Standard Deviation

The standard deviation is the square root of the variance.

The standard deviation is denoted by σ.

Standard Deviation FormulaStatistical Standard Deviation

Standard Deviation for Grouped Data

Standard Deviation for Grouped Data FormulaGrouped Data Standard Deviation

To simplify the calculation, use the following expressions that are equivalent to the formulas above:

Standard Deviation for Grouped Data FormulaStandard Deviation for Grouped Data

Standard Deviation for Grouped Data FormulaGrouped Data Standard Deviation


Examples

Calculate the standard deviation of the following distribution:

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

Standard Deviation Calculation

Standard Deviation Example

Calculate the standard deviation of the distribution for 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

Standard Deviation Operations

Standard Deviation Calcuations


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:

Standard Deviation Properties

If the samples have different size:

Standard Deviation Samples


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.