Standard Deviation Calculator: Step by Step

Q: Should I use sample or population standard deviation?

Use the sample standard deviation (s) when your data are a sample drawn from a larger group you want to draw conclusions about, which is the usual case in research and homework. Use the population standard deviation (σ) only when your data include every member of the group you care about, such as the test scores of every student in one class.

Q: How do I calculate standard deviation in Excel or Google Sheets?

Use =STDEV.S(A1:A20) for a sample and =STDEV.P(A1:A20) for a population. Google Sheets also accepts =STDEV() for a sample and =STDEVP() for a population. This calculator gives the same answers and shows the working.

How to use the standard deviation calculator

Type or paste your numbers into the box. Commas, spaces, tabs and new lines all work as separators.
Choose Sample if your data are a sample from a larger group (the usual case), or Population if they cover the entire group.
Click Calculate to get the standard deviation, variance, mean and more.
Follow the step-by-step table to check your own work, and download the chart if you need it.

The standard deviation formula

Standard deviation measures how spread out values are around their mean. There are two versions:

Sample: s = Σ(x − x)²n − 1

Population: σ = Σ(x − μ)²N

s, σ: Sample standard deviation, population standard deviation
x: Each value in your data
x, μ: The sample mean, the population mean (add up the values and divide by how many there are)
Σ: "Add up" over every value
n, N: The number of values in the sample, or in the population

The part under the square root is the variance (s² or σ²). The sum on top is called the sum of squares.

Worked example

Six friends score 4, 8, 6, 5, 3 and 7 on a 10-point quiz. Find the sample standard deviation.

Find the mean
x = 4 + 8 + 6 + 5 + 3 + 76 = 336 = 5.5
Subtract the mean from each value, then square

x x − x (x − x)²

4 −1.5 2.25

8 2.5 6.25

6 0.5 0.25

5 −0.5 0.25

3 −2.5 6.25

7 1.5 2.25

Sum 0 17.5
Divide the sum of squares by n − 1 to get the variance
s² = 17.56 − 1 = 3.5
Take the square root
s = 3.5 = 1.871

x	x − x	(x − x)²
4	−1.5	2.25
8	2.5	6.25
6	0.5	0.25
5	−0.5	0.25
3	−2.5	6.25
7	1.5	2.25
Sum	0	17.5

What it means: a typical quiz score is about 1.9 points away from the average of 5.5. If these six friends were the only people you cared about (a population), you'd divide by 6 instead and get σ = 1.708.

How to interpret standard deviation

A small standard deviation means values cluster tightly around the mean; a large one means they're spread out. Think of it as the typical distance of a value from the mean, in the same units as your data.

For data that are roughly bell-shaped, the empirical rule says:

About 68% of values fall within 1 standard deviation of the mean
About 95% fall within 2 standard deviations
About 99.7% fall within 3 standard deviations

To judge whether a standard deviation is big or small, compare it with the mean using the coefficient of variation (CV), which this calculator reports when the mean is positive. To see how far a single value is from the mean in standard-deviation units, use the z-score calculator.

Standard deviation is sensitive to outliers, because squaring the deviations gives extreme values extra weight. If your data have a few very large or small values, check the table to see how much they contribute.

Frequently asked questions

Should I use sample or population standard deviation?

Use the sample standard deviation (s) when your data are a sample drawn from a larger group you want to draw conclusions about, which is the usual case in research and homework. Use the population standard deviation (σ) only when your data include every member of the group you care about, such as the test scores of every student in one class.

Why does the sample formula divide by n − 1?

A sample’s values tend to sit a little closer to their own mean than to the true population mean, so dividing by n would underestimate the spread. Dividing by n − 1 (called Bessel’s correction) makes the sample variance an unbiased estimate of the population variance. The difference matters most for small samples.

What is the difference between variance and standard deviation?

Variance is the average squared distance from the mean; standard deviation is its square root. They describe the same spread, but standard deviation is in the same units as your data (for example minutes or dollars), which makes it much easier to interpret.

What counts as a high or low standard deviation?

It depends on the scale of your data, so compare it with the mean. The coefficient of variation (standard deviation ÷ mean) helps: under about 10% is usually considered low variability, and over about 30% high. A standard deviation of 5 is large for shoe sizes but tiny for house prices.

Can a standard deviation be zero or negative?

It can be zero, which happens only when every value is the same. It can never be negative, because it is the square root of an average of squared numbers.

How do I calculate standard deviation in Excel or Google Sheets?

Use =STDEV.S(A1:A20) for a sample and =STDEV.P(A1:A20) for a population. Google Sheets also accepts =STDEV() for a sample and =STDEVP() for a population. This calculator gives the same answers and shows the working.