### 8Statistics

This chapter describes the statistical functions provided by the Science Collection. The basic statistical functions include functions to compute the mean, variance, and standard deviation, More advanced functions allow you to calculate absolute deviation, skewness, and kurtosis, as well as the median and arbitrary percentiles. The algorithms use recurrance relations to compute average quantities in a stable way, without large intermediate values that might overflow.

The functions described in this chapter are defined in the "statistics.rkt" file in the Science Collection and are made available using the form:

 (require (planet williams/science/statistics))

#### 8.1Running Statistics

A running statistics object accumulates a minimal set of statistics (n, min, max, mean, variance, and standard deviation) for a set of data values. A running statistics object does not require that a sequence (e.g., list or vector) of the data value be maintained.

 (statistics? x) → boolean? x : any/c
Returns #t is x is a running statistics object.

 (make-statistics) → statistics?
Returns a new, empty running statistics object.

 (statistics-reset! s) → void? s : statistics?
Resets the running statistics object s.

 (statistics-tally! s x) → void? s : statistics? x : real?
Updates the running statistice object s with the value of x.

 (statistics-n s) → exact-nonnegative-integer? s : statistics?
Returns the number of values that have been added to the running statistics object s. This value is zero initially and after a call to statistics-reset!.

 (statistics-min s) → real? s : statistics?
Returns the minimum value that has been added to the running statistics object s. This value is +inf.0 initially and after a call to statistics-reset!.

 (statistics-max s) → real? s : statistics?
Returns the maximum value that has been added to the running statistics object s. This value is -inf.0 initially and after a call to statistics-reset!.

 (statistics-mean s) → real? s : statistics?
Returns the arithmetic mean of the values that have been added to the running statistics object s. This value is zero initially and after a call to statistics-reset!.

 (statistics-variance s) → real? s : statistics?
Returns the estimated, or sample, variance of the values that have been added to the running statistics object s. This value is zero initially and after a call to statistics-reset!.

 (statistics-standard-deviation s) → real? s : statistics?
Returns the standard deviation of the values that have been added to the running statistics object s. This is the square root of the value returned by statistics-variance.

#### 8.2Running Statistics Example

This example generated 100 random numbers between 0.0 and 10.0 and maintains running statistics on the values.

 #lang racket (require (planet williams/science/statistics) (planet williams/science/random-distributions)) (define (test) (let ((stat (make-statistics))) (for ((i (in-range 100))) (statistics-tally! stat (random-flat 0.0 10.0))) (printf "Running Statistics Example~n") (printf "                 n = ~a~n" (statistics-n stat)) (printf "               min = ~a~n" (statistics-min stat)) (printf "               max = ~a~n" (statistics-max stat)) (printf "              mean = ~a~n" (statistics-mean stat)) (printf "          variance = ~a~n" (statistics-variance stat)) (printf "standard deviation = ~a~n" (statistics-standard-deviation stat)))) (test)

Produces the following results.

 Running Statistics Example n = 100 min = 0.11100957474903939 max = 9.938914540059452 mean = 5.466640451797567 variance = 8.677003172428925 standard deviation = 2.945675333846031

#### 8.3Mean, Standard Deviation, and Variance

 (mean data) → real? data : sequence-of-real? (unchecked-mean data) → real? data : sequence-of-real?
Returns the arithmetic mean of "data".

(mean-and-variance data)
 real? (>=/c 0.0)
data : sequence-of-real?
(unchecked-mean-and-variance data)
 real? (>=/c 0.0)
data : sequence-of-real?
Returns the aritnmetic mean and the estimated, or sample, variance of data as multiple values. These values are computed in a single pass through data.

 (variance data mean) → (>=/c 0.0) data : sequence-of-real? mean : real? (unchecked-variance data mean) → (>=/c 0.0) data : sequence-of-real? mean : real? (variance data) → (>=/c 0.0) data : sequence-of-real? (unchecked-variance data) → (>=/c 0.0) data : sequence-of-real?
Returns the estimated, or sample, variance of data relative to the given value of mean. If mean is not provided, the variance is relative to the arithmetic mean and is computed in a single pass through data.

