Standard Deviation Calculator

Find the standard deviation of a data set, sample or population, with a step-by-step table.

Data	12, 15, 18, 22, 22, 25, 30, 31, 35, 41
Data is a	Sample

Data

Separate numbers with commas, spaces or new lines.

Data is a

Sample standard deviation (s)

9.1463411021

On average, values sit about 9.146341 from the mean of 25.1. The variance is 83.655556.

Mean: 25.1
Variance: 83.6555555556
Sum of squared deviations: 752.9
Within 1 SD of the mean: 6 of 10
Population SD (n): 8.6769810418

Your data and one standard deviation either side

Mean 25.1Mean − 1 SD 15.9537Mean + 1 SD 34.2463

Working

x	x − mean	(x − mean)²
12	-13.1	171.61
15	-10.1	102.01
18	-7.1	50.41
22	-3.1	9.61
22	-3.1	9.61
25	-0.1	0.01
30	4.9	24.01
31	5.9	34.81
35	9.9	98.01
41	15.9	252.81

How this was worked out

1
Add the squared deviations
Σ = 752.9
2
Divide by n − 1
752.9 ÷ 9 = 83.65555556
3
Take the square root
√83.65555556 = 9.1463411

About the Standard Deviation Calculator

Standard deviation measures how spread out numbers are around their mean, in the same units as the data. A small SD means values cluster tightly; a large one means they're scattered.

For roughly bell-shaped data, about 68% of values fall within 1 SD of the mean, 95% within 2 SD and 99.7% within 3 SD.

The formula

s = √( Σ(x − x̄)² ÷ (n − 1) ), σ = √( Σ(x − μ)² ÷ N )

n − 1: sample: Bessel's correction for estimating from a sample
N: population: every member is included

Worked example

2, 4, 4, 4, 5, 5, 7, 9 (population)

Mean = 5
Squared deviations sum to 32
σ² = 32 ÷ 8 = 4, σ = 2

Questions people ask

Why divide by n − 1?

A sample's own mean sits closer to its values than the true population mean does, so dividing by n underestimates the spread. n − 1 corrects that.

Is variance ever negative?

No. It's a sum of squares, so it's zero or positive.