Pythagorist
Statistics

Standard Deviation Calculator

Find the standard deviation and variance of a data set. Choose between population and sample standard deviation — both shown with complete step-by-step working.

Mean

5.4

Variance

5.84

Std Dev (σ)

2.416609

How it works

  1. Calculate the mean

    xˉ=4+7+2+9+55=5.4\bar{x} = \frac{4 + 7 + 2 + 9 + 5}{5} = 5.4

    Mean = sum ÷ count = 27 ÷ 5 = 5.4.

    Result:5.4

  2. Find deviations from the mean

    Subtract the mean (5.4) from each value: -1.4, 1.6, -3.4, 3.6, -0.4.

  3. Square each deviation

    Squared deviations: 1.96, 2.56, 11.56, 12.96, 0.16.

  4. Calculate the variance (population)

    σ2=29.25=5.84\sigma^2 = \frac{29.2}{5} = 5.84

    Sum of squared deviations = 29.2. Divide by 5 (N for population) = 5.84.

    Result:5.84

  5. Take the square root for standard deviation

    σ=5.84=2.416609\sigma = \sqrt{5.84} = 2.416609

    Standard deviation = √variance = √5.84 = 2.416609.

    Result:2.416609

Standard Deviation

σ=∑(x−xˉ)2n\sigma = \sqrt{\frac{\sum(x - \bar{x})^2}{n}}
σσStandard deviation (population)
xxEach data value
xˉx̄Mean of the data
nnNumber of values (use n−1 for sample)

Example substitution

s=(2−5)2+(5−5)2+(8−5)23=6≈2.449s = \sqrt{\frac{(2-5)^2+(5-5)^2+(8-5)^2}{3}} = \sqrt{6} \approx 2.449

For {2, 5, 8}: mean = 5, sum of squared deviations = 18, variance = 6, SD ≈ 2.449.

Worked examples

Calculate the population standard deviation of: 2, 4, 4, 4, 5, 5, 7, 9.

  1. 1
    Find the mean: Sum = 40, n = 8. Mean = 40 ÷ 8 = 5
  2. 2
    Find deviations from mean: −3, −1, −1, −1, 0, 0, 2, 4
  3. 3
    Square each deviation: 9, 1, 1, 1, 0, 0, 4, 16
  4. 4
    Sum squared deviations and divide by n: Sum = 32. Variance = 32 ÷ 8 = 4
  5. 5
    Take the square root: SD = √4 = 2
Answer: Population standard deviation σ = 2

Frequently asked questions

What is standard deviation?+

Standard deviation measures how spread out data values are from the mean. A low SD indicates values clustered near the mean; a high SD indicates they are spread out.

What is the difference between population and sample standard deviation?+

Population SD divides by n; sample SD divides by n−1. Use sample SD when your data is a sample from a larger population.

How do you calculate standard deviation step by step?+

1. Find the mean. 2. Subtract mean from each value and square the result. 3. Find the average of these squared differences (variance). 4. Take the square root.

What does a high standard deviation mean?+

A high standard deviation means data is widely spread around the mean. A low standard deviation means data is clustered close to the mean.

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