 (standard-deviation data mean) → (>=/c 0.0) data : sequence-of-real? mean : real? (unchecked-standard-deviation data mean) → (>=/c 0.0) data : sequence-of-real? mean : real? (standard-deviation data) → (>=/c 0.0) data : sequence-of-real? (unchecked-standard-deviation data) → (>=/c 0.0) data : sequence-of-real?
Returns the estimated, or sample, standard deviation of datarelative to the given value of mean. If mean is not provided, the standard deviation is relative to the arithmetic mean and is computed in a single pass through data. The standard deviation is defined as the square root of the variance.

 (sum-of-squares data mean) → (>=/c 0.0) data : sequence-of-real? mean : real? (unchecked-sum-of-squares data mean) → (>=/c 0.0) data : sequence-of-real? mean : real? (sum-of-squares data) → (>=/c 0.0) data : sequence-of-real? (unchecked-sum-of-squares data) → (>=/c 0.0) data : sequence-of-real?
Returns the total sum of squates of data aout the mean. If mean is not provided, it is calculated by a call to (mean data).

(variance-with-fixed-mean data mean)  (>=/c 0.0)
data : sequence-of-real?
mean : real?
 (unchecked-variance-with-fixed-mean data mean) → (>=/c 0.0)
data : sequence-of-real?
mean : real?
Returns an unbiased estimate of the variance of data when the population mean, mean, of the underlying distribution is known a priori.

 (standard-deviation-with-fixed-mean data mean) → (>=/c 0.0)
data : sequence-of-real?
mean : real?
 (unchecked-standard-deviation-with-fixed-mean data mean)
(>=/c 0.0)
data : sequence-of-real?
mean : real?
Returns the standard deviation of data for a fixed population mean, mean. The result is the square root of the variance-with-fixed-mean function.

#### 8.4Absolute Deviation

 (absolute-deviation data mean) → (>=/c 0.0) data : sequence-of-real? mean : real? (unchecked-absolute-deviation data mean) → (>=/c 0.0) data : sequence-of-real? mean : real? (absolute-deviation data) → (>=/c 0.0) data : sequence-of-real? (unchecked-absolute-deviation data) → (>=/c 0.0) data : sequence-of-real?
Returns the absolute devistion of data relative to the given value of the mean, mean. If mean is not provided, it is calculated by a call to (mean data). This function is also useful if you want to calculate the absolute deviation to any value other than the mean, such as zero or the median.

#### 8.5Higher Moments (Skewness and Kurtosis)

 (skew data mean sd) → real? data : sequence-of-real? mean : real? sd : (>=/c 0.0) (unchecked-skew data mean sd) → real? data : sequence-of-real? mean : real? sd : (>=/c 0.0) (skew data) → real? data : sequence-of-real? (unchecked-skew data) → real? data : sequence-of-real?
Returns the skewness of data using the given values of the mean, mean, and standard deviation, sd. The skewness measures the symmetry of the tails of a distribution. If mean and sd are not provided, they are calculated by a call to mean-and-variance.

 (kurtosis data mean sd) → real? data : sequence-of-real? mean : real? sd : (>=/c 0.0) (unchecked-kurtosis data mean sd) → real? data : sequence-of-real? mean : real? sd : (>=/c 0.0) (kurtosis data) → real? data : sequence-of-real? (unchecked-kurtosis data) → real? data : sequence-of-real?
Returns the kurtosis of data using the given values of the mean, mean, and standard deviation, sd. The kurtosis measures how sharply peaked a distribution is relative to its width. If mean and sd are not provided, they are calculated by a call to mean-and-variance.

