Standard deviation measures how spread out numbers are around their mean. A small value means the data is bunched together, and a large one means it is spread wide.
Try it with your own problem ↓
The steps
- Find the mean of the data.
- Subtract the mean from every value to get each deviation.
- Square every deviation, so negatives don't cancel positives.
- Add the squares, then divide: by n for a whole population, or by n − 1 for a sample. This is the variance.
- Take the square root of the variance.
A worked example
Find the standard deviation of 2, 4, 6, 8.
- Mean: (2 + 4 + 6 + 8) ÷ 4 = 5.
- Deviations: −3, −1, 1, 3.
- Squares: 9, 1, 1, 9. Their sum is 20.
- Population variance: 20 ÷ 4 = 5. Standard deviation: √5 ≈ 2.236.
- Sample variance: 20 ÷ 3 ≈ 6.667. Standard deviation: √6.667 ≈ 2.582.
Population standard deviation = √5 ≈ 2.236. Sample standard deviation = ≈ 2.582.
Try it with your own problem
Open the free The Dot Plot with this example already filled in. It shows every step, and you can type over it with your own numbers.
Work it out: standard deviation of 2 4 6 8Opens The Dot Plot with this example
Or try these
- A different set: 4, 8, 6, 5, 7
- Mean absolute deviation: 2 4 6 8
- Mean and spread of 10, 12, 23, 23, 16, 23, 21, 16
Common mistakes
- Forgetting to square. The deviations always add to 0, so you can't use them as they are.
- Forgetting the square root at the end. The variance is in squared units.
- Using n when you should use n − 1. If the data is a sample from a bigger group, divide by n − 1.
- Using the wrong mean. Double-check the mean first, since every later step depends on it.
Questions people ask
What is the difference between population and sample standard deviation?
Use the population one when your data is every member of the group. Use the sample one (dividing by n − 1) when your data is a sample used to estimate a larger group.
What is a "good" standard deviation?
There isn't a good or bad value. It only means something compared to the mean and to other data in the same units.
What does the 68-95-99.7 rule say?
For bell-shaped data, about 68% of values fall within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3.
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