1492 - Maximum value of the distribution
1493 int number
1495 - Number of values to be returned
1496 ::math::statistics::random-gamma alpha beta number
1499 Return a list of "number" random values satisfying a Gamma dis‐
1500 tribution with given shape and rate parameters.
1501 float alpha
1503 - Shape parameter
1504 float beta
1506 - Rate parameter
1507 int number
1509 - Number of values to be returned
1510 ::math::statistics::random-poisson mu number
1513 Return a list of "number" random values satisfying a Poisson
1514 distribution with given mean.
1515 float mu
1517 - Mean of the distribution
1518 int number
1520 - Number of values to be returned
1521 ::math::statistics::random-chisquare df number
1524 Return a list of "number" random values satisfying a chi square
1525 distribution with given degrees of freedom.
1526 float df
1528 - Degrees of freedom
1529 int number
1531 - Number of values to be returned
1532 ::math::statistics::random-student-t df number
1535 Return a list of "number" random values satisfying a Student's t
1536 distribution with given degrees of freedom.
1537 float df
1539 - Degrees of freedom
1540 int number
1542 - Number of values to be returned
1543 ::math::statistics::random-beta a b number
1546 Return a list of "number" random values satisfying a Beta dis‐
1547 tribution with given shape parameters.
1548 float a
1550 - First shape parameter
1551 float b
1553 - Second shape parameter
1554 int number
1556 - Number of values to be returned
1557 ::math::statistics::random-weibull scale shape number
1560 Return a list of "number" random values satisfying a Weibull
1561 distribution with given scale and shape parameters.
1562 float scale
1564 - Scale parameter
1565 float shape
1567 - Shape parameter
1568 int number
1570 - Number of values to be returned
1571 ::math::statistics::random-gumbel location scale number
1574 Return a list of "number" random values satisfying a Gumbel dis‐
1575 tribution with given location and scale parameters.
1576 float location
1578 - Location parameter
1579 float scale
1581 - Scale parameter
1582 int number
1584 - Number of values to be returned
1585 ::math::statistics::random-pareto scale shape number
1588 Return a list of "number" random values satisfying a Pareto dis‐
1589 tribution with given scale and shape parameters.
1590 float scale
1592 - Scale parameter
1593 float shape
1595 - Shape parameter
1596 int number
1598 - Number of values to be returned
1599 ::math::statistics::random-cauchy location scale number
1602 Return a list of "number" random values satisfying a Cauchy dis‐
1603 tribution with given location and scale parameters.
1604 float location
1606 - Location parameter
1607 float scale
1609 - Scale parameter
1610 int number
1612 - Number of values to be returned
1613 ::math::statistics::histogram-uniform xmin xmax limits number
1616 Return the expected histogram for a uniform distribution.
1617 float xmin
1619 - Minimum value of the distribution
1620 float xmax
1622 - Maximum value of the distribution
1623 list limits
1625 - Upper limits for the buckets in the histogram
1626 int number
1628 - Total number of "observations" in the histogram
1629 ::math::statistics::incompleteGamma x p ?tol?
1632 Evaluate the incomplete Gamma integral
1633 1 / x p-1
1636 P(p,x) = -------- | dt exp(-t) * t
1637 Gamma(p) / 0
1638 float x
1641 - Value of x (limit of the integral)
1642 float p
1644 - Value of p in the integrand
1645 float tol
1647 - Required tolerance (default: 1.0e-9)
1648 ::math::statistics::incompleteBeta a b x ?tol?
1651 Evaluate the incomplete Beta integral
1652 float a
1654 - First shape parameter
1655 float b
1657 - Second shape parameter
1658 float x
1660 - Value of x (limit of the integral)
1661 float tol
1663 - Required tolerance (default: 1.0e-9)
1664 ::math::statistics::estimate-pareto values
1667 Estimate the parameters for the Pareto distribution that comes
1668 closest to the given values. Returns the estimated scale and
1669 shape parameters, as well as the standard error for the shape
1670 parameter.
1671 list values
1673 - List of values, assumed to be distributed according to
1674 a Pareto distribution
1675 TO DO: more function descriptions to be added
1678
1680 The data manipulation procedures act on lists or lists of lists:
1681 ::math::statistics::filter varname data expression
1683 Return a list consisting of the data for which the logical
1684 expression is true (this command works analogously to the com‐
1685 mand foreach).
1686 string varname
1688 - Name of the variable used in the expression
1689 list data
1691 - List of data
1692 string expression
1694 - Logical expression using the variable name
1695 ::math::statistics::map varname data expression
1698 Return a list consisting of the data that are transformed via
1699 the expression.
