Standard Deviation Calculator (sample & population)

Understanding the standard deviation calculator — concept and principle

A standard deviation calculator measures how spread out a set of numbers is around its mean. It returns both the sample standard deviation s and the population standard deviation σ, together with the variance, mean, coefficient of variation and standard error. It is used for test results, lab measurements, investment returns, quality control and any time you need to say not just "how much on average" but "how consistent".

How it is calculated

1. Mean: x̄ = Σx ÷ n.
2. Sum of squared deviations: SS = Σ(x − x̄)².
3. Sample variance s² = SS ÷ (n − 1); sample SD s = √s².
4. Population SD σ = √(SS ÷ n).
5. Coefficient of variation CV = s ÷ x̄ × 100%.
6. Standard error of the mean SE = s ÷ √n.

The SD has the same unit as the data; the variance is in squared units; the CV has no unit. At least 2 values are required.

The principle

Each deviation x − x̄ says how far one value is from the centre. Deviations always sum to zero, so they are squared before averaging; the square root then brings the result back to the original unit. Dividing by n − 1 instead of n (Bessel's correction) is used when the data are a sample from a larger population: the sample mean is fitted to the same data, which makes deviations slightly too small, and n − 1 compensates so that s² is an unbiased estimate of the population variance. Use σ (divide by n) only when your data are the entire population.

Worked example

1. Data: 12, 15, 11, 19, 22, 15, 18, 25, 14, 16 (n = 10). Sum = 167, so x̄ = 16.7.
2. Squared deviations: 22.09, 2.89, 32.49, 5.29, 28.09, 2.89, 1.69, 68.89, 7.29, 0.49; SS = 172.1.
3. s² = 172.1 ÷ 9 ≈ 19.1222; s = √19.1222 ≈ 4.3729.
4. σ = √(172.1 ÷ 10) = √17.21 ≈ 4.1485.
5. CV = 4.3729 ÷ 16.7 × 100 ≈ 26.19%; SE = 4.3729 ÷ √10 ≈ 1.3828.

How to read the result

For roughly bell-shaped data, about two-thirds of values lie within one SD of the mean and about 95% within two. Here that is roughly 12.3 to 21.1. The CV lets you compare spread between data sets with different units or scales; a lower CV means more consistent values. The SE describes the precision of the mean, not the spread of individual values, and shrinks as n grows.

Common mistakes

• Using population SD for a sample, which understates spread, especially for small n.
• Confusing SD (spread of data) with SE (uncertainty of the mean).
• Reporting a CV for data that can be negative or have a mean near zero, where it is meaningless.

Frequently asked questions

Which SD does Excel use?

STDEV.S divides by n − 1 (sample) and STDEV.P divides by n (population).

Can the standard deviation be negative?

No. It is a square root of a sum of squares, so it is zero or positive; zero means every value is the same.

What is a high standard deviation?

There is no fixed threshold. Judge it relative to the mean (the CV) or against values typical for your field.

Sources: NIST/SEMATECH e-Handbook of Statistical Methods, Section 1.3.5.6 Measures of Scale · Moore, D. S., McCabe, G. P. and Craig, B. A., Introduction to the Practice of Statistics (W. H. Freeman)