The population variance \(\sigma^2\) (pronounced sigma squared) of a discrete set of numbers is expressed by the following formula: 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 standard deviation of a normal distribution enables us to calculate confidence intervals. Therefore, if all values of a dataset are the same, the standard deviation and variance are zero. The smaller the variance and standard deviation, the more the mean value is indicative of the whole dataset. Where a dataset is more dispersed, values are spread further away from the mean, leading to a larger variance and standard deviation. In datasets with a small spread all values are very close to the mean, resulting in a small variance and standard deviation. They summarise how close each observed data value is to the mean value. The variance and the standard deviation are measures of the spread of the data around the mean.
