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## Statistics

Still under testing! (But the documented procedures below should all work.)

This library is a port of Larry Hunter's Lisp statistics library to chicken scheme.

The library provides a number of formulae and methods taken from the book "Fundamentals of Biostatistics" by Bernard Rosner (5th edition).

#### Statistical Distributions

To use this library, you need to understand the underlying statistics. In brief:

The Binomial distribution is used when counting discrete events in a series of trials, each of which events has a probability p of producing a positive outcome. An example would be tossing a coin n times: the probability of a head is p, and the distribution gives the expected number of heads in the n trials. The binomial distribution is defined as B(n, p).

The Poisson distribution is used to count discrete events which occur with a known average rate. A typical example is the decay of radioactive elements. A poisson distribution is defined Pois(mu).

The Normal distribution is used for real-valued events which cluster around a specific mean with a symmetric variance. A typical example would be the distribution of people's heights. A normal distribution is defined N(mean, variance).

### Provided Functions

#### Utilities

[procedure] (average-rank value sorted-values)

returns the average position of given value in the list of sorted values: the rank is based from 1.

```> (average-rank 2 '(1 2 2 3 4))
5/2```
[procedure] (beta-incomplete x a b)
[procedure] (bin-and-count items n)

Divides the range of the list of items into n bins, and returns a vector of the number of items which fall into each bin.

```> (bin-and-count '(1 1 2 3 3 4 5) 5)
#(2 1 2 1 1)```
[procedure] (combinations n k)

returns the number of ways to select k items from n, where the order does not matter.

[procedure] (factorial n)

returns the factorial of n.

[procedure] (find-critical-value p-function p-value)
[procedure] (fisher-z-transform r)

returns the transformation of a correlation coefficient r into an approximately normal distribution.

[procedure] (gamma-incomplete a x)
[procedure] (gamma-ln x)
[procedure] (permutations n k)

returns the number of ways to select k items from n, where the order does matter.

[procedure] (random-normal mean sd)

returns a random number distributed with specified mean and standard deviation.

[procedure] (random-pick items)

returns a random item from the given list of items.

[procedure] (random-sample n items)

returns a random sample from the list of items without replacement of size n.

[procedure] (sign n)

returns 0, 1 or -1 according to if n is zero, positive or negative.

[procedure] (square n)

#### Descriptive statistics

These functions provide information on a given list of numbers, the items. Note, the list does not have to be sorted.

[procedure] (mean items)

returns the arithmetic mean of the items (the sum of the numbers divided by the number of numbers).

`(mean '(1 2 3 4 5)) => 3`
[procedure] (median items)

returns the value which separates the upper and lower halves of the list of numbers.

`(median '(1 2 3 4)) => 5/2`
[procedure] (mode items)

returns two values. The first is a list of the modes and the second is the frequency. (A mode of a list of numbers is the most frequently occurring value.)

```> (mode '(1 2 3 4))
(1 2 3 4)
1
> (mode '(1 2 2 3 4))
(2)
2
> (mode '(1 2 2 3 3 4))
(2 3)
2```
[procedure] (geometric-mean items)

returns the geometric mean of the items (the result of multiplying the items together and then taking the nth root, where n is the number of items).

`(geometric-mean '(1 2 3 4 5)) => 2.60517108469735`
[procedure] (range items)

returns the difference between the biggest and the smallest value from the list of items.

`(range '(5 1 2 3 4)) => 4`
[procedure] (percentile items percent)

returns the item closest to the percent value if the items are sorted into order; the returned item may be in the list, or the average of adjacent values.

```(percentile '(1 2 3 4) 50) => 5/2
(percentile '(1 2 3 4) 67) => 3```
[procedure] (variance items)
[procedure] (standard-deviation items)
[procedure] (coefficient-of-variation items)

returns 100 * (std-dev / mean) of the items.

`(coefficient-of-variation '(1 2 3 4)) => 51.6397779494322`
[procedure] (standard-error-of-the-mean items)

returns std-dev / sqrt(length items).

` (standard-error-of-the-mean '(1 2 3 4)) => 0.645497224367903`
[procedure] (mean-sd-n items)

returns three values, one for the mean, one for the standard deviation, and one for the length of the list.

```> (mean-sd-n '(1 2 3 4))
5/2
1.29099444873581
4```

#### Distributional functions

[procedure] (binomial-probability n k p)

returns the probability that the number of positive outcomes for a binomial distribution B(n, p) is k.

