Functions for descriptive statistics: what changed

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5	2
6	3		Descriptive statistics aim at characterising empirical data by summative parameters (and also by tables and plots).
7	4
8	5		## Standard functions defined in math unit
9	6
10	7		The unit [`math`](https://www.freepascal.org/docs-html/rtl/math/statisticalroutines.html) of the [[RTL\|run-time library]] provides a plethora of routines for descriptive statistics.
11	8
12	9		- [`mean`](https://www.freepascal.org/docs-html/rtl/math/mean.html): Returns the mean value of an array.
13	10		- [`stdDev`](https://www.freepascal.org/docs-html/rtl/math/stddev.html): Returns the (sample) standard deviation of an array.
14	11		- [`popNStdDev`](https://www.freepascal.org/docs-html/rtl/math/popnstddev.html): Returns the (population) standard deviation of an array.
15	12		- [`meanAndStdDev`](https://www.freepascal.org/docs-html/rtl/math/meanandstddev.html): Returns mean and standard deviation of an array.
16	13		- [`momentSkewKurtosis`](https://www.freepascal.org/docs-html/rtl/math/momentskewkurtosis.html): Returns the first four moments of an array.
17	14		- [`variance`](https://www.freepascal.org/docs-html/rtl/math/variance.html): Returns the (sample) variance of an array.
18	15		- [`popnVariance`](https://www.freepascal.org/docs-html/rtl/math/popnvariance.html): Returns the (population) variance of an array.
19	16		- [`totalVariance`](https://www.freepascal.org/docs-html/rtl/math/totalvariance.html): Returns the total variance of an array.
20	17		- [`sumOfSquares`](https://www.freepascal.org/docs-html/rtl/math/sumofsquares.html): Returns the sum of squares of an array.
21	18		- [`sum`](https://www.freepascal.org/docs-html/rtl/math/sum.html): Returns the sum of values of an array.
22	19		- [`sumsAndSquares`](https://www.freepascal.org/docs-html/rtl/math/sumsandsquares.html): Returns sum and sum of squares of the values in an array.
23	20		- [`MinValue`](https://www.freepascal.org/docs-html/rtl/math/minvalue.html): Returns the smallest of the values in an array.
24	21		- [`MaxValue`](https://www.freepascal.org/docs-html/rtl/math/maxvalue.html): Returns the largest of the values in an array.
25	22
26	23		These functions expect an array of predefined length (e.g. `array[1..100] of float`) or a 0-based open array (e.g. `array of extended`) with subsequent call of the [`setLength`](https://www.freepascal.org/docs-html/rtl/system/setlength.html) procedure.
27	24
28	25		## Standard functions defined in other units
29	26
30	27		- [`length`](https://www.freepascal.org/docs-html/rtl/system/length.html): Delivers the length (n) of an array.
31	28
32	29		## Custom functions
33	30
34	31		Some functions aren't defined in the RTL. The subsequent section lists source code of some commonly used measures for centrality and dispersion. Where not otherwise specified the code is provided with a BSD license.
35	32
36	33		Together with other useful statistical code expanded and thoroughly tested versions of these functions are also available in the [QUANTUM SALIS](http://quantum-salis.sf.net/) project.
37	34
38	35		### Median
39	36
40	37		The term median denotes the 50% quantile of a sample, i.e. the value separating the higher half of a data vector from its lower half. It can be calculated with:
41	38
42	39		``` pascal
43	40		type
44	41		TExtArray = array of Extended;
45	42
46	43		// Based on Shell Sort - avoiding recursion allows for sorting
47	44		// of very large arrays, too
48	45		function sortExtArray(const data: TExtArray): TExtArray;
49	46		var
50	47		data2: TExtArray;
51	48		arrayLength, i, j, k: longint;
52	49		h: extended;
53	50		begin
54	51		arrayLength := high(data);
55	52		data2 := copy(data, 0, arrayLength + 1);
56	53		k := arrayLength div 2;
57	54		while k > 0 do
58	55		begin
59	56		for i := 0 to arrayLength - k do
60	57		begin
61	58		j := i;
62	59		while (j >= 0) and (data2[j] > data2[j + k]) do
63	60		begin
64	61		h := data2[j];
65	62		data2[j] := data2[j + k];
66	63		data2[j + k] := h;
67	64		if j > k then
68	65		dec(j, k)
69	66		else
70	67		j := 0;
71	68		end;
72	69		end;
73	70		k := k div 2
74	71		end;
75	72		result := data2;
76	73		end;
77	74
78	75		function median(const data: TExtArray): extended;
79	76		var
80	77		centralElement: integer;
81	78		sortedData: TExtArray;
82	79		begin
83	80		sortedData := sortExtArray(data);
84	81		centralElement := length(sortedData) div 2;
85	82		if odd(length(sortedData)) then
86	83		result := sortedData[centralElement]
87	84		else
88	85		result := (sortedData[centralElement - 1] + sortedData[centralElement]) / 2;
89	86		end;
90	87		```
91	88
92	89		Of course, the function `sortExtArray` may be replaced with another sorting algorithm, e.g. QuickSort. The Shell Sort algorithm presented here has the advantage that it is able to sort very large vectors even on machines with a very small amount of memory (albeit with the expense of slightly reduced speed compared to QuickSort).
93	90
94	91		### Standard error of the mean
95	92
96	93		The standard error of the mean (SEM) is a measure that estimates how precisely the true mean of the population can be known.
97	94
98	95		Calculation of SEM is simple:
99	96
100	97		``` pascal
101	98		function sem(const data: array of Extended): extended;
102	99		begin
103	100		sem := stddev(data) / sqrt(length(data));
104	101		end;
105	102		```
106	103
107	104		### Coefficient of variation
108	105
109	106		The coefficient of variation (CoV or CV), also known as relative standard deviation (RSD), is a measure of dispersion that is standardised with respect to the data's mean.
110	107
111	108		It can be calculated with:
112	109
113	110		``` pascal
114	111		{**
115	112		calculates the coefficient of variation (CV or CoV) of a vector of extended
116	113		}
117	114		function cv(const data: array of Extended): extended;
118	115		begin
119	116		result := stddev(data) / mean(data);
120	117		end;
121	118		```
122	119
123	120		---
124	121		From the Free Pascal wiki, page "Functions for descriptive statistics", as it stood on 2020-11-04 after a change by Jwdietrich, out of the wiki's dump of 2024-05-26. Copyright stays with the author who wrote it, according to the old wiki's [[Licensing]] page. If you have concerns about this material, [message the admin](/wiki-contact).

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