Calculation of Standard Deviation
Follow the steps below to calculate the standard deviation.
1. Find the mean.
2. Find the difference between each data item and the mean.
3. Square each of the differences.
4. Find the average (mean) of these squared differences.
5. Take the square root of this average.
The table shows the calculations for these data: 5, 8, 10, 14, and 18.
Below the table to the right, step 4 of the computation of standard deviation is broken down into
substeps: (4) adding the squares of the differences and (5) dividing by the number of data items.
1. Find the mean. 11
2. Find the difference between each data item and the mean. (See Table: Column 2)
3. Square each of the differences. (See Table: Column 3)
4. Find the average (mean) of these squared differences. (104/5 = 20.8)
5. Take the square root of this average. √20.8 ~ 4.6
x
5
8
10
14
18
x–x
–6
–3
1
3
7
(xi ! x )2
36
9
1
9
49
(4) The sum of data items = 55
The sum of data squared differences = 104
The number of data items = 5
The mean of the squared differences = 20.8
x = 55 ÷ 5 = 11
(5) σ (standard deviation) = √20.8 ~ 4.6
Standard deviation involves just two changes from Kai and Mai Spread Data in the formula for
mean absolute deviation:
 Replacing |xi – x| with (xi - x )2
 Taking the square root of the final expression
Thus, the formula for standard deviation is
The symbol usually used for standard deviation is the lowercase form of the Greek letter sigma
σ. Look for it on your calculators. Remember that the uppercase Greek sigma, Σ, is the
summation symbol.
Standard Deviation and the Normal Distribution
The following facts hold true whenever a set of data is normally distributed:
 Approximately 68% of all results will be within one standard deviation of the mean.
 Approximately 95% of all results will be within two standard deviations of the mean.
The areas shown in the next diagram illustrate these facts using
The darkly shaded area stretches from one standard deviation below the mean to one standard
deviation above the mean and is approximately 68% of the total area under the curve. The two
lightly shaded areas represent data between one and two standard deviations from the mean.
The total shaded area stretches from two standard deviations below the mean to two standard
deviations above the mean and comprises approximately 95% of the total area under the curve.
The standard deviation provides a good rule of thumb for deciding whether something is “rare.”