Standard Deviation Calculator

Standard deviation measures how spread out data is around its mean. A small value means the data is close together; a large value means it is spread out. This calculator finds the standard deviation and variance for a whole population or for a sample, showing each squared deviation.

Try:

How to use the standard deviation calculator

  1. Type or paste your data values.
  2. Choose population (all the data) or sample (part of it).
  3. Press Calculate to see the standard deviation, variance and steps.

Formula

σ=∑(xi−μ)2N\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}
s=∑(xi−xˉ)2n−1s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}

Worked example: 2, 4, 4, 4, 5, 5, 7, 9 (population)

  • Data values: 2, 4, 4, 4, 5, 5, 7, 9
  • Data type: Population (all data)

✓ Answer checked

σ≈2\sigma \approx 2
Variance
σ2=4\sigma^2 = 4
Mean
55
Count
8
Sum of squared deviations
3232
−3−2−1012341: -312: -123: -134: -145: 056: 067: 278: 48
Each bar is a value's deviation from the mean (5). Standard deviation measures the typical size of these bars.
Step-by-step working (5 steps)

  1. Find the mean

    Add the 8 values and divide by 8.

    xˉ=5\bar{x} = 5

  2. Find each deviation and square it

    Subtract the mean from each value, then square the result.

    (2−5)2=9(4−5)2=1(4−5)2=1(4−5)2=1(5−5)2=0(5−5)2=0(7−5)2=4(9−5)2=16\begin{gathered}(2 - 5)^2 = 9\\(4 - 5)^2 = 1\\(4 - 5)^2 = 1\\(4 - 5)^2 = 1\\(5 - 5)^2 = 0\\(5 - 5)^2 = 0\\(7 - 5)^2 = 4\\(9 - 5)^2 = 16\end{gathered}

  3. Add the squares

    This is the sum of squared deviations.

    ∑(xi−xˉ)2=32\sum (x_i - \bar{x})^2 = 32

  4. Divide by N Variance

    For a whole population, divide by the count.

    σ2=328=4\sigma^2 = \frac{32}{8} = 4

  5. Take the square root

    The standard deviation is the square root of the variance.

    σ=4≈2\sigma = \sqrt{4} \approx 2
Formulas used
σ=∑(xi−μ)2N\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}
How this was checked
  • ✓ A different one-pass method (Welford’s algorithm) gives the same standard deviation.

Treated as a whole population (divides by N). Decimal rounded to 10 significant figures.

Frequently asked questions

What is the difference between sample and population standard deviation?

Population standard deviation divides by N, the number of values. Sample standard deviation divides by n − 1, which corrects for estimating from part of the data.

How do I calculate standard deviation?

Find the mean, subtract it from each value, square the results, add them, divide by N (or n − 1), and take the square root.

What is variance?

Variance is the standard deviation squared: the average of the squared deviations from the mean.

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