Title: Measures of spread
Course: UGQuants (SSPS) > Section: Univariate Analysis

Measures of spread

While measures of central tendency – mean, median and mode – offer us a very simple summary or ‘model’ of a particular variable, a clear weakness is that this simplicity is at the expense of detail: they do not reflect the complexity and nuances of the variable. For nominal and ordinal variables we can largely overcome this simply by presenting a frequency table that shows the distribution of all values for the variable. However, for variables measured at an interval or ratio level, which often have a great many possible values, presenting a table of these values is not very useful. For example, a list of the distribution of personal income for hundreds or perhaps thousands of individuals would be hopelessly large and inaccessible, and not very informative since most values would only be taken by a very small number of cases.

But an advantage of ratio or interval variables is that we have many other ways of representing the data beyond simply stating central tendency. One of these involves measuring the spread of the data. These measures give us a much more detailed picture of the distribution of the data for the variable (or the ‘shape’ of the data). Another way to think of them is that they give us an indication of the adequacy of our measures of central tendency as summaries or ‘models’ of our data. They show how good a ‘fit’ the mean is for our data as a whole. Broadly speaking, the more widely spread our data (the greater variety or heterogeneity within them) the less good a summary or ‘fit’ the mean, e.g., is likely to be.

Animal | Height in Metres
Cat | 0.22 | 0.24 | 0.25 | 0.25 | 0.27
Dog | 0.09 | 0.10 | 0.25 | 0.25 | 0.54

Images courtesy of Vlado/freedigitalphotos.net

Both have exactly the same measures of central tendency (work these out yourself!):

Mean = 24.6
Median = 0.25
Mode = 0.25

But clearly the heights of the dogs is much more varied and ‘spread out’!

Two common and closely related ways to measure spread are the variance and the standard deviation. It should be noted that, like the mean itself, these measures do not only provide a useful way of understanding the detail of data for one variable, but are also fundamental to many more complicated approaches to analysing data.