#### Up to date source on GitHub

## Introduction

While working on another project, I found myself needing to calculate basic statistics on various sets of data of various underlying types. LINQ has `Count`

, `Min`

, `Max`

, and `Average`

, but no other statistical aggregates. As I always do in a case like this, I started with Google, figuring someone else must have written some handy extension methods for this already. There are plenty of statistical and numerical processing packages out there, but what I want is a simple and lightweight implementation for the basic stats: variance (sample and population), standard deviation (sample and population), covariance, Pearson (chi squared), range, median, and mode.

## Background

I've modeled the API on the various overloads of `Enumerable.Average`

, so you are able to use these methods on the same types of collections that those methods accept. Hopefully, this will make the usage familiar and easy to use.

That means overloads for collections of the common numerical data types and their `Nullable`

counter parts, as well as convenient selector overloads.

public static decimal? StandardDeviation(this IEnumerable<decimal?> source);
public static decimal StandardDeviation(this IEnumerable<decimal> source);
public static double? StandardDeviation(this IEnumerable<double?> source);
public static double StandardDeviation(this IEnumerable<double> source);
public static float? StandardDeviation(this IEnumerable<float?> source);
public static float StandardDeviation(this IEnumerable<float> source);
public static double? StandardDeviation(this IEnumerable<int?> source);
public static double StandardDeviation(this IEnumerable<int> source);
public static double? StandardDeviation(this IEnumerable<long?> source);
public static double StandardDeviation(this IEnumerable<long> source);
public static decimal? StandardDeviation<TSource>
(this IEnumerable<TSource> source, Func<TSource, decimal?> selector);
public static decimal StandardDeviation<TSource>
(this IEnumerable<TSource> source, Func<TSource, decimal> selector);
public static double? StandardDeviation<TSource>
(this IEnumerable<TSource> source, Func<TSource, double?> selector);
public static double StandardDeviation<TSource>
(this IEnumerable<TSource> source, Func<TSource, double> selector);
public static float? StandardDeviation<TSource>
(this IEnumerable<TSource> source, Func<TSource, float?> selector);
public static float StandardDeviation<TSource>
(this IEnumerable<TSource> source, Func<TSource, float> selector);
public static double? StandardDeviation<TSource>
(this IEnumerable<TSource> source, Func<TSource, int?> selector);
public static double StandardDeviation<TSource>
(this IEnumerable<TSource> source, Func<TSource, int> selector);
public static double? StandardDeviation<TSource>
(this IEnumerable<TSource> source, Func<TSource, long?> selector);
public static double StandardDeviation<TSource>
(this IEnumerable<TSource> source, Func<TSource, long> selector);

All of the overloads that take a collection of `Nullable`

types only include actual values in the calculated result. For example:

public static double? StandardDeviation(this IEnumerable<double?> source)
{
IEnumerable<double> values = source.Coalesce();
if (values.Any())
return values.StandardDeviation();
return null;
}

where the Coalesce method is:

public static IEnumerable<T> Coalesce<T>(this IEnumerable<T?> source) where T : struct
{
Debug.Assert(source != null);
return source.Where(x => x.HasValue).Select(x => (T)x);
}

### A Note About Mode

Since a distribution of values may not have a mode, all of the `Mode`

methods return a `Nullable`

type. For instance, in the series `{ 1, 2, 3, 4 }`

, no single value appears more than once. In cases such as this, the return value will be `null`

.

In the case where there are multiple modes, `Mode`

returns the maximum mode (i.e., the value that appears the most times). If there is a tie for the maximum mode, it returns the smallest value in the set of maximum modes.

There are also two methods for calculating all modes in a series. These return an `IEnumerable`

of all of the modes in descending order of modality.

## The Statistics Calculations

Links, descriptions, and mathematical images from Wikipedia.

### Variance

Variance is the measure of the amount of variation of all the scores for a variable (not just the extremes which give the range).

Sample variance is typically denoted by the lower case sigma squared: **σ**^{2}.

public static double Variance(this IEnumerable<double> source)
{
int n = 0;
double mean = 0;
double M2 = 0;
foreach (double x in source)
{
n = n + 1;
double delta = x - mean;
mean = mean + delta / n;
M2 += delta * (x - mean);
}
return M2 / (n - 1);
}

### Standard Deviation

The Standard Deviation of a statistical population, a data set, or a probability distribution is the square root of its variance.

