How to Calculate Standard Deviation (Step by Step)

What standard deviation tells you

Standard deviation puts a single number on how spread out your data is around its mean. A small value means the numbers cluster tightly; a large one means they scatter widely. Because it is in the same units as the data — euros, centimetres, seconds — it reads naturally as a typical distance from the average.

The steps, worked through

Take 2, 4, 4, 4, 5, 5, 7, 9. The mean is 40 ÷ 8 = 5. The squared differences from 5 are 9, 1, 1, 1, 0, 0, 4, 16, which sum to 32. Dividing by 8 gives a variance of 4, and its square root is 2 — the standard deviation. So values sit, on average, about 2 units from the mean of 5.

Population vs sample (n or n − 1)

If your numbers are the entire group you care about, divide by n. If they are a sample used to estimate a larger population, divide by n − 1 instead — this Bessel's correction slightly increases the result to make up for the extra uncertainty of a sample. Choosing the wrong one is the most common standard-deviation mistake.

Reading it: the 68–95–99.7 rule

For roughly bell-shaped (normal) data, about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. So with a mean of 5 and a standard deviation of 2, most values (95%) land between 1 and 9. This rule turns the number into a quick sense of what is typical and what is unusual.

Frequently asked questions

What's the difference between variance and standard deviation?

Variance is the average of the squared differences from the mean; standard deviation is its square root, which puts the spread back in the original units.

When do I use n vs n − 1?

Use n when your data is the whole population, and n − 1 when it is a sample estimating a larger population.

What is a 'good' standard deviation?

There isn't one in absolute terms — it depends on the scale of the data. Compare it to the mean (the coefficient of variation) to judge whether the spread is large or small.