Standard Deviation Calculator
Enter a list of numbers to get the standard deviation and variance, either for a sample or for a whole population, with a table of the squared deviations used in the calculation.
Result
- Sample variance (s²)
- 4.571429
- Mean
- 5
- Count (n)
- 8
- Sum of squared deviations
- 32
- Standard error of the mean
- 0.755929
- Population SD for comparison
- 2
| x | x − mean | (x − mean)² |
|---|---|---|
| 2 | -3 | 9 |
| 4 | -1 | 1 |
| 4 | -1 | 1 |
| 4 | -1 | 1 |
| 5 | 0 | 0 |
| 5 | 0 | 0 |
| 7 | 2 | 4 |
| 9 | 4 | 16 |
What is standard deviation?
Standard deviation measures how spread out a set of numbers is around its mean. A low value means the values sit close to the average; a high value means they are scattered. Because it is expressed in the same units as the data, it is the most widely used measure of dispersion in science, finance and quality control.
Formulas
Population standard deviation: σ = √[ Σ(xᵢ − μ)² ÷ N ]
Sample standard deviation: s = √[ Σ(xᵢ − x̄)² ÷ (n − 1) ]
The only difference is the divisor. Dividing by n − 1 (Bessel's correction) makes the sample version an unbiased estimate of the population variance. Variance is the quantity under the square root.
Worked example with 2, 4, 4, 4, 5, 5, 7, 9. The mean is 40 ÷ 8 = 5. The deviations are −3, −1, −1, −1, 0, 0, 2, 4 and their squares are 9, 1, 1, 1, 0, 0, 4, 16, which sum to 32. As a sample, the variance is 32 ÷ 7 ≈ 4.571 and the standard deviation is √4.571 ≈ 2.138. Treated as a population, the variance is 32 ÷ 8 = 4 and the standard deviation is exactly 2.
How to use this calculator
- Paste your numbers into the box; any mix of commas, spaces or line breaks works.
- Choose sample if the data is a subset of a larger group, or population if it is the whole group.
- Read the standard deviation, variance and mean. The table underneath lists each value's deviation so you can follow the arithmetic.
Interpreting the result
- Compare the standard deviation with the mean. A SD of 2 is large if the mean is 5 but tiny if the mean is 5,000.
- For bell-shaped data, roughly 68% of values fall within ±1 SD of the mean and 95% within ±2 SD.
- The standard error (SD ÷ √n) tells you how precisely the sample mean estimates the population mean; it shrinks as you collect more data.
For the mean, median and mode on their own, use the average calculator.
Frequently asked questions
Should I use sample or population standard deviation?
Use population if your numbers are the entire group you care about, such as the test scores of every student in one class. Use sample if the numbers are a subset meant to represent a larger group. The sample version divides by n − 1, which corrects for the fact that a sample tends to underestimate the spread.
What does the standard deviation actually tell me?
It is the typical distance between a value and the mean, in the same units as the data. For roughly bell-shaped data about 68% of values lie within one standard deviation of the mean and about 95% within two.
What is the difference between variance and standard deviation?
Variance is the average of the squared deviations, so its units are squared (dollars² or cm²). Standard deviation is the square root of the variance, which brings the measure back to the original units and makes it easier to interpret.
Can the standard deviation be zero or negative?
It is zero only when every value is identical. It can never be negative, because it is a square root of a sum of squares. A very small value simply means the data is tightly clustered around the mean.