#### 8.6Autocorrelation

 (lag-1-autocorrelation data mean) → real? data : nonempty-sequence-of-real? mean : real? (unchecked-lag-1-autocorrelation data mean) → real? data : nonempty-sequence-of-real? mean : real? (lag-1-autocorrelation data) → real? data : nonempty-sequence-of-real? (unchecked-lag-1-autocorrelation data) → real? data : nonempty-sequence-of-real?
Returns the lag-1 autocorrelation of data using the given value of the mean, mean. If mean is not provided, it is calculated by a call to (mean data).

#### 8.7Covariance

(covariance data1 data2 mean1 mean2)  real?
data1 : nonempty-sequence-of-real?
data2 : nonempty-sequence-of-real?
mean1 : real?
mean2 : real?
 (unchecked-covariance data1 data2 mean1 mean2) → real?
data1 : nonempty-sequence-of-real?
data2 : nonempty-sequence-of-real?
mean1 : real?
mean2 : real?
(covariance data1 data2)  real?
data1 : nonempty-sequence-of-real?
data2 : nonempty-sequence-of-real?
(unchecked-covariance data1 data2)  real?
data1 : nonempty-sequence-of-real?
data2 : nonempty-sequence-of-real?
Returns the covariance of data1 and data2 using the given values of mean1 and mean2. If the values of mean1 and mean2 are not given, they are calculated using calls to (mean data1) and (mean data2), respectively.

#### 8.8Correlation

 (correlation data1 data2) → real? data1 : nonempty-sequence-of-real? data2 : nonempty-sequence-of-real? (unchecked-correlation data1 data2) → real? data1 : nonempty-sequence-of-real? data2 : nonempty-sequence-of-real?
Returns the Pearson correlation coefficient between data1 and data2.

#### 8.9Weighted Samples

 (weighted-mean weights data) → real? weights : sequence-of-real? data : sequence-of-real? (unchecked-weighted-mean weights data) → real? weights : sequence-of-real? data : sequence-of-real?
Returns the weighted mean of data using weights, weights.

(weighted-variance weights data wmean)  (>=/c 0.0)
weights : sequence-of-real?
data : sequence-of-real?
wmean : real?
 (unchecked-weighted-variance weights data wmean) → (>=/c 0.0)
weights : sequence-of-real?
data : sequence-of-real?
wmean : real?
(weighted-variance weights data)  (>=/c 0.0)
weights : sequence-of-real?
data : sequence-of-real?
(unchecked-weighted-variance weights data)  (>=/c 0.0)
weights : sequence-of-real?
data : sequence-of-real?
Returns the weighted variance of data using weights, weights, and the given weighted mean, wmean. If wmean is not provided, it is calculated by a call to (weighted-mean weights data).

 (weighted-standard-deviation weights data wmean) → (>=/c 0.0)
weights : sequence-of-real?
data : sequence-of-real?
wmean : real?
 (unchecked-weighted-standard-deviation weights data wmean) → (>=/c 0.0)
weights : sequence-of-real?
data : sequence-of-real?
wmean : real?
(weighted-standard-deviation weights data)  (>=/c 0.0)
weights : sequence-of-real?
data : sequence-of-real?
 (unchecked-weighted-standard-deviation weights data) → (>=/c 0.0)
weights : sequence-of-real?
data : sequence-of-real?
Returns the weighted standard deviation of data using weights, weights. The standard deviation is defined as the square root of the variance. The result is the square root of the corresponding weighted-variance function.

 (weighted-variance-with-fixed-mean weights data wmean) → (>=/c 0.0)
weights : sequence-of-real?
data : sequence-of-reals?
wmean : real?
 (unchecked-weighted-variance-with-fixed-mean weights data wmean)
(>=/c 0.0)
weights : sequence-of-real?
data : sequence-of-reals?
wmean : real?
Returns an unbiased estimate of the weighted variance of data using weights, weights, when the weighted population mean, wmean, of the underlying population is known a priori.

