# 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:

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 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.