Standard Deviation
The standard deviation is the square root of the variance.
The standard deviation is denoted by σ.
Standard Deviation for Grouped Data
To simplify the calculation, use the following expressions that are equivalent to the formulas above:
Examples
Calculate the standard deviation of the following distribution:
9, 3, 8, 8, 9, 8, 9, 18
Calculate the standard deviation of the distribution for 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 Standard Deviation
1 The standard deviation is always positive or in the event that the values are equal, it is zero.
2 If all values of the variable are added by the same number the standard deviation does not change.
3 If all values of the variable are multiplied by the same number the standard deviation is multiplied by the square of that number.
4 If there are multiple distributions with the same mean and their standard deviations are known, the total standard deviation can be calcuated.
If all samples have the same size:
If the samples have different size:
Observations on the Standard Deviation
1 The standard deviation, like the mean and variance, is an index very sensitive to extreme scores.
2 In cases where the mean cannot be found, it is not possible to find the standard deviation.
3 The smaller the standard deviation is, the greater the concentration of data will be around the mean.