Standard Deviation Calculator

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About the Standard Deviation Calculator

The standard deviation calculator lets you calculate the standard deviation for your data (population or sample).

The calculator will tell you not just the standard deviation, but also how to calculate it.

Usage Guide

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i. Valid Inputs

Your input needs to be a list of numbers. Each number can be a real number in any format — integers, decimals, fractions, or even mixed numbers. Here are a few examples.

Whole numbers or decimals → $\hspace{0.2em} 2 \hspace{0.2em}$ , $\hspace{0.2em} -4.25 \hspace{0.2em}$ , $\hspace{0.2em} 0 \hspace{0.2em}$ , $\hspace{0.2em} 0.33 \hspace{0.2em}$
Fractions → $\hspace{0.2em} 2/3 \hspace{0.2em}$ , $\hspace{0.2em} -1/5 \hspace{0.2em}$
Mixed numbers → $\hspace{0.2em} 5 \hspace{0.5em} 1/4 \hspace{0.2em}$

ii. Example

If you would like to see an example of the calculator's working, just click the "example" button.

iii. Solutions

As mentioned earlier, the calculator won't just tell you the answer but also the steps you can follow to do the calculation yourself. The "show/hide solution" button would be available to you after the calculator has processed your input.

iv. Share

We would love to see you share our calculators with your family, friends, or anyone else who might find it useful.

By checking the "include calculation" checkbox, you can share your calculation as well.

Here's a quick overview of what standard deviation is and how to calculate it.

Standard Deviation

Standard deviation is a measure of variation or dispersion in a set of data points. It's a measure of how much the observations or values in a dataset deviate from the mean.

Now before we can move ahead, it's important to mention there are two slightly different formulas for the standard deviation depending on the type of data one is working with.

Standard Deviation (Population)

When you have the data for the whole population being studied, standard deviation $\hspace{0.2em} (\sigma) \hspace{0.2em}$ is given by the formula

\sigma \hspace{0.2em} = \hspace{0.2em} \sqrt{ \frac{ \Sigma (x - \mu)^2}{n} }

Here $\hspace{0.2em} x \hspace{0.2em}$ represents an individual data point, $\hspace{0.2em} \mu \hspace{0.2em}$ is the population mean, and $\hspace{0.2em} n \hspace{0.2em}$ is the number of data points.

Example

The ages (in years) of Molly's kids are $\hspace{0.2em} 11 \hspace{0.2em}$ , $\hspace{0.2em} 9 \hspace{0.2em}$ , $\hspace{0.2em} 6 \hspace{0.2em}$ , and $\hspace{0.2em} 2 \hspace{0.2em}$ . Calculate their mean age and also the standard deviation.

Solution

Let's start by find the mean age $\hspace{0.2em} \mu \hspace{0.2em}$ . It will also help us in the calculation of standard deviation.

\begin{align*} \mu \hspace{0.2em} &= \hspace{0.2em} \frac{\Sigma \hspace{0.05em} x}{n} \\[1em] &= \hspace{0.2em} \frac{11 + 9 + 6 + 2}{4} \\[1em] &= \hspace{0.2em} 7 \end{align*}

So the mean age is $\hspace{0.2em} 7 \hspace{0.2em}$ years.

Alright, let's calculate the standard deviation $\hspace{0.2em} \sigma \hspace{0.2em}$ . We have the formula

\sigma \hspace{0.2em} = \hspace{0.2em} \sqrt{ \frac{ \Sigma (x - \mu)^2 }{n} }

Now,

\begin{align*} &\Sigma (x - \mu)^2 \\[1em] = \hspace{0.25em} & (11 - 7)^2 + (9 - 7)^2 + (6 - 7)^2 + (2 - 7)^2 \\[1em] = \hspace{0.25em} & 4^2 + 2^2 + (-1)^2 + (-5)^2 \\[1em] = \hspace{0.25em} & 46 \end{align*}

Substituting this in the formula for $\hspace{0.2em} \sigma \hspace{0.2em}$ , we get

\begin{align*} \sigma \hspace{0.2em} &= \hspace{0.2em} \sqrt{ \frac{46}{4} } \\[1.5em] &\approx \hspace{0.2em} 3.39 \end{align*}

The standard deviation in kids' ages is $\hspace{0.2em} 3.39 \hspace{0.2em}$ years.

We used the formula for population standard deviation because we had the data (ages) for the entire population (the four kids).

Standard Deviation (Sample)

Often, it's not possible, or practical, to get the data for the whole population we want to study. So we choose an appropriate sample to represent the population.

In such cases, the formula for the standard deviation $\hspace{0.2em} s \hspace{0.2em}$ is

s \hspace{0.2em} = \hspace{0.2em} \sqrt{ \frac{ \Sigma (x - \overline{x})^2}{n - 1} }

Here, $\hspace{0.2em} x \hspace{0.2em}$ represents an individual data point, $\hspace{0.2em} \overline{x} \hspace{0.2em}$ is the sample mean, and $\hspace{0.2em} n \hspace{0.2em}$ is the number of data points.

Example

Of the $\hspace{0.2em} 54 \hspace{0.2em}$ adults in a housing society, $\hspace{0.2em} 6 \hspace{0.2em}$ were randomly selected for a study. Their weights (in pounds) were found to be $\hspace{0.2em} 198 \hspace{0.2em}$ , $\hspace{0.2em} 154 \hspace{0.2em}$ , $\hspace{0.2em} 173 \hspace{0.2em}$ , $\hspace{0.2em} 167 \hspace{0.2em}$ , $\hspace{0.2em} 195 \hspace{0.2em}$ , and $\hspace{0.2em} 185 \hspace{0.2em}$ . Calculate the standard deviation in the weights of adults in the society.

Solution

This time we have the data only for a sample of the population and not the whole population ( $\hspace{0.2em} 6 \hspace{0.2em}$ out of $\hspace{0.2em} 54 \hspace{0.2em}$ adults). So. we'll use the formula for sample standard deviation.

The formula for sample standard deviation is —

s \hspace{0.2em} = \hspace{0.2em} \sqrt{ \frac{ \Sigma (x - \overline{x})^2}{n - 1} }

Again, we start by find the mean $\hspace{0.2em} \overline{x} \hspace{0.2em}$ .

\begin{align*} \overline{x} \hspace{0.2em} &= \hspace{0.2em} \frac{\Sigma \hspace{0.05em} x}{n} \\[1em] &= \hspace{0.2em} \frac{198 + 154 + 173 + 167 + 195 + 185}{6} \\[1em] &\approx \hspace{0.2em} 178.67 \end{align*}

Next,

\begin{align*} &\Sigma (x - \overline{x})^2 \\[1em] \approx \hspace{0.4em} & (198 - 178.67)^2 + ... + (185 - 178.67)^2 \\[1em] \approx \hspace{0.4em} & 1457.33 \end{align*}

Substituting this in the formula for $\hspace{0.2em} s \hspace{0.2em}$ , we get

\begin{align*} s \hspace{0.2em} &= \hspace{0.2em} \sqrt{ \frac{1457.33}{6 - 1} } \\[1.5em] &= \hspace{0.2em} 17.07 \end{align*}

The standard deviation is $\hspace{0.2em} 17.07 \hspace{0.2em}$ pounds.