 (weighted-standard-deviation-with-fixed-mean weights data wmean)
(>=/c 0.0)
weights : sequence-of-real?
data : sequence-of-real?
wmean : real?
 (unchecked-weighted-standard-deviation-with-fixed-mean weights data wmean)
(>=/c 0.0)
weights : sequence-of-real?
data : sequence-of-real?
wmean : real?
Returns the weighted standard deviation of data using weights, weights, with a fixed population mean, wmean. The result is the square root of the weighted-variance-with-fixed-mean function.

 (weighted-absolute-deviation weights data wmean) → (>=/c 0.0)
weights : sequence-of-real?
data : sequence-of-real?
wmean : real?
 (unchecked-weighted-absolute-deviation weights data wmean) → (>=/c 0.0)
weights : sequence-of-real?
data : sequence-of-real?
wmean : real?
(weighted-absolute-deviation weights data)  (>=/c 0.0)
weights : sequence-of-real?
data : sequence-of-real?
 (unchecked-weighted-absolute-deviation weights data) → (>=/c 0.0)
weights : sequence-of-real?
data : sequence-of-real?
Returns the weighted absolute devistion of data using weights, weights, relative to the given value of the weighted mean, wmean. If wmean is not provided, it is calculated by a call to (weighted-mean weights data). This function is also useful if you want to calculate the weighted absolute deviation to any value other than the mean, such as zero or the weighted median.

(weighted-skew weights data wmean wsd)  (>=/c 0.0)
weights : sequence-of-real?
data : sequence-of-real?
wmean : real?
wsd : (>=/c 0.0)
 (unchecked-weighted-skew weights data wmean wsd) → (>=/c 0.0)
weights : sequence-of-real?
data : sequence-of-real?
wmean : real?
wsd : (>=/c 0.0)
(weighted-skew weights data)  (>=/c 0.0)
weights : sequence-of-real?
data : sequence-of-real?
(unchecked-weighted-skew weights data)  (>=/c 0.0)
weights : sequence-of-real?
data : sequence-of-real?
Returns the weighted skewness of data using weights, weights, using the given values of the weighted mean, wmean, and weighted standard deviation, wsd. The skewness measures the symmetry of the tails of a distribution. If wmean and wsd are not provided, they are calculated by calls to (weighted-mean weights data) and (weighted-standard-deviation weights data wmean).

(weighted-kurtosis weights data wmean wsd)  (>=/c 0.0)
weights : sequence-of-real?
data : sequence-of-real?
wmean : real?
wsd : (>=/c 0.0)
 (unchecked-weighted-kurtosis weights data wmean wsd) → (>=/c 0.0)
weights : sequence-of-real?
data : sequence-of-real?
wmean : real?
wsd : (>=/c 0.0)
(weighted-kurtosis weights data)  (>=/c 0.0)
weights : sequence-of-real?
data : sequence-of-real?
(unchecked-weighted-kurtosis weights data)  (>=/c 0.0)
weights : sequence-of-real?
data : sequence-of-real?
Returns the weighted kurtosis of data using weights, weights, using the given values of the weighted mean, wmean, and weighted standard deviation, wsd. The kurtosis measures how sharply peaked a distribution is relative to its width. If wmean and wsd are not provided, they are calculated by calls to (weighted-mean weights data) and (weighted-standard-deviation weights data wmean).

#### 8.10Maximum and Minimum

 (maximum data) → real? data : nonempty-sequence-of-real? (unchecked-maximum data) → real? data : nonempty-sequence-of-real?
Returns the maximum value in data.

 (minimum data) → real? data : nonempty-sequence-of-real? (unchecked-minimum data) → real? data : nonempty-sequence-of-real?
Returns the minimum value in data.

(minimum-maximum data)
 real? real?
data : nonempty-sequence-of-real?
(unchecked-minimum-maximum data)
 real? real?
data : nonempty-sequence-of-real?
Returns the minimum and maximum values on data as multiple values.

