## Desmos - Statistics

## Overview

The calculator provides several functions for computing statistical properties from lists of data, performing basic statistical tests, counting combinations and permutations, and working with distributions. These functions are accessible from the "Stats" and "Dist" sections of the "functions" menu in the keypad, or can be typed directly into the expressions list using a keyboard.

### General statistical functions

**total( list) **or

**total(**

*a*,*b*,*c*, ...)Output the sum of a list of numbers.

**length( list) **or

**length(**,

*a*,*b**,*

**c****...)**

Output the length of a list of numbers

**mean( list) **or

**mean(**or

*a*,*b*,*c*, ...)**mean(**

*distribution*)Output the mean of a list of numbers. This function will also return the mean of a distribution, if it exists. See the section on distributions below.

**median( list) **or

**median(**or

*a*,*b*,*c*, ...)**median(**

*distribution*)Output the median of a list of numbers. This function will also return the median of a distribution, if it exists. See the section on distributions below.

**min( list) **or

**min(**

*a*,*b*,*c*, ...)Output the minimum value contained in a list of numbers.

**max( list) **or

**max(**

*a*,*b*,*c*, ...)Output the maximum value contained in a list of numbers.

**quartile( list, q)**

Output the *q*th quartile of *list*. *q* must be a number between 0 and 4 (inclusive), otherwise the result will be undefined. Note that this function uses the Moore and McCabe method, which discards the median in odd-length data sets before computing the upper and lower quartiles.

**quantile( list, q) **or

**quantile(**

*distribution*,*q*)Output the *q*th quantile of *list*. *q* must be a number between 0 and 1 (inclusive), otherwise the result will be undefined. Passing a distribution as the first argument to **quantile** allows you to evaluate its inverse CDF. See the section on distributions below.

**stdev( list) **or

**stdev(**or

*a*,*b*,*c*, ...)**stdev(**

*distribution*)Output the *sample* standard deviation of a list of numbers. This function will also return the standard deviation of a distribution, if it exists. See the section on distributions below.

**stdevp( list) **or

**stdevp(**

*a*,*b*,*c*, ...)Output the *population* standard deviation of a list of numbers.

**mad( list) **or

**mad(**

*a*,*b*,*c*, ...)Output the* *mean absolute deviation of a list of numbers.

**var( list) **or

**var(**or

*a*,*b*,*c*, ...)**var(**

*distribution*)Output the* sample* variance of a list of numbers. This function will also return the variance for a distribution, if it exists. See the section on distributions below. Note that, while not on the keypad, a **varp** function is also available to compute the *population* variance.

**cov( list1, list2)**

Output the covariance between two lists of numbers.

**corr( list1, list2)**

Output the Pearson correlation coefficient between two lists of numbers.

**nCr( n, r)**

Output the number of *r*-sized combinations (unordered arrangements) that can be selected from a set of size *n*.

**nPr( n, r)**

Output the number of *r*-sized permutations (ordered arrangements) that can be selected from a set of size *n*.

**n****!**

Output the factorial of *n*.

### Statistical tests

**ttest( list, value = 0)**

Perform a one-sample *t*-test of whether the mean of the population from which *list* is sampled differs from *value* (the null hypothesis). The output includes *p*-values for both the one-tailed versions (labeled "less than" and "greater than") and the two-tailed version (labeled "not equal") of the test. Note that if the second argument is omitted the hypothesized mean defaults to 0.

**tscore( list, value = 0)**

Output the raw test statistic used in the one-sample **ttest** function.

**ittest( list1, list2)**

Perform an independent (unpaired) two-sample t-test of whether the mean of the population from which *list1 *is sampled differs from the mean of the population from which *list2* is sampled. The output includes *p*-values for both the one-tailed versions (labeled "less than" and "greater than") and the two-tailed version (labeled "not equal") of the test. Note that, while the sample sizes may differ (*list1* and *list2* need not have equal length), this test *does* assume that the underlying populations have equal variance.

# Distributions

The calculator can plot the probability density functions (PDFs), probability mass functions (PMFs), and cumulative distribution functions (CDFs) of several common statistical distributions, as well as compute cumulative probabilities for those distributions.

## Plotting

Each of the following functions will plot a distribution's PDF or PMF.

**normaldist( mean = 0, standard deviation = 1)**

Plot the PDF of a normal distribution with the given *mean* and *standard deviation*. Note that if the second argument is omitted the *standard deviation* defaults to 1, and if both arguments are omitted the *mean* also defaults to 0.

**tdist( degrees of freedom)**

Plot the PDF of a Student's *t*-distribution with the given *degrees of freedom*. Note that *degrees of freedom* must be greater than 0.

**poissondist( mean)**

Plot the PMF of a Poisson distribution with the given *mean*. Note that *mean* must be greater than 0.

**binomialdist( trials, probability = 0.5)**

Plot the PMF of a binomial distribution given a number of (independent) *trials* and a *probability* of success on each trial. Note that *trials* must be a nonnegative integer and *probability* must be a number between 0 and 1 (inclusive).

## Computing cumulative probabilities

When using any of the above functions to plot a PDF/PMF, a checkbox labeled "Find Cumulative Probability (CDF)" will appear. If that box is checked, the calculator will output the cumulative probability between the values in the "Min" and "Max" input fields. It will also display a visualization of the cumulative probability, either as a shaded region under the curve (for continuous distributions) or as a series of vertical segments and points (for discrete distributions).

Normal Distribution:

Binomial Distribution:

## Other functions for use with distributions

The top-level distribution functions offer a simple way to plot PDFs and PMFs and compute cumulative probabilities, but the calculator also provides some functions for working with distribution PDFs/PMFs and CDFs inside of other expressions. Once you have created a distribution, you are able to access its **.pdf()** and **.cdf() **functions.

Note that for discrete distributions there is a difference between what the calculator will plot for the top-level distribution function and what it will plot for the **.pdf()** function. When using the **.pdf()** and **.cdf()** functions, a discrete PMF or CDF will be plotted as a step function rather than as a series of points.

**distribution****.pdf( value)**

Evaluate *distribution*'s PDF/PMF at the given *value*. If *value* is numeric, the calculator will output a numeric evaluation. If *value* is an expression that depends on a free variable, the calculator will plot the PDF/PMF as a function of *value*.

**distribution****.cdf( value)**

Evaluate *distribution*'s CDF at the given *value*. If *value* is numeric, the calculator will output a numeric evaluation. If *value* is an expression that depends on a free variable, the calculator will plot the CDF as a function of *value*. For example, **normaldist(0,1).cdf(2) **will output the probability that a random variable from a standard normal distribution has a value less than or equal to 2.

**distribution****.cdf( lower, upper)**

Compute *distribution*'s cumulative probability between *lower* and *upper*. For example, **normaldist(0,1).cdf(-1, 1)** will output the probability that a random variable from a standard normal distribution has a value between -1 and 1.

Note that for discrete distributions **d****.pdf( x)** will round

*x*to the nearest integer, and a plot of

**d****.pdf(**will look like a piecewise-constant function. To plot a set of points instead, you could use a table or a point list:

*x*)

**R****=[0â€¦10]**,

**(**.

*R*,*d*.pdf(*R*))The **.pdf()** and **.cdf()** functions let you combine distributions in interesting ways. For example, by plotting the difference between their PDFs, it's possible to see that a *t*-distribution approaches a standard normal distribution as its number of degrees of freedom increases: