Title: Visualising the spread of data
Course: UGQuants (SSPS) > Section: Univariate Analysis

Visualising the spread of data

Measures of data spread such as standard deviation provide us with rather more information about ratio and interval level variables in addition to measures of central tendency such as the mean but they still exclude a lot of detail about the ‘shape’ of the data. Although we can use other calculations to produce different numerical representations of the shape of our data, as the term ‘shape’ implies, particularly for interval and ratio level variables with a relatively large number of possible values, presenting visual representations of data through graphs, figures and diagrams is very valuable too.

For example, data of this kind are often represented as a curved line. One way of doing this is to produce a histogram. The nature and mechanics of producing histograms is explained within the module on data visualisation, along with some examples. But it is useful at this point to reflect briefly on the significance of different ‘shapes’ of data as represented by curved lines. One way of describing the shape of a distribution is its skew.

Skewed and Normal distributions

Source: http://whatilearned.wikia.com/wiki/Describing_Data_Sets

The image at the top left of the figure above shows an example of a right- or positive skewed distribution. Large numbers of cases tend to take values towards the lower end of the range of values, with increasing smaller numbers of cases taking higher values. The longer ‘tail’ of the data is towards the right, towards the higher values. In positively skewed data the mode will be lower than the median which will in turn be lower than the mean. Income is a good real life example of data that is often positively skewed, because typically we find large numbers of people on low incomes and a small group on very high incomes. This is because there is an obvious limit or ‘floor’ below which income cannot fall (i.e. 0 income) but no upper limit to how much individuals can earn.

The image at the top right of the figure above shows an example of a left- or negative skewed distribution. Large numbers of cases tend to take values towards the upper end of the range of values, with increasing smaller numbers of cases taking lower values. The longer ‘tail’ of the data is towards the left, towards the lower values. The mode is therefore higher than the median, which is in turn higher than the mean. Age at retirement would be an example of this kind of skew, because most people now retire in their mid-60s but smaller numbers of people retire earlier and even smaller numbers retire when they are comparatively young.

The image at the bottom of the figure above (labelled ‘No Skew’) is in fact the most important of all. This is an example of a perfectly normal distribution represented by a perfectly symmetrical bell-shaped curve where the mode, median and mean coincide and there are increasingly fewer numbers of cases the further we move along the two ‘tails’ towards the extreme data points.

This is important partly because certain types of analysis assume that data for a variable are normally (or at least near normally) distributed. We encounter these in other parts of the course.

But even more importantly we can think of the normal distribution as the shape of randomness or chance. It is quite easy to prove to yourself that this is the case. Try taking a coin from your pocket, tossing it 10 times and recording how many ‘heads’ and how many ‘tails’ you get. Now repeat the exercise a few times or, better still, get your friends to do the same. You will find that the more times you do the exercise the more your results will resemble a normal distribution. The most likely random outcome (5 heads and 5 tails) will, if you have the stamina (or the quantity of friends!) to do the exercise a great many times, be the most common outcome, ‘tailing off’ towards the least likely outcomes (0 heads or tails).

But what relevance does coin-tossing have for social and political science? It provides an example of the fact that we can accurately predict the overall pattern of outcomes from a series of random events or ‘trials’. This in turn allows us to predict the probability of achieving a particular outcome.