Guide

How to calculate standard deviation

Find the mean, subtract it from each value, square those deviations, average them (divide by n − 1 for a sample, or N for a population), then take the square root. The result is standard deviation in the original units.

The problem

The formula looks long because it is several small jobs glued together. If you skip a job — especially the divisor — you get a number that looks like SD but answers a different question.

Step-by-step

Find the mean. Add the values and divide by how many there are.
Find each deviation. Subtract the mean from every value. Some will be negative; that is expected.
Square the deviations. Negatives disappear, and large misses weigh more.
Add the squared deviations.
Divide. Sample: divide by n − 1. Population: divide by N. That quotient is variance.
Take the square root. That is s (sample) or σ (population).

Worked example: 10, 34, 23, 54, 9

Five values. Sum = 130. Mean μ = 130 / 5 = 26. This walkthrough is the population path (divide by N = 5), which the in-app lesson uses to reach σ ≈ 16.75. Sample SD on the same list is a little larger because the divisor is 4 instead of 5.

Deviations from 26: −16, 8, −3, 28, −17. Squares: 256, 64, 9, 784, 289. Sum of squares = 1402. Variance = 1402 / 5 = 280.4. σ = √280.4 ≈ 16.75.

Useful tips

You need at least two values for a meaningful spread in this calculator.
The two-pass method (mean first, then deviations) stays accurate when values are large. A computational shortcut exists; algebraically it matches, but it is easier to lose precision on a four-function calculator.
Commas, spaces, or new lines are all valid ways to paste a list.

Common mistakes

Dividing a sample by n instead of n − 1.
Stopping at variance and calling it standard deviation.
Averaging the unsquared deviations — they cancel toward zero.

How Standard Deviation Calculator helps

Paste the list, choose Sample or Population, tap Calculate, then Show solution. The table is the same working you would write on paper. Load the worked example from Examples if you want 10, 34, 23, 54, 9 already entered.

Calculate this example on Android

Show solution working table for a standard deviation calculation on Android