Standard deviation is typically denoted by the lower case sigma: **σ**.

public static double StandardDeviation(this IEnumerable<double> source)
{
return Math.Sqrt(source.Variance());
}

### Median

Median is the number separating the higher half of a sample, a population, or a probability distribution, from the lower half.

public static double Median(this IEnumerable<double> source)
{
var sortedList = from number in source
orderby number
select number;
int count = sortedList.Count();
int itemIndex = count / 2;
if (count % 2 == 0) return (sortedList.ElementAt(itemIndex) +
sortedList.ElementAt(itemIndex - 1)) / 2;
return sortedList.ElementAt(itemIndex);
}

### Mode

Mode is the value that occurs the most frequently in a data set or a probability distribution.

public static T? Mode<T>(this IEnumerable<T> source) where T : struct
{
var sortedList = from number in source
orderby number
select number;
int count = 0;
int max = 0;
T current = default(T);
T? mode = new T?();
foreach (T next in sortedList)
{
if (current.Equals(next) == false)
{
current = next;
count = 1;
}
else
{
count++;
}
if (count > max)
{
max = count;
mode = current;
}
}
if (max > 1)
return mode;
return null;
}

### Range

Range is the length of the smallest interval which contains all the data.

public static double Range(this IEnumerable<double> source)
{
return source.Max() - source.Min();
}

### Covariance

Covariance is a measure of how much two variables change together.

public static double Covariance(this IEnumerable<double> source, IEnumerable<double> other)
{
int len = source.Count();
double avgSource = source.Average();
double avgOther = other.Average();
double covariance = 0;
for (int i = 0; i < len; i++)
covariance += (source.ElementAt(i) - avgSource) * (other.ElementAt(i) - avgOther);
return covariance / len;
}

### Pearson's Chi Square Test

Pearson's chi square test is used to assess two types of comparisons: tests of goodness of fit, and tests of independence.

In other words, it is a measure of how well a sample distribution matches a predicted distribution or the degree of correlation between two sample distributions. Pearson's is often used in scientific applications to test the validity of hypotheses.

public static double Pearson(this IEnumerable<double> source,
IEnumerable<double> other)
{
return source.Covariance(other) / (source.StandardDeviationP() *
other.StandardDeviationP());
}

## Using the Code

The included Unit Tests should provide plenty of examples for how to use these methods, but at its simplest, they behave like other enumerable extension methods. The following program...

static void Main(string[] args)
{
IEnumerable<int> data = new int[] { 1, 2, 5, 6, 6, 8, 9, 9, 9 };
Console.WriteLine("Count = {0}", data.Count());
Console.WriteLine("Average = {0}", data.Average());
Console.WriteLine("Median = {0}", data.Median());
Console.WriteLine("Mode = {0}", data.Mode());
Console.WriteLine("Sample Variance = {0}", data.Variance());
Console.WriteLine("Sample Standard Deviation = {0}", data.StandardDeviation());
Console.WriteLine("Population Variance = {0}", data.VarianceP());
Console.WriteLine("Population Standard Deviation = {0}",
data.StandardDeviationP());
Console.WriteLine("Range = {0}", data.Range());
}

... produces:

Count = 9
Average = 6.11111111111111
Median = 6
Mode = 9
Sample Variance = 9.11111111111111
Sample Standard Deviation = 3.01846171271247
Population Variance = 8.09876543209877
Population Standard Deviation = 2.8458329944146
Range = 8

## Points of Interest

I didn't spend much time optimizing the calculations, so be careful if you are evaluating extremely large data sets. If you come up with an optimization in any of the attached code, drop me a note and I'll update the source.

Hopefully, you'll find this code handy the next time you need some simple statistics calculation.

## History

- Version 1.0 - Initial upload, 9/19/2009.
- Version 1.1 - Added Covariance and Pearson as well as a couple of fixes/optimizations, 10/26/2009.
- Version 1.2 - Updated variance implementation and added GitHub and NuGet links 12/3/2013