Standard Deviation Calculator

How it works

Standard deviation measures how far, on average, the numbers in a list sit from their mean. A small standard deviation means the numbers cluster tightly around the average; a large one means they’re spread out widely.

This standard deviation calculator works through the same five steps you’d use with pen and paper:

1. Mean = sum of all numbers ÷ count of numbers
2. Deviation = each number − the mean
3. Sum of squared deviations = add up every deviation, squared
4. Variance = sum of squared deviations ÷ (n − 1 for a sample, or n for a population)
5. Standard deviation = √variance

Squaring each deviation in step 3 matters for two reasons: it stops positive and negative deviations from cancelling each other out (which would otherwise always sum to zero), and it weights larger deviations more heavily than smaller ones, so a few big outliers pull the result up more than a lot of small ones would.

Sample vs population — why divide by n − 1?

Use population standard deviation only when your numbers are the entire group you care about — every student in one specific class, every transaction in a full financial year. Divide the sum of squared deviations by n, the full count.

Use sample standard deviation whenever your numbers are a subset drawn from a larger population you’re trying to describe — a survey of 200 customers standing in for your whole customer base, or one class’s test results standing in for a whole year level’s ability. Divide by n − 1 instead.

The n − 1 adjustment is called Bessel’s correction, and it exists for a subtle reason: a sample’s mean is calculated from the very same data used to measure the spread, so that data ends up looking slightly more tightly clustered around its own mean than the true, wider population actually is. Dividing by the smaller number n − 1, instead of n, inflates the result just enough to correct for that built-in bias, giving a more honest estimate of how spread out the real population is likely to be. When in doubt, sample (n − 1) is the safer default — it’s what Excel’s STDEV.S, Google Sheets and most statistics software use unless you tell them otherwise.

Worked example

Mr Delaney, a PE teacher at a high school in Perth, runs a one-minute sit-up test with a small group of 8 Year 10 students during a fitness unit. Their results are:

12, 15, 9, 21, 15, 18, 14, 10

Step 1 — Mean. Add the 8 results and divide by 8:

(12 + 15 + 9 + 21 + 15 + 18 + 14 + 10) ÷ 8 = 114 ÷ 8 = 14.25

Step 2 and 3 — Deviation and squared deviation for each student:

Sit-ups	Deviation from mean	Deviation squared
12	−2.25	5.0625
15	0.75	0.5625
9	−5.25	27.5625
21	6.75	45.5625
15	0.75	0.5625
18	3.75	14.0625
14	−0.25	0.0625
10	−4.25	18.0625

Adding up the last column gives a sum of squared deviations of 111.5.

Step 4 — Variance. Mr Delaney treats his 8 students as a sample representing the wider Year 10 cohort, not the entire group he’s interested in, so he divides by n − 1 = 7:

111.5 ÷ 7 = 15.93 (sample variance)

Step 5 — Standard deviation. Take the square root of the variance:

√15.93 ≈ 3.99 sit-ups (sample standard deviation)

So the class averaged 14.25 sit-ups, with a typical spread of just under 4 sit-ups either side of that average. If Mr Delaney instead treated these 8 results as the entire group he cared about — say, this was the whole cohort and he had no interest in generalising beyond it — he’d use the population formula instead, dividing by n = 8:

111.5 ÷ 8 = 13.94 (population variance), so √13.94 ≈ 3.73 (population standard deviation)

The population figure is always a little smaller than the sample figure for the same data, because dividing by a larger number (n instead of n − 1) produces a smaller result.