 (maximum-index data) → exact-nonnegative-integer? data : nonempty-sequence-of-real? (unchecked-maximum-index data) → exact-nonnegative-integer? data : nonempty-sequence-of-real?
Returns the index of the maximum value in data. When there are several equal maximum elements, the index of the first one is chosen.

 (minimum-index data) → exact-nonnegative-integer? data : nonempty-sequence-of-real? (unchecked-minimum-index data) → exact-nonnegative-integer? data : nonempty-sequence-of-real?
Returns the index of the minimum value in data. When there are several equal minimum elements, the index of the first one is chosen.

(minimum-maximum-index data)
 exact-nonnegative-ineger? exact-nonnegative-integer?
data : nonempty-sequence-of-real?
(unchecked-minimum-maximum-index data)

 exact-nonnegative-ineger? exact-nonnegative-integer?
data : nonempty-sequence-of-real?
Returns the indices of the minimum and maximum values in data as multiple values. When there are several equal minimum or maximum elements, the index of the first ones are chosen.

#### 8.11Median and Quantiles

Thw median and quantile functions described in this section operate on sorted data. The contracts for these functions enforce this. Also, for convenience we use quantiles measured on a scale of 0 to 1 instead of percentiles, which use a scale of 0 to 100).

 (median-from-sorted-data sorted-data) → real? sorted-data : nonempty-sorted-vector-of-real? (unchecked-median-from-sorted-data sorted-data) → real? sorted-data : nonempty-sorted-vector-of-real?
Returns the median value of sorted-data. When the dataset has an odd number of elements, the median is the value of element (n - 1)/2. When the dataset has an even number of elements, the median is the mean of the two nearest middle values, elements (n - 1)/2 and n/2.

(quantile-from-sorted-data sorted-data f)  real?
sorted-data : nonempty-sorted-vector-of-real?
f : (real-in 0.0 1.0)
 (unchecked-quantile-from-sorted-data sorted-data f) → real?
sorted-data : nonempty-sorted-vector-of-real?
f : (real-in 0.0 1.0)
Returns a quantile value of sorted-data. The quantile is determined by the value f, a fraction between 0 and 1. For example to compute the 75th percentile, f should have the value 0.75.

The quantile is found by interpolation using the formula:

quantile = 1 - delta(x[i]) + delta(x(i + 1))

where i is floor((n - 1) × f) and delta is (n - 1) × f - 1.

#### 8.12Statistics Example

This example generates two vectors from a unit Gaussian distribution and a vector of conse squared weighting data. All of the vectors are of length 1,000. Thes data are used to test all of the statistics functions.

