Standard Deviation Explained with Examples

Standard deviation sounds technical, but the idea is simple: it measures how spread out numbers are around their mean. A small standard deviation means the numbers cluster together; a large one means they are scattered.

The method, step by step

Take the classic data set: $2,4,4,4,5,5,7,9$.

1. Find the mean. $(2+4+4+4+5+5+7+9)÷8=40÷8=5$.
2. Subtract the mean from each value, and square each difference. The differences are $-3,-1,-1,-1,0,0,2,4$; squaring gives $9,1,1,1,0,0,4,16$. (Squaring matters: without it, positive and negative differences would cancel out.)
3. Average the squared differences. That average is the variance: $(9+1+1+1+0+0+4+16)÷8=32÷8=4$.
4. Take the square root. The standard deviation is $\sqrt{4}=2$. The square root brings the units back to the original data — variance is in squared units, which is why we report the standard deviation instead.

So for this data set: mean $5$, variance $4$, standard deviation $2$.

Population vs sample — the one decision to get right

• Population (you have all the data): divide by $n$, as above. Standard deviation $=2$.
• Sample (your data is a sample of something bigger): divide by $n-1$ instead. Variance $=32÷7\approx 4.57$, standard deviation $\approx 2.14$.

If a question does not say, check whether it describes a full group or a sample from one — that single choice changes the answer.

Why not just use the range?

The range ($9-2=7$ here) uses only two values and ignores everything between. Standard deviation uses every value, so it describes the whole data set, not just its extremes. For the mean, median and mode behind this example, see Mean, Median, Mode Explained.

Common mistakes

• Forgetting to square before averaging — the differences cancel otherwise.
• Forgetting the final square root — that leaves you with the variance, not the standard deviation.
• Dividing by the wrong number — $n$ for a population, $n-1$ for a sample.

Practice it

Want to check your work? Try the Standard Deviation Calculator — every step shown — or the Statistics Calculator for mean, median, mode and quartiles together. Then try similar problems on the practice page.