```> (do-ec (: i 0 11)
(format #t "i = ~d P = ~f~&" i (binomial-probability 10 i 0.5)))
i = 0 P = 0.0009765625
i = 1 P = 0.009765625
i = 2 P = 0.0439453125
i = 3 P = 0.1171875
i = 4 P = 0.205078125
i = 5 P = 0.24609375
i = 6 P = 0.205078125
i = 7 P = 0.1171875
i = 8 P = 0.0439453125
i = 9 P = 0.009765625
i = 10 P = 0.0009765625```
[procedure] (binomial-cumulative-probability n k p)

returns the probability that less than k positive outcomes occur for a binomial distribution B(n, p).

```> (do-ec (: i 0 11)
(format #t "i = ~d P = ~f~&" i (binomial-cumulative-probability 10 i 0.5)))
i = 0 P = 0.0
i = 1 P = 0.0009765625
i = 2 P = 0.0107421875
i = 3 P = 0.0546875
i = 4 P = 0.171875
i = 5 P = 0.376953125
i = 6 P = 0.623046875
i = 7 P = 0.828125
i = 8 P = 0.9453125
i = 9 P = 0.9892578125
i = 10 P = 0.9990234375```
[procedure] (binomial-ge-probability n k p)

returns the probability of k or more positive outcomes for a binomial distribution B(n, p).

[procedure] (binomial-le-probability n k p)

returns the probability k or fewer positive outcomes for a binomial distribution B(n, p).

[procedure] (poisson-probability mu k)

returns the probability of k events occurring when the average is mu.

```> (do-ec (: i 0 20)
(format #t "P(X=~2d) = ~,4f~&" i (poisson-probability 10 i)))
P(X= 0) = 0.0000
P(X= 1) = 0.0005
P(X= 2) = 0.0023
P(X= 3) = 0.0076
P(X= 4) = 0.0189
P(X= 5) = 0.0378
P(X= 6) = 0.0631
P(X= 7) = 0.0901
P(X= 8) = 0.1126
P(X= 9) = 0.1251
P(X=10) = 0.1251
P(X=11) = 0.1137
P(X=12) = 0.0948
P(X=13) = 0.0729
P(X=14) = 0.0521
P(X=15) = 0.0347
P(X=16) = 0.0217
P(X=17) = 0.0128
P(X=18) = 0.0071
P(X=19) = 0.0037```
[procedure] (poisson-cumulative-probability mu k)

returns the probability of less than k events occurring when the average is mu.

```> (do-ec (: i 0 20)
(format #t "P(X=~2d) = ~,4f~&" i (poisson-cumulative-probability 10 i)))
P(X= 0) = 0.0000
P(X= 1) = 0.0000
P(X= 2) = 0.0005
P(X= 3) = 0.0028
P(X= 4) = 0.0103
P(X= 5) = 0.0293
P(X= 6) = 0.0671
P(X= 7) = 0.1301
P(X= 8) = 0.2202
P(X= 9) = 0.3328
P(X=10) = 0.4579
P(X=11) = 0.5830
P(X=12) = 0.6968
P(X=13) = 0.7916
P(X=14) = 0.8645
P(X=15) = 0.9165
P(X=16) = 0.9513
P(X=17) = 0.9730
P(X=18) = 0.9857
P(X=19) = 0.9928```
[procedure] (poisson-ge-probability mu k)

returns the probability of k or more events occurring when the average is mu.

[procedure] (normal-pdf x mean variance)

returns the likelihood of x given a normal distribution with stated mean and variance.

```> (do-ec (: i 0 11)
(format #t "~3d ~,4f~&" i (normal-pdf i 5 4)))
0 0.0088
1 0.0270
2 0.0648
3 0.1210
4 0.1760
5 0.1995
6 0.1760
7 0.1210
8 0.0648
9 0.0270
10 0.0088```
[procedure] (convert-to-standard-normal x mean variance)

returns a value for x rescaling the given normal distribution to a standard N(0, 1).

```> (convert-to-standard-normal 5 6 2)
-1/2```
[procedure] (phi x)

returns the cumulative distribution function (CDF) of the standard normal distribution.

