Mean and Standard Deviation Calculator
Paste your numbers and the results update instantly. Switch between sample and population, and see every step of the work.
Your data
Answer
Results appear here once you enter at least one number.
- Standard deviation (s)
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- Mean (x̄)
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- Variance (s²)
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- Count (n)
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- Sum (Σx)
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- Sum of squares (SS)
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Solution: step by step
- Add the values and divide by the count to get the mean.
- Subtract the mean from each value, then square the result.
Deviations from the mean i xᵢ xᵢ − mean (xᵢ − mean)² - Add the squared differences to get the sum of squares.
- Divide by n − 1 to get the variance.
- Take the square root of the variance.
What standard deviation tells you
Standard deviation measures how spread out a data set is. A low value means the data points sit close to the mean. A high value means they are scattered widely around it. It is the square root of the variance, so it is in the same units as your data.
Use the sample option when your numbers are a subset of a larger group. Use the population option when you have every value in the group.
Formulas used
Mean
μ or x̄ = Σxᵢ ÷ nSum of squares
SS = Σ(xᵢ − mean)²Variance
Population: σ² = SS ÷ n Sample: s² = SS ÷ (n − 1)Standard deviation
Population: σ = √(SS ÷ n) Sample: s = √(SS ÷ (n − 1))Accepted data formats
All of these are read as the same list: 42, 54, 65, 47, 59, 40, 53.
| Type | Example input |
|---|---|
| Column (new lines) | 42 54 65 47 59 40 53 |
| Comma separated | 42, 54, 65, 47, 59, 40, 53 |
| Spaces | 42 54 65 47 59 40 53 |
| Mixed delimiters | 42 54 65,,, 47,,59, 40 53 |
Terms
- Sum
- The total of all values, x₁ + x₂ + … + xₙ.
- Count (n)
- The number of values in the data set.
- Mean
- The sum divided by the count. Also called the average.
- Sum of squares (SS)
- The sum of the squared differences between each value and the mean.
- Variance
- The sum of squares divided by n (population) or n − 1 (sample).
- Standard deviation
- The square root of the variance.
Frequently asked questions
What is the difference between sample and population standard deviation?
Population standard deviation divides the sum of squares by n. Sample standard deviation divides by n − 1, which corrects the bias that comes from estimating the mean from the same data.
How do you calculate standard deviation?
Find the mean, subtract it from each value, square the differences, add them to get the sum of squares, divide by n or n − 1 to get the variance, then take the square root.
What does a high standard deviation mean?
The values are spread far from the mean. A low standard deviation means they cluster close to it.
Why do I need at least two values for a sample?
The sample formula divides by n − 1. With one value that is zero, so the result is undefined.