What a standard deviation actually tells you

Two sets of numbers can have exactly the same mean and look nothing alike. A standard deviation is the one-number answer to "how spread out are they" — roughly, the typical distance of a value from the mean.

Both sets below are invented for this example. Both have a mean of exactly 50:

• Set A: 48, 49, 50, 51, 52
• Set B: 10, 30, 50, 70, 90

The mean cannot tell them apart. The standard deviation puts Set A at about 1.41 and Set B at about 28.28.

The method, on paper

Five steps, and none of them is hard.

1. Find the mean.
2. Subtract the mean from each value to get its deviation.
3. Square each deviation.
4. Average the squares. That average is the variance.
5. Take the square root. That is the standard deviation.

Step 3 exists because the deviations always sum to zero — the positives and negatives cancel exactly, every time, by construction. Squaring removes the signs. Step 5 undoes the squaring so the answer comes back in the original units instead of units squared.

A full worked example

Take 4, 8, 15, 16, 23, 42.

The mean: 4 + 8 + 15 + 16 + 23 + 42 = 108, and 108 ÷ 6 = 18.

The deviations: −14, −10, −3, −2, 5, 24. They sum to zero, as they must — a useful check that you have the mean right.

The squares: 196, 100, 9, 4, 25, 576, which total 910.

Now the divisor, and this is the part with two answers.

• Divide by n = 6: variance 910 ÷ 6 = 151.67, and the square root is 12.32.
• Divide by n − 1 = 5: variance 910 ÷ 5 = 182, and the square root is 13.49.

Sample against population, and why n−1 exists

If those six numbers are the whole group you care about — every item, not a selection — divide by n. That is the population standard deviation.

If they are a sample drawn from something larger, and you want to estimate the spread of that larger thing, divide by n − 1. That is the sample standard deviation.

The reason for the smaller divisor: you measured the deviations from the sample's own mean, and a sample's mean always sits closer to that sample's own values than the true underlying mean does. So the squared deviations come out systematically too small, and dividing by n would hand you a spread that is too low every single time. Using n − 1 corrects the variance for that. The correction is large on small sets — here it moved 12.32 to 13.49 — and fades away as n grows, because the gap between n and n − 1 shrinks in relative terms.

To convert between them without redoing the sums, multiply the population figure by the square root of n ÷ (n − 1). Here: 12.32 × the square root of (6 ÷ 5) = 13.49. Same number, no re-adding.

The number means nothing on its own

A standard deviation of 5 is enormous next to a mean of 6 and negligible next to a mean of 6,000. It carries the units of the data, so it can only be read next to the mean.

Two honest limits beyond that. First, it is built from squared deviations, so a single far-out value affects it far more than it affects the mean — in the worked example the lone 42 contributed 576 of the 910, well over half the total from one of six values. Second, one number cannot describe a shape. Two sets can share a mean and a standard deviation and still be arranged completely differently, so the summary is a starting point, not a description.

There is no such thing as a standard deviation that is too big or too small in the abstract. Whether a given spread is acceptable depends entirely on what the data is and what you need from it, and no rule of thumb substitutes for that.

Doing it without the paper

The free Standard Deviation tool takes a comma-separated list and returns the mean, the variance and the standard deviation in your browser, keeping nothing.

One thing to know before you use the output: it computes the population figure — it divides by n, not by n − 1, and it labels the result accordingly. On the six numbers above it gives a mean of 18, a variance of 151.67 and a standard deviation of 12.32. If your list is a sample and you need the sample figure, apply the conversion above: multiply by the square root of n ÷ (n − 1) to get 13.49. Which of the two you want is a question about where your numbers came from, and only you can answer it.