Question

Give an example of a real-world situation where we might need to know the mean AND standard deviation of a distribution. In other words, under what circumstances might we be concerned with...

1) the center of the distribution AND

2) the spread of the distribution (i.e., how far away data points are from the mean)

HINT: When might looking exclusively at the mean be misleading or uninformative?

Answer 1

In real life, it is quite possible to have two sets of observations with the same mean (average) but differing considerably in their measurements about the average.

Standard deviation is the measure of spread of the data about an average. Standard Deviation of a given set of observations is defined as the positive square root of the arithmetic mean of the squares of deviations of the observations from their arithmetic mean. It is an absolute measure of dispersion. It depends upon the unit of measurement.

IQ tests are made so that the median IQ is 100. This means that half the people would score less than 100, and half would score more.

But suppose we want to know what per cent of the population has an IQ less than 80. To answer this, we need to know the standard deviation of the scores. The standard deviation tells how much the data is spread out.

For an IQ test the standard deviation is 15. What this means is that:

About 68% of the population scores within 1 SD (15 pts) of the median. In other words, about 2/3rds of the population has an IQ between 85 and 115.

About 96% of the population scores within 2 SD's (30 pts) of the median. In other words, about 96% of the population has an IQ between 70 to 130. This also means that about 2% of the population scores above 130.

Another example: The range of daily maximum temperatures for cities near the coast is smaller than for cities inland. While two cities may each have the same average maximum temperature, the standard deviation of the daily maximum temperature for the coastal city will be less than that of the inland city as, on any particular day, the actual maximum temperature is more likely to be further from the average maximum temperature for the inland city than for the coastal one. If one was planning to move to a city with a temperate climate, one would be interested in the standard deviation.

Another example is in sport. In any sport, there will be teams that rate highly at some things and poorly at others. Chances are, the teams leading in the standings will not show such disparity, but will perform well in most categories. The lower the standard deviation of their ratings in each category, the more balanced and consistent the team. Teams with a higher standard deviation will likely be more unpredictable. For example, a team that is consistently bad in most categories will have a low standard deviation. A team that is consistently good in most categories will also have a low standard deviation. However, a team with a high standard deviation might be the type of team that scores a lot (strong offense) but also concedes a lot (weak defense), or, vice versa, that might have a poor offense but compensates by being difficult to score on.