 #lang racket (require (planet williams/science/random-distributions/gaussian) (planet williams/science/statistics) (planet williams/science/math)) (define (naive-sort! data) (let loop () (let ((n (vector-length data)) (sorted? #t)) (do ((i 1 (+ i 1))) ((= i n) data) (when (< (vector-ref data i) (vector-ref data (- i 1))) (let ((t (vector-ref data i))) (vector-set! data i (vector-ref data (- i 1))) (vector-set! data (- i 1) t) (set! sorted? #f)))) (unless sorted? (loop))))) (let ((data1 (make-vector 1000)) (data2 (make-vector 1000)) (w     (make-vector 1000))) (for ((i (in-range 1000))) (vector-set! data1 i (random-unit-gaussian)) (vector-set! data2 i (random-unit-gaussian)) (vector-set! w i (expt (cos (- (* 2.0 pi (/ i 1000.0)) pi)) 2))) (printf "Statistics Example~n") (printf "                                mean = ~a~n" (mean data1)) (printf "                            variance = ~a~n" (variance data1)) (printf "                  standard deviation = ~a~n" (standard-deviation data1)) (printf "                   variance from 0.0 = ~a~n" (variance-with-fixed-mean data1 0.0)) (printf "         standard deviation from 0.0 = ~a~n" (standard-deviation-with-fixed-mean data1 0.0)) (printf "                  absolute deviation = ~a~n" (absolute-deviation data1)) (printf "         absolute deviation from 0.0 = ~a~n" (absolute-deviation data1 0.0)) (printf "                                skew = ~a~n" (skew data1)) (printf "                            kurtosis = ~a~n" (kurtosis data1)) (printf "               lag-1 autocorrelation = ~a~n" (lag-1-autocorrelation data1)) (printf "                          covariance = ~a~n" (covariance data1 data2)) (printf "                       weighted mean = ~a~n" (weighted-mean w data1)) (printf "                   weighted variance = ~a~n" (weighted-variance w data1)) (printf "         weighted standard deviation = ~a~n" (weighted-standard-deviation w data1)) (printf "          weighted variance from 0.0 = ~a~n" (weighted-variance-with-fixed-mean w data1 0.0)) (printf "weighted standard deviation from 0.0 = ~a~n" (weighted-standard-deviation-with-fixed-mean w data1 0.0)) (printf "         weighted absolute deviation = ~a~n" (weighted-absolute-deviation w data1)) (printf "weighted absolute deviation from 0.0 = ~a~n" (weighted-absolute-deviation w data1 0.0)) (printf "                       weighted skew = ~a~n" (weighted-skew w data1)) (printf "                   weighted kurtosis = ~a~n" (weighted-kurtosis w data1)) (printf "                             maximum = ~a~n" (maximum data1)) (printf "                             minimum = ~a~n" (minimum data1)) (printf "              index of maximum value = ~a~n" (maximum-index data1)) (printf "              index of minimum value = ~a~n" (minimum-index data1)) (naive-sort! data1) (printf "                              median = ~a~n" (median-from-sorted-data data1)) (printf "                        10% quantile = ~a~n" (quantile-from-sorted-data data1 0.1)) (printf "                        20% quantile = ~a~n" (quantile-from-sorted-data data1 0.2)) (printf "                        30% quantile = ~a~n" (quantile-from-sorted-data data1 0.3)) (printf "                        40% quantile = ~a~n" (quantile-from-sorted-data data1 0.4)) (printf "                        50% quantile = ~a~n" (quantile-from-sorted-data data1 0.5)) (printf "                        60% quantile = ~a~n" (quantile-from-sorted-data data1 0.6)) (printf "                        70% quantile = ~a~n" (quantile-from-sorted-data data1 0.7)) (printf "                        80% quantile = ~a~n" (quantile-from-sorted-data data1 0.8)) (printf "                        90% quantile = ~a~n" (quantile-from-sorted-data data1 0.9)))

Produces the following output:

 Statistics Example mean = 0.03457693091555611 variance = 1.0285343857083435 standard deviation = 1.0141668431320083 variance from 0.0 = 1.028701415474174 standard deviation from 0.0 = 1.014249188056946 absolute deviation = 0.7987180852601665 absolute deviation from 0.0 = 0.7987898146946209 skew = 0.04340293467117837 kurtosis = 0.17722452271702993 lag-1 autocorrelation = 0.0029930889831972143 covariance = 0.005782911085590894 weighted mean = 0.05096139259270008 weighted variance = 1.0500293763787367 weighted standard deviation = 1.0247094107007786 weighted variance from 0.0 = 1.0510513958491579 weighted standard deviation from 0.0 = 1.0252079768755011 weighted absolute deviation = 0.8054378524718832 weighted absolute deviation from 0.0 = 0.8052440544958938 weighted skew = 0.046448729539282155 weighted kurtosis = 0.3050060704791675 maximum = 3.731148814104969 minimum = -3.327265864298485 index of maximum value = 502 index of minimum value = 476 median = 0.019281803306206644 10% quantile = -1.243869878615807 20% quantile = -0.7816243947573505 30% quantile = -0.4708703241429585 40% quantile = -0.2299309332835332 50% quantile = 0.019281803306206644 60% quantile = 0.30022966479982344 70% quantile = 0.5317978807508836 80% quantile = 0.832291888537874 90% quantile = 1.3061151234700463
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