Standard deviation calculator with working

Data: 2, 4, 4, 4, 5, 5, 7, 9.

1. Mean $= 40 \div 8 = 5$.
2. Differences from the mean, squared: 9, 1, 1, 1, 0, 0, 4, 16.
3. Sum of squares $= 32$.
4. Population variance $= 32 \div 8 = 4$, so $\sigma = 2$.
5. Sample variance $= 32 \div 7 \approx 4.571$, so $s \approx 2.138$.

A small standard deviation means the values are bunched close to the mean; a large one means they're spread out. Two classes might both average 65% in a test, but a class with a standard deviation of 5 scored much more consistently than one with 15. For data that's roughly bell-shaped (normally distributed), about 68% of values lie within one standard deviation of the mean.

The variance is the standard deviation squared (the step before the square root). It's useful in calculations, but its units are squared (cm², for example), so standard deviation is easier to interpret because it's in the same units as the data.

• Range: largest − smallest. Quick, but affected by one extreme value.
• Interquartile range: the spread of the middle half of the data, used with box plots.
• Standard deviation: uses every value.

For the mean, median, mode and range, use our mean, median and mode calculator.

• Forgetting to square the differences (the positive and negative ones would cancel out).
• Forgetting the square root at the end (that gives the variance).
• Using $n$ when the question asks for a sample standard deviation (use $n - 1$).

Standard deviation isn't part of GCSE Maths; it's taught in A level Maths, statistics courses and the IB.

Scientific calculators have a statistics mode: enter the data, then read $\bar{x}$, $\sigma x$ (population) and $sx$ (sample). Our calculator shows the steps behind those buttons, which is what written answers need.