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Variance Calculator

Variance is the average squared distance from the mean. For 2, 4, 4, 4, 5, 5, 7 and 9 the mean is 5 and the squared deviations add up to 32. The population variance is 32 ÷ 8 = 4; the sample variance divides by n − 1 instead: 32 ÷ 7 = 4.57. The standard deviation is the square root: 2 or 2.14.

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Results

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Variance (sample)
4.5714
Standard deviation2.1381
Mean5
Count8
Sum of squares Σ(x − mean)²32
Variance (population), for comparison4
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Working
  1. Mean = 40 ÷ 8 = 5
  2. Squared deviations: (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
  3. Sum of squares = 32
  4. Variance = 32 ÷ (n − 1) = 32 ÷ 7 = 4.5714
  5. Standard deviation = √4.5714 = 2.1381
Results are estimates for planning and material ordering only. They are not a substitute for structural design or approval by a qualified engineer. Read the full disclaimer.
Formula


mean = Σx ÷ n
Sum of squares SS = Σ(x − mean)²
Population variance σ² = SS ÷ n
Sample variance s² = SS ÷ (n − 1)
Standard deviation = √variance


Worked Example


2, 4, 4, 4, 5, 5, 7, 9:


    

  1. Mean = 40 ÷ 8 = 5
    2. 

  2. Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16; SS = 32
    3. 

  3. Population variance = 32 ÷ 8 = 4; sample variance = 32 ÷ 7 = 4.571
    4. 

  4. Standard deviations: 2 and 2.138
    5. 



Sample or Population?






Your data
Divide by
Example






Part of a larger group
n − 1 (sample)
6 test cubes from a pour of 80 m³




Every value there is
n (population)
The final level of every pile in a project






Frequently Asked Questions
How do you calculate variance?
Find the mean, subtract it from each value, square each result and add them up. Divide by n for the population variance or by n − 1 for the sample variance.
What is the difference between sample and population variance?
Population variance divides by n and is used when you have every value. Sample variance divides by n − 1, which corrects for estimating the spread of a larger group from part of it.
How are variance and standard deviation related?
Standard deviation is the square root of the variance. It is in the same units as the data, so it is easier to read: a variance of 4 MPa² is a standard deviation of 2 MPa.
Can variance be negative?
No. It is a sum of squares, so it is zero when all values are equal and positive otherwise.
Why square the deviations?
So that values above and below the mean do not cancel out, and so bigger deviations count for more.
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