What is standard deviation?
Standard deviation is a number that tells you how spread out a set of values is. A small one means most values sit close to the average. A big one means they’re scattered far from it. Two classes can both average 75 on a test and still be nothing alike: in one, everyone scored between 70 and 80, and in the other, half the class got 100 and half got 50.
What the number means
Think of it as roughly the typical distance between a value and the average. If a class averages 75 with a standard deviation of 5, a typical score is about 5 points from 75. With a standard deviation of 25, a typical score is about 25 points away.
It comes in the same units as the data. Test scores give you points, heights give you inches, wait times give you minutes.
Twenty students, one test, and the class average stays at 75 the whole time. Drag the slider and watch the scores spread out while the standard deviation climbs.
This is the same exercise that’s in the app.
- Class average
- 75
- Standard deviation
- 3.8
- Lowest to highest
- 67 to 83
Same average every time. The standard deviation is the part the average leaves out.
One, two and three standard deviations
When data follows the familiar bell curve, standard deviation tells you where most of it lands. About 68 percent of values fall within one standard deviation of the average, about 95 percent within two, and about 99.7 percent within three.
So a value two standard deviations above average is unusual but not rare. Only about 2.5 percent of values sit that high, or about one in forty.
How it’s worked out
Find the average
Add the values up and divide by how many there are.
Measure each distance
Subtract the average from each value, then square each result so the negatives don’t cancel out the positives.
Average the squares
Add up the squared distances and divide by how many values there are. That number is the variance.
Take the square root
The square root of the variance is the standard deviation, back in the data’s own units.
Percentiles: the other way to describe spread
A percentile tells you what share of values fall below a given one. Scoring in the 90th percentile means you did better than about 90 percent of people. Percentiles don’t need a bell curve, which makes them handy for lopsided data like incomes or wait times.
Questions people ask
What is a good standard deviation?
There isn’t one number. It depends on what you’re measuring and why. A standard deviation of five minutes is tiny for a flight’s arrival time and huge for a train that’s supposed to come every six minutes. Compare it to the average and to what matters to you.
Why do some formulas divide by n minus 1?
When you’re working from a sample instead of everyone, dividing by one less than the count gives a slightly bigger, fairer estimate of the spread in the whole group. Spreadsheets offer both: STDEV.S for a sample and STDEV.P for everyone.
Is standard deviation the same as variance?
They’re close relatives. Variance is the average squared distance from the mean, and standard deviation is its square root. Standard deviation is easier to read because it’s in the same units as your data.
Learn it by doing it
Module 8, “Spread, Shape and Percentiles,” teaches spread and percentiles in four five-minute lessons.
Data to AI is five-minute lessons on how AI works, starting with the data on your phone, for curious teens, parents and grandparents. No math needed. The first lesson runs right here in your browser.