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# Tally­Cowan

2017-08-02 04:10:07
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## Introduction

A tally is a member of a disjoint type whose purpose is to keep running descriptive statistics on observations, consisting of one or two Scheme numbers, that are injected into it. It can compute a variety of simple statistics such as the arithmetic mean, the variance, and the standard deviation. It does not record the observations themselves, and therefore cannot compute the median or the mode. It is also possible to inject a secondary observation at the same time as the primary one; this does not affect the basic statistics, but is used for covariance and correlation computations.

A histogram is a type of tally which also maintains a number of bins decided in advance, and counts how many observations fall into each bin. It can return bin counts and indicate the current median and modal bins. The bins are of equal width.

## Constructors

(make-tally [ key ])

Returns a newly allocated tally with no observations. The key is an arbitrary Scheme object useful for distinguishing this tally from other tallies; the default key is #f.

(make-histogram low high width [ key ])

Returns a newly allocated histogram with no observations. It will have sufficient bins to count observations from low (inclusive) to high (exclusive) with the specified width. Observations that fall on a bin boundary are always placed in the higher bin. There are two additional bins for observations lower than low and higher than high. The key is as for make-tally.

## Injectors

(tally! tally value)

Injects value into tally and updates all descriptive statistics.

(tally-weighted! tally value weight)

Injects value into tally, assigning it a weight of weight for the purpose of calculating weighted sums and weighted averages, and updates all descriptive statistics.

(tally-correlated! tally value1 value2)

Injects value1 into tally, tracking value2 separately for the purpose of calculating covariance and correlation.

(tally-weighted-correlated! tally value1 weight value2)

Injects value1 into tally, assigning it a weight of weight and tracking value2 separately for the purpose of calculating covariance and correlation.

## Generators

(generator->tally! tally generator)

Injects the values returned by generator into tally and updates all descriptive statistics.

(weighted-generators->tally! tally generatorv generatorw)

Injects the values returned by generatorv into tally, assigning each a weight returned from generatorw for the purpose of calculating weighted sums and weighted averages, and updates all descriptive statistics. Tallying stops when either generator is exhausted, at which point the other generator is in an undefined state.

(correlated-generators->tally! tally generatorv1 generatorv2)

Injects the values returned by generatorv1 into tally, tracking the alues returned by generatorv2 separately for the purpose of calculating covariance and correlation. Tallying stops when either generator is exhausted, at which point the other generator is in an undefined state.

(weighted-correlated-generators->tally! tally generatorv1 generatorw generatorv2)

Injects the values returned by generatorv1 into tally, weighted by the values returned by generatorw and tracking the alues returned by generatorv2 separately for the purpose of calculating covariance and correlation. Tallying stops when any generator is exhausted, at which point the other generators are in an undefined state.

## Basic statistics

(tally-key tally)

Returns the key of tally.

(tally-count tally)

Returns the number of observations injected so far as an exact integer.

(tally-sum tally)

Returns the sum of all observations injected so far.

(tally-sum-squares tally)

Returns the sum of the squares of all observations injected so far.

(tally-product tally)

Returns the product of all observations injected so far.

(tally-harmonic-sum tally)

Returns the harmonic sum of all observations injected so far. This is the sum of the reciprocals of the observation values. If any observation value is zero, returns +nan.0.

(tally-mean tally)

Returns the arithmetic mean of all observations injected so far. This is the quotient of the sum and the count, or +nan.0 if the count is zero.

(tally-geometric-mean tally)

Returns the geometric mean of all observations injected so far. This is the nth root of the product, where n is the count of observations.

(tally-harmonic-mean tally)

Returns the harmonic mean of all observations injected so far. This is the count divided by the harmonic sum, or +nan.0 if the count is zero.

(tally-max tally)

Returns the maximum observation injected so far. If no observations have been injected, returns -inf.0.

(tally-min tally)

Returns the minimum observation injected so far. If no observations have been injected, returns +inf.0.

(tally-weighted-sum tally)

Returns the weighted sum (that is, the sum of observations multiplied by their weights, treating a missing weight as 1) of the observations and weights injected so far.

(tally-weighted-mean tally)

Returns the weighted arithmetic mean (that is, the arithmetic mean) of observations multiplied by their weights, treating a missing weight as 1) of the observations and weights injected so far. If no observations have been injected, returns +nan.0.

## Range and variance

(tally-range tally)

Returns the difference between the maximum and the minimum. If no observations have been injected, returns +nan.0.

(tally-sample-variance tally)

Returns the sample variance of the observations injected so far. If no observations have been injected, returns +nan.0.

(tally-population-variance tally)

Returns the population variance of the observations injected so far. If no observations have been injected, returns +nan.0.

(tally-sample-standard-deviation tally)

Returns the sample standard deviation of the observations injected so far. If no observations have been injected, returns +nan.0.

(tally-population-standard-deviation tally)

Returns the population standard deviation of the observations injected so far. If no observations have been injected, returns +nan.0.

## Covariance and correlation

(tally-sample-covariance tally)

Returns the sample covariance of the pairs of observations injected so far. If no observations have been injected, returns +nan.0.

(tally-population-covariance tally)

Returns the population covariance of the pairs of observations injected so far. If no observations have been injected, returns +nan.0.

(tally-correlation tally)

Returns the correlation based on the pairs of observations injected so far.

## Histogram values

Note: A tally that is not a histogram looks to these functions like a histogram whose low, high, and width values are all zero.

(histogram-low histogram)

(histogram-high histogram)

(histogram-width histogram)

Return the basic parameters of histogram.

(histogram-bin-count histogram)

Return the number of bins in histogram.

(histogram-bins histogram)

Return a vector whose elements are the counts in the bins in histogram. It is an error to modify this vector.

(histogram-bin-ref histogram index)

Return the number of observations in the histogram bin numbered index. It is an error if index is less than zero, or greater than or equal to the number of bins.

(histogram-underflow histogram)

Return the number of observations less than the low parameter of histogram.

(histogram-overflow histogram)

Return the number of observations greater than or equal to the high parameter of histogram.

(histogram-modal-bins histogram)

Return a list of the indexes to the bins that contain the maximum number of observations. The underflow and overflow bins are excluded. Note that the modal bin does not necessarily contain the modal value; if the width of a bin is 1.0 and the values are {1, 1.1, 1.2, 1.3, 1.4, 1.5, 2.0, 2.0, 2.0}, then the modal bin is 1.0-2.0 whereas the modal value, 2.0, is in the non-modal bin 2.0-3.0.

(histogram-median-interval histogram)

Return two values, a lower (inclusive) and upper (exclusive) bound for the median observation. Normally the upper and lower bounds are one bin width apart, but if the number of items in bins lower than k for some k is equal to the number of items in higher bins, the values will be two bin widths apart. The underflow and overflow bins are excluded.