# Standard Deviation Calculator

Created by Jasmine J Mah
Reviewed by Bogna Szyk and Jack Bowater
Last updated: Jun 05, 2023

The standard deviation calculator shows you how to calculate the mean and standard deviation of a dataset. If you are learning statistics, it is essential to learn how to find the standard deviation because it is very widely used.

You'll love the special features of our standard deviation calculator:

• It works as a population or sample standard deviation calculator.
• We'll show you the steps for easy understanding.
• It's excellent as a learning tool or as a calculator for small datasets.
• The definition and formula for standard deviation are explained below.

## What is standard deviation?

The standard deviation is a measure of the variability in a dataset. In other words, the standard deviation describes how "spread-out" the data is around the mean. This calculator deals with separate data points, but we also have a dedicated grouped data standard deviation calculator for ranged data.

A high standard deviation indicates that a dataset is more spread out.

A low standard deviation indicates that the data is more tightly clustered around the mean or less spread out.

Can you imagine what a standard deviation looks like? While you can calculate the standard deviation for any dataset, it can be helpful to visualize the standard deviation for normally distributed data. The empirical rule states that for any dataset which approximates a normal distribution, about 68% of the data will fall within one standard deviation from the mean, shown in the figure below.

Not only is standard deviation a widely-used measure of variation, it forms the basis of other tools which characterize variation, including the quantities calculated by the relative standard deviation calculator and the confidence interval calculator.

## Standard deviation formula

The mathematical definition for standard deviation (σ) is the positive square root of the variance ($\sigma^2$):

$\mathrm{variance} = \sigma^2 \\ \mathrm{standard \ deviation} = \sqrt{\sigma^2} = \sigma$

The standard deviation equation seems simple, but how do you calculate variance?

Variance is defined as the average squared difference from the mean for all data points. It is written as:

$\sigma^2 = \frac{1}{N}\displaystyle\sum_{i}^N (x_i - \mu)^2$

where:

• $\sigma^2$ - Variance;
• $\mu$ - Mean; and
• $x_i$ - The ith data point out of $N$ total data points.

You can calculate variance in three steps:

1. Find the difference from the mean for each point. Use the formula:
$x_i - \mu$

2. Square the difference from the mean for each point:
$(x_i - \mu)^2$

3. Find the average of the squared differences from the mean which you found in step 2:
$\frac{1}{N}\sum (x_i - \mu)^2$
This is the variance for population data. Note that this step is slightly different for sample data (see next section).

Now we recall that the standard deviation is the (positive) square root of variance, so the complete standard deviation equation (for population data) becomes:

$\sigma = \sqrt{\frac{1}{N}\displaystyle\sum_{i}^N (x_i - \mu)^2}$

## Population vs. sample standard deviation formula

In many scientific experiments, only a sample of a population is measured for practical reasons. This sample allows us to make inferences about the population. However, when sample data is used to estimate the variance of a population, the variance formula $\sigma^2 = \frac{1}{N}\sum (x_i - \mu)^2$ underestimates the variance of the population.

To avoid underestimating the variance of a population (and consequently, the standard deviation), we replace $N$ with $N - 1$ in the formulas for variance and standard deviation, when sample data is used. This adjustment is known as Bessels' correction.

The sample variance formula becomes:

$s^2 = \frac{1}{N-1}\sum (x_i - \={x})^2$

and the complete standard deviation formula becomes:

$s = \sqrt{\frac{1}{N-1}\sum (x_i - \={x})^2}$

where:

• $s^2$ - Estimate of variance;
• $s$ - Estimate of standard deviation; and
• $\={x}$ (pronounced as "x-bar") - Sample mean.

## Example calculation

Let's say we have a sample dataset with seven numbers: 2, 4, 5, 6, 6, 9, 10. How do we calculate standard deviation? Follow these steps:

1. Calculate the mean

To calculate the mean (x̄), divide the sum of all numbers by the number of data points:
$\={x} = \frac{2 + 4 + 5 + 6 + 6 + 9 + 10}{7} = 6$.

2. Calculate the squared differences from the mean

Now that we know the mean (x̄ = 6), we will calculate the squared difference from the mean for each data point:
$(x_i - \={x})^2$.

For the first point with a value of 2, the calculation would be:
$(2-6)^2 = (-4)^2 = 16$.

The calculated squared differences from the mean for all data points are shown in the table below:

xi

(xi - x̄)2

2

16

4

4

5

1

6

0

6

0

9

9

10

16

3. Calculate the variance and standard deviation

Since we are using sample data, we calculate variance using the sample variance equation and the squared differences from the mean we found in step 2:

$s^2 = \frac{1}{N-1}\sum (x_i - \={x})^2$,
which gives
$s^2 = \frac{16 + 4 + 1 + 0 + 0 + 9 + 16}{7 - 1} = 7.6667$.

The standard deviation (s) is the square root of the variance, so our final step is:

$s = \sqrt{7.6667} = 2.7689$.

The standard deviation of the sample dataset was 2.8. Now that you know how to find the standard deviation try calculating it yourself, then check your answer using our calculator!

🔎 Did you know? Standard deviation is one of the measures of dispersion and coefficient of dispersion, concepts that help us understand the spread of our data.

## How to find standard deviation by hand?

If you are calculating standard deviation with a handheld calculator, there is an easier formula you should use to use to calculate variance. This alternative formula is mathematically equivalent but easier to type into a calculator.

The easy-to-type formula for variance (for population data) is:

$\sigma^2 = \frac{\sum(x_i^2) - (\sum x_i)^2}{N}$

The easy-to-type formula for sample variance is:

$s^2 = \frac{\sum(x_i^2) - (\sum x_i)^2}{N-1}$

To find standard deviation, you would first calculate variance using either of the formulas above. Then, the standard deviation would be the square root of variance.

For example, with a sample dataset of 1, 2, 4, 6, the calculation for sample variance would be:
$\sum(x_i^2) = (1^2 + 2^2 + 4^2 + 6^2) = 57$
$(\sum x_i)^2 = \frac{(1 + 2 + 4 + 6)^2}{4} = \frac{169}{4} = 42.25$

which give

$\sigma^2 = \frac{57 - 42.25}{4-1} = 4.9167$.

The standard deviation would then be the square root of the variance:

$\sqrt{4.9167} \approx 2.2$

Try it yourself, then check your answer with our standard deviation calculator!

## Summary of variables and equations

Table 1. Variables for population data

Variable

Symbol

Equation

Number of observations

$N$

Population mean

$\mu$

$\frac{1}{N}\sum x_i$

Sum of squares

$\mathrm{SS}$

$\sum(x_i - \mu)^2$

Variance

$\sigma^2$

$\frac{\mathrm{SS}}{N}$

Standard deviation

$\sigma$

$\sqrt{\sigma^2}$

Table 2. Variables for sample data

Variable

Symbol

Equation

Sample mean

$\={x}$

$\frac{1}{N}\sum x_i$

Sum of squares

$\mathrm{SS}$

$\sum (x_i - \={x})^2$

Sample variance

$s^2$

$\frac{SS}{N-1}$

Standard deviation

$s$

$\sqrt{s^2}$

Jasmine J Mah
Data (You may enter up to 30 numbers)
x₁
x₂
Settings
Dataset type
sample
Steps to show
summary
Results
 Observations (N): 0 Mean (x̄): 0 Standard deviation (s): 0
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