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:
x_{i} | f_{i} | x_{i} · f_{i} | x_{i}^{2} · f_{i} | |
---|---|---|---|---|
[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 |
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.