Measures of spread

The variance and the standard deviation are measures of the spread of the data around the mean.  They summarise how close each observed data value is to the mean value. 

In datasets with a small spread all values are very close to the mean, resulting in a small variance and standard deviation. Where a dataset is more dispersed, values are spread further away from the mean, leading to a larger variance and standard deviation.

The smaller the variance and standard deviation, the more the mean value is indicative of the whole dataset. Therefore, if all values of a dataset are the same, the standard deviation and variance are zero.

The standard deviation of a normal distribution enables us to calculate confidence intervals.  In a normal distribution, about 68% of the values are within one standard deviation either side of the mean and about 95% of the scores are within two standard deviations of the mean.

The population variance \(\sigma^2\) (pronounced sigma squared) of a discrete set of numbers is expressed by the following formula:

\(\sigma^2 = \frac{\displaystyle\sum_{i=1}^{N} (X_i-\mu)^2}{N}\)

where:
\(X_i\) represents the ith unit, starting from the first observation to the last
\(\mu\) represents the population mean
\(N\) represents the number of units in the population

The variance of a sample \(s^2\) (pronounced s squared) is expressed by a slightly different formula:

\(s^2 = \frac{\displaystyle\sum_{i=1}^{n} (x_i-\bar{x})^2}{n-1}\)

where:
\(x_i\) represents the ith unit, starting from the first observation to the last
\(\bar{x}\) represents the sample mean
\(n\) represents the number of units in the sample

The standard deviation is the square root of the variance. The standard deviation for a population is represented by \(\sigma\), and the standard deviation for a sample is represented by \(s\).

Calculating the population variance \(\sigma^2\) and standard deviation \(\sigma\)

Dataset A

Calculate the population mean \((\mu)\) of Dataset A.
(4 + 5 + 5 + 5 + 6 + 6 + 6 + 6 + 7 + 7 + 7 + 8) / 12
mean \((\mu)\) = 6

Calculate the deviation of the individual values from the mean by subtracting the mean from each value in the dataset
\(X_i - \mu\) = -2, -1, -1, -1, 0, 0, 0, 0, 1, 1, 1, 2

Square each individual deviation value
\((X_i - \mu)^2\) = 4, 1, 1, 1, 0, 0, 0, 0, 1,1,1, 4

Calculate the mean of the squared deviation values
\(\frac{\displaystyle\sum_{i=1}^{N} (X_i-\mu)^2}{N}\) =
(4 + 1 +1 +1 + 0 + 0 + 0 + 0 +1 +1 +1 + 4) / 12

Variance \(\sigma^2\)= 1.17

Calculate the square root of the variance

Standard deviation \(\sigma\) = 1.08

Dataset B

Calculate the population mean \((\mu)\) of Dataset B.
(1 + 2 + 3 + 4 + 5 + 6 + 6 + 7 + 8 + 9 + 10 + 11) / 12
mean \((\mu)\) = 6

Calculate the deviation of the individual values from the mean by subtracting the mean from each value in the dataset
\(X_i - \mu\) = -5, -4, -3, -2, -1, 0, 0, 1, 2, 3, 4, 5,

Square each individual deviation value
\((X_i - \mu)^2\) = 25, 16, 9, 4, 1, 0, 0, 1, 4, 9, 16, 25

Calculate the mean of the squared deviation values
\(\frac{\displaystyle\sum_{i=1}^{N} (X_i-\mu)^2}{N}\) =
(25 + 16 + 9 + 4 + 1 + 0 + 0 + 1 + 4 + 9 + 16 + 25) / 12

Variance \(\sigma^2\) = 9.17

Calculate the square root of the variance

Standard deviation \(\sigma\) = 3.03

The larger variance and standard deviation in Dataset B further demonstrates that Dataset B is more dispersed than Dataset A.