1700 string varname
1702 - Name of the variable used in the expression
1703 list data
1705 - List of data
1706 string expression
1708 - Expression to be used to transform (map) the data
1709 ::math::statistics::samplescount varname list expression
1712 Return a list consisting of the counts of all data in the sub‐
1713 lists of the "list" argument for which the expression is true.
1714 string varname
1716 - Name of the variable used in the expression
1717 list data
1719 - List of sublists, each containing the data
1720 string expression
1722 - Logical expression to test the data (defaults to
1723 "true").
1724 ::math::statistics::subdivide
1727 Routine PM - not implemented yet
1728
1731 The following simple plotting procedures are available:
1732 ::math::statistics::plot-scale canvas xmin xmax ymin ymax
1734 Set the scale for a plot in the given canvas. All plot routines
1735 expect this function to be called first. There is no automatic
1736 scaling provided.
1737 widget canvas
1739 - Canvas widget to use
1740 float xmin
1742 - Minimum x value
1743 float xmax
1745 - Maximum x value
1746 float ymin
1748 - Minimum y value
1749 float ymax
1751 - Maximum y value
1752 ::math::statistics::plot-xydata canvas xdata ydata tag
1755 Create a simple XY plot in the given canvas - the data are shown
1756 as a collection of dots. The tag can be used to manipulate the
1757 appearance.
1758 widget canvas
1760 - Canvas widget to use
1761 float xdata
1763 - Series of independent data
1764 float ydata
1766 - Series of dependent data
1767 string tag
1769 - Tag to give to the plotted data (defaults to xyplot)
1770 ::math::statistics::plot-xyline canvas xdata ydata tag
1773 Create a simple XY plot in the given canvas - the data are shown
1774 as a line through the data points. The tag can be used to manip‐
1775 ulate the appearance.
1776 widget canvas
1778 - Canvas widget to use
1779 list xdata
1781 - Series of independent data
1782 list ydata
1784 - Series of dependent data
1785 string tag
1787 - Tag to give to the plotted data (defaults to xyplot)
1788 ::math::statistics::plot-tdata canvas tdata tag
1791 Create a simple XY plot in the given canvas - the data are shown
1792 as a collection of dots. The horizontal coordinate is equal to
1793 the index. The tag can be used to manipulate the appearance.
1794 This type of presentation is suitable for autocorrelation func‐
1795 tions for instance or for inspecting the time-dependent behav‐
1796 iour.
1797 widget canvas
1799 - Canvas widget to use
1800 list tdata
1802 - Series of dependent data
1803 string tag
1805 - Tag to give to the plotted data (defaults to xyplot)
1806 ::math::statistics::plot-tline canvas tdata tag
1809 Create a simple XY plot in the given canvas - the data are shown
1810 as a line. See plot-tdata for an explanation.
1811 widget canvas
1813 - Canvas widget to use
1814 list tdata
1816 - Series of dependent data
1817 string tag
1819 - Tag to give to the plotted data (defaults to xyplot)
1820 ::math::statistics::plot-histogram canvas counts limits tag
1823 Create a simple histogram in the given canvas
1824 widget canvas
1826 - Canvas widget to use
1827 list counts
1829 - Series of bucket counts
1830 list limits
1832 - Series of upper limits for the buckets
1833 string tag
1835 - Tag to give to the plotted data (defaults to xyplot)
1836
1839 The following procedures are yet to be implemented:
1840 · F-test-stdev
1842 · interval-mean-stdev
1844 · histogram-normal
1846 · histogram-exponential
1848 · test-histogram
1850 · test-corr
1852 · quantiles-*
1854 · fourier-coeffs
1856 · fourier-residuals
1858 · onepar-function-fit
1860 · onepar-function-residuals
1862 · plot-linear-model
1864 · subdivide
1866
1868 The code below is a small example of how you can examine a set of data:
1869 # Simple example:
1871 # - Generate data (as a cheap way of getting some)
1872 # - Perform statistical analysis to describe the data
1873 #
1874 package require math::statistics
1875 #
1877 # Two auxiliary procs
1878 #
1879 proc pause {time} {
1880 set wait 0
1881 after [expr {$time*1000}] {set ::wait 1}
1882 vwait wait
1883 }
1884 proc print-histogram {counts limits} {
1886 foreach count $counts limit $limits {
