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# Standard Deviation Calculator

To get unlimited answers, .

## Standard Deviation Lesson

#### Lesson Contents

### What is the Standard Deviation?

**In statistics, the standard deviation is a measure of how dispersed the values in a data set are.** It is the square root of the variance. The formula for population standard deviation is given as:

$$\sigma ={\sqrt {{\frac {1}{N}}\sum _{i=1}^{N}(x_{i}-\mu )^{2}}}$$

And the formula for sample standard deviation is given as:

$$s ={\sqrt {{\frac {1}{N-1}}\sum _{i=1}^{N}(x_{i}-\bar{x} )^{2}}}$$

Where *N* is the number of values in the data set, *x _{i}* is the data point of index

*i*,

*μ*is the population mean, and

*x̄*is the sample mean.

### Why we Calculate the Standard Deviation of a Data Set

**A large standard deviation means the values are very dispersed**, on average the difference between values is large. **A small standard deviation means the values are not very dispersed** and have small differences between them on average.

**We can create a distribution chart based on the dispersion of a data set.** For example, we use the standard deviation to validate if a data set is

*normally distributed*. A set is normally distributed if most of the data falls within 1, 2, and 3 standard deviations of the mean (see empirical rule).

## How the Calculator Works

The standard deviation calculator on this page is written in the programming language *JavaScript* (JS). Since JS runs in your device’s internet browser, there is **no waiting** on communications to a server or page refreshes. Therefore, you get your answer nearly instantly.

When you click the “calculate” button, the inputted data set gets formatted into a JS array. That array is then fed to a library of math functions, where the actual standard deviation calculation is performed. After rounding the answer, the standard deviation is printed to the output area in the calculator.