The variance is the arithmetic mean of the squared deviations from the mean of a statistical distribution.
The variance is denoted by .
Variance for Grouped Data
To simplify the calculation of the variance, use the following expressions which are equivalent to the formulas above:
Calculate the variance of the following distribution:
9, 3, 8, 8, 9, 8, 9, 18
Calculate the variance of the distribution of the following table:
|xi||fi||xi · fi||xi2 · fi|
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:
If the samples have different size:
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.