1887 if { $limit != {} } {
1888 puts [format "<%12.4g\t%d" $limit $count]
1889 set prev_limit $limit
1890 } else {
1891 puts [format ">%12.4g\t%d" $prev_limit $count]
1892 }
1893 }
1894 }
1895 #
1897 # Our source of arbitrary data
1898 #
1899 proc generateData { data1 data2 } {
1900 upvar 1 $data1 _data1
1901 upvar 1 $data2 _data2
1902 set d1 0.0
1904 set d2 0.0
1905 for { set i 0 } { $i < 100 } { incr i } {
1906 set d1 [expr {10.0-2.0*cos(2.0*3.1415926*$i/24.0)+3.5*rand()}]
1907 set d2 [expr {0.7*$d2+0.3*$d1+0.7*rand()}]
1908 lappend _data1 $d1
1909 lappend _data2 $d2
1910 }
1911 return {}
1912 }
1913 #
1915 # The analysis session
1916 #
1917 package require Tk
1918 console show
1919 canvas .plot1
1920 canvas .plot2
1921 pack .plot1 .plot2 -fill both -side top
1922 generateData data1 data2
1924 puts "Basic statistics:"
1926 set b1 [::math::statistics::basic-stats $data1]
1927 set b2 [::math::statistics::basic-stats $data2]
1928 foreach label {mean min max number stdev var} v1 $b1 v2 $b2 {
1929 puts "$label\t$v1\t$v2"
1930 }
1931 puts "Plot the data as function of \"time\" and against each other"
1932 ::math::statistics::plot-scale .plot1 0 100 0 20
1933 ::math::statistics::plot-scale .plot2 0 20 0 20
1934 ::math::statistics::plot-tline .plot1 $data1
1935 ::math::statistics::plot-tline .plot1 $data2
1936 ::math::statistics::plot-xydata .plot2 $data1 $data2
1937 puts "Correlation coefficient:"
1939 puts [::math::statistics::corr $data1 $data2]
1940 pause 2
1942 puts "Plot histograms"
1943 .plot2 delete all
1944 ::math::statistics::plot-scale .plot2 0 20 0 100
1945 set limits [::math::statistics::minmax-histogram-limits 7 16]
1946 set histogram_data [::math::statistics::histogram $limits $data1]
1947 ::math::statistics::plot-histogram .plot2 $histogram_data $limits
1948 puts "First series:"
1950 print-histogram $histogram_data $limits
1951 pause 2
1953 set limits [::math::statistics::minmax-histogram-limits 0 15 10]
1954 set histogram_data [::math::statistics::histogram $limits $data2]
1955 ::math::statistics::plot-histogram .plot2 $histogram_data $limits d2
1956 .plot2 itemconfigure d2 -fill red
1957 puts "Second series:"
1959 print-histogram $histogram_data $limits
1960 puts "Autocorrelation function:"
1962 set autoc [::math::statistics::autocorr $data1]
1963 puts [::math::statistics::map $autoc {[format "%.2f" $x]}]
1964 puts "Cross-correlation function:"
1965 set crossc [::math::statistics::crosscorr $data1 $data2]
1966 puts [::math::statistics::map $crossc {[format "%.2f" $x]}]
1967 ::math::statistics::plot-scale .plot1 0 100 -1 4
1969 ::math::statistics::plot-tline .plot1 $autoc "autoc"
1970 ::math::statistics::plot-tline .plot1 $crossc "crossc"
1971 .plot1 itemconfigure autoc -fill green
1972 .plot1 itemconfigure crossc -fill yellow
1973 puts "Quantiles: 0.1, 0.2, 0.5, 0.8, 0.9"
1975 puts "First: [::math::statistics::quantiles $data1 {0.1 0.2 0.5 0.8 0.9}]"
1976 puts "Second: [::math::statistics::quantiles $data2 {0.1 0.2 0.5 0.8 0.9}]"
1977 If you run this example, then the following should be clear:
1980 · There is a strong correlation between two time series, as dis‐
1982 played by the raw data and especially by the correlation func‐
1983 tions.
1984 · Both time series show a significant periodic component
1986 · The histograms are not very useful in identifying the nature of
1988 the time series - they do not show the periodic nature.
1989
1991 This document, and the package it describes, will undoubtedly contain
1992 bugs and other problems. Please report such in the category math ::
1993 statistics of the Tcllib Trackers
1994 [http://core.tcl.tk/tcllib/reportlist]. Please also report any ideas
1995 for enhancements you may have for either package and/or documentation.
1996 When proposing code changes, please provide unified diffs, i.e the out‐
1998 put of diff -u.
1999 Note further that attachments are strongly preferred over inlined
2001 patches. Attachments can be made by going to the Edit form of the
2002 ticket immediately after its creation, and then using the left-most
2003 button in the secondary navigation bar.
2004
2006 data analysis, mathematics, statistics
2007
2009 Mathematics
2010tcllib 1 math::statistics(n)