Title: Standard deviation
Course: UGQuants (SSPS) > Section: Univariate Analysis

Standard deviation

Standard deviation may be understood in exactly the same way as variance – i.e. as representing the ‘average’ extent to which all values for a particular variable differ from the mean value for that variable. It is also calculated in exactly the same way as is variance, save that the formula for standard deviation corrects for the initial squaring of residuals and thus represents a more intuitive representation of ‘average’ spread of values around the mean value. The formula is therefore:

√[sum of (mean value – case value)^2 / number of cases]

Again, as with variance, for a sample rather than a population, standard deviation is calculated by dividing by the number of cases minus one (n – 1), and this is the formula that is used as standard in software such as SPSS.

Some examples of standard deviation are as follows:

Note that although described as ‘standard’ deviation, it is always expressed in the units by which the particular variable is measured (in the examples above: years, hours), whereas often when we talk of ‘standardization’ in quantitative analysis we refer to procedures that adjust for different units of measurement. The comparative potential of standard deviation is thus limited: we can directly compare the standard deviations of two variables measured in the same units, or for the same variable across different groups or populations, but it is much less useful to compare, e.g., the standard deviation of income in pounds with the standard deviation of age in years.These figures provide an immediate indication of how good a ‘model’ or summary the mean provides of adult ages in the UK; how much the time that people spend on paid employment varies; and the ages to which people might expect to live across the world.

However, the value and significance of standard deviation is not as limited and abstract as it might first appear. It is actually a very important concept in inferential statistics.