How to use the Standard Deviation
- Enter your values separated by commas, spaces or line breaks.
- Choose sample if the data is a subset of a larger group, population if it is the whole group.
- Read the standard deviation, variance and mean.
- Interpret the result relative to the mean, not as an absolute figure.
How the calculation works
The calculation finds the mean, takes each value's deviation from it, squares those deviations so positives and negatives do not cancel, averages them to get variance, then takes the square root to return to the original units. Squaring is what makes the measure sensitive to extreme values: a point far from the mean contributes disproportionately.
The sample and population versions differ in the divisor. Dividing by n − 1 rather than n — Bessel's correction — compensates for the fact that a sample's own mean is closer to its own values than the true population mean would be, which otherwise biases the estimate downwards. With large samples the difference is negligible; with ten values it is meaningful.
Population σ = √( Σ(x − μ)² / n ) | Sample s = √( Σ(x − x̄)² / (n − 1) )Source: Standard statistical definitions including Bessel's correction for sample variance.
Worked example
Test scores from a sample of six students: 72, 88, 65, 91, 78, 84.
- Mean = 478 / 6 = 79.67.
- Squared deviations: 58.8, 69.4, 214.7, 128.4, 2.8, 18.8 — sum 492.9... precisely 492.83.
- Sample variance = 492.83 / 5 = 98.57.
- s = √98.57 = 9.93.
Sample standard deviation ≈ 9.9 marks. About two thirds of similar students would be expected within 70 to 90.
Frequently asked questions
What counts as a high standard deviation?+
It is only meaningful relative to the mean. A deviation of 5 is large for values averaging 20 and tiny for values averaging 5,000 — the coefficient of variation, deviation divided by mean, makes the comparison fair.
What is the difference between variance and standard deviation?+
Variance is the average squared deviation, so it is in squared units; standard deviation is its square root and shares the units of your data, which makes it far easier to interpret.
Does the 68-95-99.7 rule always apply?+
Only for roughly normal distributions. Skewed data — waiting times, incomes — does not follow those proportions, so check the shape before quoting them.
Which version should I choose if I am unsure?+
Choose sample. Most real data sets are samples of something larger, and the sample formula gives the less biased estimate of the underlying spread.
Last reviewed August 31, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.