Variance

The variance is the arithmetic mean of the squared deviations from the mean of a statistical distribution.

The variance is denoted by signo.

Variance FormulaStatistical Variance

Variance for Grouped Data

Variance for Grouped Data FormulaVariance for Grouped Data


To simplify the calculation of the variance, use the following expressions which are equivalent to the formulas above:

Variance for Grouped Data FormulaGrouped Data Variance

Examples

Calculate the variance of the following distribution:

9, 3, 8, 8, 9, 8, 9, 18

Variance Calculations

Statistical Variance Operations


Calculate the variance of the distribution of the following table:

  xi fi xi · fi xi2 · fi
[10, 20) 15 1 15 225
[20, 30) 25 8 200 5,000
[30,40) 35 10 350 12,250
[40, 50) 45 9 405 18,225
[50, 60 55 8 440 24,200
[60,70) 65 4 260 16,900
[70, 80) 75 2 150 11,250
    42 1,820 88,050

Variance of the Distribution

Variance Operations


Properties of the Variance

1 The variance is always positive or in the event that the values are equal, the variance is zero.

2 If all values of the variable are added by the same number, the variance does not change.

3 If all values of the variable are multiplied by the same number, the variance is multiplied by the square of that number.

4 If there are multiple distributions with the same mean and their variances are known, the total variance can be calculated.

If all samples have the same size:

Variance Properties

If the samples have different size:

Variance


Observations on the Variance

1 The variance, like the average, is an index sensitive to extreme scores.

2 In cases where the mean cannot be found, it will not be possible to find the variance.

3 The variance is not expressed in the same units as the data since the deviations are squared.