# Variance

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:  #### Examples

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 [10, 20) 15 1 15 225 25 8 200 5,000 35 10 350 12,250 45 9 405 18,225 55 8 440 24,200 65 4 260 16,900 75 2 150 11,250 42 1,820 88,050  # 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.