```> (do-ec (: x -2 2 0.4)
(format #t "~4,1f ~,4f~&" x (phi x)))
-2.0 0.0228
-1.6 0.0548
-1.2 0.1151
-0.8 0.2119
-0.4 0.3446
0.0 0.5000
0.4 0.6554
0.8 0.7881
1.2 0.8849
1.6 0.9452```
[procedure] (z percentile)

returns the inverse of the standard normal distribution. Input is a percentile, between 0.0 and 1.0.

[procedure] ( t-distribution degrees-of-freedom percentile)

returns the point in the t-distribution given the degrees-of-freedom and percentile. degrees-of-freedom must be a positive integer, and percentile a value between 0.0 and 1.0.

[procedure] (chi-square degrees-of-freedom percentile)

returns the point at which chi-square distribution has percentile to its left, using given degrees-of-freedom.

[procedure] (chi-square-cdf x degrees-of-freedom)

returns the probability that a random variable is to the left of x using the chi-square distribution with given degrees-of-freedom.

#### Confidence intervals

These functions report bounds for an observed property of a distribution: the bounds are tighter as the confidence level, alpha, varies from 0.0 to 1.0.

[procedure] (binomial-probability-ci n p alpha)

returns two values, the upper and lower bounds on an observed probability p from n trials with confidence (1-alpha).

```> (binomial-probability-ci 10 0.8 0.9)
0.724273681640625
0.851547241210938
; 2 values```
[procedure] (poisson-mu-ci k alpha)

returns two values, the upper and lower bounds on the poisson parameter if k events are observed; the bound is for confidence (1-alpha).

```> (poisson-mu-ci 10 0.9)
8.305419921875
10.0635986328125
; 2 values```
[procedure] (normal-mean-ci mean standard-deviation k alpha)

returns two values, the upper and lower bounds on the mean of the normal distibution of k events are observed; the bound is for confidence (1-alpha).

```> (normal-mean-ci 0.5 0.1 10 0.8)
0.472063716520217
0.527936283479783
; 2 values```
[procedure] (normal-mean-ci-on-sequence items alpha)

returns two values, the upper and lower bounds on the mean of the given items, assuming they are normally distributed; the bound is for confidence (1-alpha).

```> (normal-mean-ci-on-sequence '(1 2 3 4 5) 0.9)
2.40860081649174
3.59139918350826
; 2 values```
[procedure] (normal-variance-ci standard-deviation k alpha)

returns two values, the upper and lower bounds on the variance of the normal distibution of k events are observed; the bound is for confidence (1-alpha).

[procedure] (normal-variance-ci-on-sequence items alpha)

returns two values, the upper and lower bounds on the variance of the given items, assuming they are normally distributed; the bound is for confidence (1-alpha).

[procedure] normal-sd-ci standard-deviation k alpha)

returns two values, the upper and lower bounds on the standard deviation of the normal distibution of k events are observed; the bound is for confidence (1-alpha).

[procedure] (normal-sd-ci-on-sequence sequence items)

returns two values, the upper and lower bounds on the standard deviation of the given items, assuming they are normally distributed; the bound is for confidence (1-alpha).

#### Hypothesis testing

##### (parametric)
• z-test
• z-test-on-sequence
• t-test-one-sample
• t-test-one-sample-on-sequence
• t-test-paired
• t-test-paired-on-sequences
• t-test-two-sample
• t-test-two-sample-on-sequences
• f-test
• chi-square-test-one-sample
• binomial-test-one-sample
• binomial-test-two-sample
• fisher-exact-test
• mcnemars-test
• poisson-test-one-sample
##### (non parametric)
• sign-test
• sign-test-on-sequence
• wilcoxon-signed-rank-test
• wilcoxon-signed-rank-test-on-sequences
• chi-square-test-rxc
• chi-square-test-for-trend

#### Sample size estimates

• t-test-one-sample-sse
• t-test-two-sample-sse
• t-test-paired-sse
• binomial-test-one-sample-sse
• binomial-test-two-sample-sse
• binomial-test-paired-sse
• correlation-sse

#### Correlation and regression

• linear-regression
• correlation-coefficient
• correlation-test-two-sample
• correlation-test-two-sample-on-sequences
• spearman-rank-correlation

#### Significance test functions

• t-significance
• f-significance

### Authors

Peter Lane wrote the scheme version of this library. The original Lisp version was written by Larry Hunter.