Standard Deviation

What you will learn

• Say what standard deviation measures.
• Calculate it step by step.
• Compare the spread of two sets of data.

Before you start: you should know how to find the mean and the range. See Range and Outliers.

The idea

Two classes both have a mean test score of $70$. In one class, everyone scored between $65$ and $75$; in the other, scores ranged from $30$ to $100$. The averages match, but the classes are completely different. We need a measure of spread.

The range (largest minus smallest) is crude because it depends on just two values. Standard deviation uses every value. Roughly speaking, it is the typical distance of a data point from the mean.

Building it step by step

1. Find the mean.
2. Find each value’s deviation from the mean ($x - \mu$).
3. Square each deviation. This makes them all positive and gives larger deviations more weight.
4. Average the squared deviations. This is the variance.
5. Take the square root to return to the original units. This is the standard deviation.

Population or sample?

If your data is the entire group you care about, divide by $N$ (population standard deviation, $\sigma$). If it is a sample used to estimate a bigger population, divide by $n - 1$ instead (sample standard deviation, $s$). The slightly smaller divisor corrects for the fact that a sample tends to underestimate the true spread.

Worked examples

Example 1: Population standard deviation

Find the standard deviation of $4, 8, 6, 5, 7$.

1. Mean: $\mu = \frac{4 + 8 + 6 + 5 + 7}{5} = \frac{30}{5} = 6$.
2. Deviations: $-2, 2, 0, -1, 1$.
3. Squared deviations: $4, 4, 0, 1, 1$, summing to $10$.
4. Variance: $\frac{10}{5} = 2$.
5. Standard deviation: $\sigma = \sqrt{2} \approx 1.41$.

Answer: $\sigma \approx 1.41$

Example 2: Comparing consistency

Two archers’ scores have the same mean of $8.2$. Archer A has $\sigma = 0.6$; archer B has $\sigma = 1.9$. Who is more consistent?

1. A smaller standard deviation means scores cluster more tightly around the mean.
2. $0.6 < 1.9$.

Answer: Archer A is more consistent.

Example 3: Using the normal distribution

Adult heights in a population are roughly normal with mean $170$ cm and standard deviation $8$ cm. About what percentage of people are between $162$ and $178$ cm?

1. $162 = 170 - 8$ and $178 = 170 + 8$, so this is within one standard deviation of the mean.
2. For a normal distribution, about $68\%$ of values lie within $1\sigma$, $95\%$ within $2\sigma$ and $99.7\%$ within $3\sigma$.

Answer: about $68\%$

Standard deviation is in the same units as the data (cm, £, seconds); variance is in squared units. That is why we usually report standard deviation.

Why it matters

Manufacturers use standard deviation to control quality: if bolt diameters vary too much, they won’t fit. Investors use it to measure risk, since a fund whose returns swing widely has a high standard deviation. Scientists report it as error bars, and exam boards use it to compare how spread out results are from year to year.

It also lets you say how unusual a value is. A measurement three standard deviations from the mean is rare enough to deserve a second look.