Standard Deviation, Explained (With a Worked Example)

What standard deviation measures

Standard deviation (SD) measures spread — how far, on average, the individual values in a dataset sit from their mean. It answers a single question: are the numbers bunched together, or strung out across a wide range?

A low standard deviation means the values are clustered tightly around the average. A high standard deviation means they are scattered far from it. Crucially, SD is expressed in the same units as the original data — dollars, points, centimeters — so you can read it directly. If a set of test scores has a mean of 75 and an SD of 4, a typical score lands about 4 points away from 75. If the SD were 20, scores would be all over the place.

Variance: the step before standard deviation

Standard deviation is built on a closely related quantity called the variance. Variance is the average of the squared distances between each value and the mean. Squaring those distances does two useful things: it removes negative signs (so values below the mean don't cancel values above it), and it gives extra weight to big departures.

The catch is that squaring leaves variance in squared units — "square points" or "square dollars," which mean nothing intuitively. The fix is simple: take the square root at the end. That is all standard deviation is.

Population vs sample standard deviation

There are two versions of the formula, and choosing the right one matters.

Use the population standard deviation (written σ, the Greek letter sigma) when your data is the entire group you care about — every student in a single class, every item produced in one batch. Here you divide the summed squared deviations by N, the count of values.

Use the sample standard deviation (written s) when your data is only a sample drawn from a larger population you want to learn about — 200 voters polled to estimate a whole electorate. Here you divide by N − 1 instead of N. This is called Bessel's correction. A sample tends to underestimate the true spread of its parent population because the values cluster around the sample's own mean rather than the (unknown) population mean. Dividing by the smaller N − 1 nudges the estimate upward and makes the resulting variance an unbiased estimate of the population variance.

The practical rule: if in doubt, and you are working with a sample of something bigger, use N − 1. Most calculators and spreadsheet functions default to the sample version for this reason.

The method, step by step

Whichever version you need, the procedure is the same five steps:

1. Find the mean of the data.
2. Subtract the mean from each value to get its deviation.
3. Square each deviation.
4. Average the squared deviations — divide by N for a population, or N − 1 for a sample. This is the variance.
5. Take the square root of the variance. That is the standard deviation.

A full worked example

Take this small dataset of eight values: {2, 4, 4, 4, 5, 5, 7, 9}.

Step 1 — mean. Add them up: 2 + 4 + 4 + 4 + 5 + 5 + 7 + 9 = 40. Divide by N = 8: the mean is 5.

Steps 2–3 — deviations and squares. Subtract 5 from each value, then square the result:

The squared deviations add up to 32. (Notice the plain deviations sum to zero — they always do, which is exactly why we square them first.)

Step 4 — variance. For the population version, divide by N = 8: variance = 32 ÷ 8 = 4. For the sample version, divide by N − 1 = 7: variance = 32 ÷ 7 ≈ 4.571.

Step 5 — square root. Population SD = √4 = 2. Sample SD = √4.571 ≈ 2.14.

So the same eight numbers give a population standard deviation of exactly 2 and a sample standard deviation of about 2.14 — the N − 1 divisor makes the sample figure a little larger, as expected. You can check any of these results instantly with the statistics calculator.

Interpreting the result: the 68–95–99.7 rule

For data that follows a normal distribution — the familiar symmetric bell curve — standard deviation comes with a handy yardstick called the empirical rule:

• About 68% of values fall within one SD of the mean.
• About 95% fall within two SDs.
• About 99.7% fall within three SDs.

This is what makes SD so useful for spotting the unusual. If a test has a mean of 500 and an SD of 100, a score of 700 is two standard deviations above average — in the top 2.5% or so. Expressing a value as the number of standard deviations it sits from the mean is exactly what a z-score does, which is why z-scores and standard deviation always travel together.

What it means in the real world

Standard deviation shows up everywhere a single average is not enough:

• Test scores. Two cohorts with the same average can have very different SDs. A low SD means consistent performance; a high SD signals a split between strong and struggling students.
• Quality control. Factories track the SD of a product's dimensions. A rising SD warns that a process is drifting out of tolerance long before any single part fails inspection.
• Finance. The standard deviation of an investment's returns is the standard measure of volatility, or risk. Two funds may average the same return, but the one with the higher SD swings more wildly — a bumpier, riskier ride.

When you compare datasets measured in different units, you can divide the SD by the mean to get a relative measure, or use a weighted average when some values should count more than others.

The takeaway

Standard deviation turns the vague idea of "spread" into a single number you can read in the data's own units. Find the mean, measure how far each value strays, square those distances, average them (÷N for a population, ÷N−1 for a sample), and take the square root. The result tells you whether your average is a dependable summary or a thin disguise over a lot of variation.

Frequently asked questions

What does standard deviation actually tell you?

It tells you how spread out a set of numbers is around their mean. A small standard deviation means the values are tightly clustered near the average; a large one means they are scattered far from it. Because it is measured in the same units as the data, you can read it directly — an SD of 2 points on a test means typical scores land about 2 points away from the average.

What is the difference between population and sample standard deviation?

Population SD (σ) divides the summed squared deviations by N and is used when your data is the entire group you care about. Sample SD (s) divides by N−1 — Bessel's correction — and is used when your data is a sample meant to estimate a larger population. Dividing by N−1 makes the sample estimate of variance unbiased, so it nudges the result slightly higher.

How is standard deviation related to variance?

Variance is the average of the squared deviations from the mean, and standard deviation is simply its square root. Squaring during the calculation removes negative signs and exaggerates large gaps, but it also leaves variance in squared units. Taking the square root at the end converts the answer back into the original units of the data, which is why SD is easier to interpret.

What is the 68–95–99.7 rule?

For data that follows a normal (bell-shaped) distribution, about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. It gives you a quick sense of how unusual a value is: anything beyond two standard deviations is fairly rare, and beyond three is exceptional.

This guide is general educational information. For high-stakes statistical work, confirm which convention (population or sample) your context requires.