# Measures Of Dispersion Calculator

Several measures of variation: SD, Variance, CV, SEM, MAD.

The standard deviation calculator computes the standard deviation, variance, mean, sum, SEM and MAD, with step by step calculation.
Enter comma , space or Enter after each data value.
The tool doesn't count empty cells or non-numeric cells.

Use the measures of spread calculator to obtain various statistics that demonstrate the variation in the data.

#### Population variance formula

To calculate the population variance, you need the entire dataset.

$\mathrm{Var\left(x\right)}={\sigma }^{2}=\frac{\sum _{i=1}^{n}{\left({x}_{i}-\mathrm{x̄}\right)}^{2}}{n}$

#### Sample variance formula

Following the sample variance formula

${S}^{2}=\frac{\sum _{i=1}^{n}{\left({x}_{i}-\mathrm{x̄}\right)}^{2}}{n-1}$

#### Population Standard Deviation Formula

To calculate the population standard deviation, you need the entire dataset.

$\sigma =\sqrt{\frac{\sum _{i=1}^{n}{\left({x}_{i}-\mathrm{x̄}\right)}^{2}}{n}}$

#### Sample Standard Deviation Formula

To calculate the sample standard deviation, you need the entire dataset. The sample standard deviation is the square root of the sample variance.

$S=\sqrt{\frac{\sum _{i=1}^{n}{\left({x}_{i}-\mathrm{x̄}\right)}^{2}}{n-1}}$

### What is the Coefficient of variation(CV)?

The coefficient of variation is the ratio between the standard deviation and the mean. It lacks a unit, allowing for comparisons between groups with different units.

### What is the standard deviation of the average (mean)?

The sample average is a random variable, and each time you calculate it, you will obtain a different result. The standard deviation of the sample average is referred to as SEM, which stands for the Standard Error of the Mean. SEM is smaller than the standard error of the population.

#### Sample Standard Error of the Mean (SEM) formula$\mathrm{SEM}=\frac{S}{\sqrt{n}}$

In statistics MAD may be the acronym for Median Absolute Deviation or Mean Absolute Deviation

### What is Mean Absolute Deviation?

The Mean Absolute Deviation (MAD), is a statistic that measures the data variability.
The MAD is the average absolute distances from the arithmetic mean.
It is similar to the standard deviation, but instead of the addition of squares differences, it uses the absolute differences, and obviously, there is no need to take a square root.

#### Mean Absolute Deviation (MAD) formula

$\mathrm{MAD}=\frac{\sum _{i=1}^{n}{\mathrm{|x}}_{i}-\mathrm{x̄|}}{n}$

### What is Median Absolute Deviation?

The Median Absolute Deviation (MAD), is a statistic that measures the data variability.
The MAD is the median of the absolute distances from the median.

Mdn = median(Xi)

### Glossary

n - Sample size, the number of values.
Mean - Average.
S - Sample standard deviation, if the list of values you entered is only a sample from the entire population, this is the best estimation for the Population Standard Deviation.
S2 - Sample variance, if the list of values you entered is only a sample from the entire population, this is the best estimation for the Population Variance.
σ - Population standard deviation, if the list of values you entered is the entire population, this is the exact Standard deviation.
σ2 - Population variance, if the list of values you entered is the entire population, this is the exact Variance.
MAD (Mean) - Mean absolute deviation, the average of the absolute distances from the average.
MAD (Median) - Median absolute deviation, the median of the absolute distances from the median.
SEM - Standard Error of the Mean, The standard deviation of the mean.