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Sample Variance Formula and Calculation by Hand

Sample variance is the measure of the variability in a given sample. A sample is a set of observations that are a subset of a population.

Sample Variance Formula

The following is the formula for sample variance.

s2=ni=1(xiˉx)2n1

where,

  • n is the number of observations in the sample.
  • xi is the ith observation.
  • ˉx is the mean of the obervations.

Data Example

The following is a sample of 6 students with math scores.

Data Example of Sample Variance
Data Example of Sample Variance

Calculating sample variance by hand

Step 1: the mean

The following calculates the mean of the sample. The numerator is the sum of all observations, and the denominator is the number of the sample.

ˉx=80+90+81+82+78+716=80.33

Step 2: sample variance

The following is to calculate the sample variance. Note that, the denominator is 6-1, rather than 6.

s2=(8080.33)2+(9080.33)2+(8180.33)2+(8280.33)2+(7880.33)2+(7180.33)261=37.87

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