Standard deviation of a list
Enter the numbers and choose the whole population or a sample.
- Count of values (n)8
- Mean5
- Population variance4
- Sample standard deviation (for comparison)2.1381
- Coefficient of variation40%
- Population standard deviation2
How to calculate standard deviation
The math takes four steps: find the mean, subtract the mean from each value, square the differences, and average those squares to get the variance. The square root of the variance is the standard deviation.
You can double-check the mean with the average calculator. When all values match, the standard deviation is zero because nothing spreads out.
Sample or population
Use the population version when the list holds every member of the group, like the scores of an entire class. Use the sample version when the list is a slice of a bigger group, like 30 customers pulled from a list of thousands. Dividing by n − 1 corrects the habit of a sample to understate the spread.
| Situation | Type | Divisor |
|---|---|---|
| Every score in one class | Population | n |
| A poll of some voters | Sample | n − 1 |
| All orders in a month | Population | n |
| Repeated lab measurements | Sample | n − 1 |
Worked examples
| Values | Mean | Population | Sample |
|---|---|---|---|
| 2, 4, 4, 4, 5, 5, 7, 9 | 5 | 2 | 2.1381 |
| 10, 20 | 15 | 5 | 7.0711 |
| 5, 5, 5 | 5 | 0 | 0 |
| 1, 2, 3, 4, 5 | 3 | 1.4142 | 1.5811 |
Step by step for 10 and 20: the mean is 15; the differences are −5 and +5; the squares are 25 and 25; the sum is 50. Population: 50 ÷ 2 = 25, root = 5. Sample: 50 ÷ 1 = 50, root = 7.0711.
How to read the result
A small standard deviation means values sit close to the mean. A large one means the data is scattered. The result comes in the same unit as the data, so a 2 lb spread around a 5 lb average is a lot, while a 2 lb spread around 500 lb is tiny. The coefficient of variation (standard deviation ÷ mean) compares spreads on different scales.
- For bell-shaped data, about 68% of values fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3 (the empirical rule).
- Extreme values pull the standard deviation up hard, because the differences are squared.
- Mixing sample and population in the same analysis gives different numbers, so note which one you used.
Frequently asked questions
How do you calculate standard deviation?
Find the mean, square each value's distance from the mean, average those squares to get the variance, and take the square root. For 10 and 20, the population standard deviation is 5.
What is the difference between sample and population standard deviation?
The population version divides the sum of squares by n and applies to the whole group. The sample version divides by n − 1 and applies to a sample. For 10 and 20, they are 5 and 7.0711.
What is the difference between variance and standard deviation?
Variance is the average of the squared differences from the mean, in squared units. Standard deviation is the square root of the variance, in the original unit of the data.
What does a high or low standard deviation mean?
A low standard deviation means values cluster near the mean, and a high one means the data is spread out. Judge the size against the mean with the coefficient of variation.
Which Excel function is standard deviation?
STDEV.P gives the population version and STDEV.S gives the sample version. The older STDEV function matches the sample version. Google Sheets uses the same names, and the results here match them.
Can you find standard deviation with only one value?
For a population, the answer is zero. For a sample, it does not exist, since the divisor would be n − 1 = 0. You need at least 2 values.
Sources and method
- Descriptive statistics: variance and standard deviation, population and sample