3.2 Appropriate use of mean, median and mode

## 3.2 Appropriate use of mean, median and mode

It seems that it can be relatively easy to work out the mode, median and mean. But why would anyone want to know all or any of these values? Because all of these can tell us something about a set of observations.

In general, the mode is not used very often. It is not a measure of the centre of the data in the same way that the median and mean can be. However, the mode is the only measure of centre appropriate for nominal data. For example, if we were looking at the most frequently purchased food item in a certain snack bar in 1999, it does not make sense to talk about the median food or the mean food but it does make sense to say that the most frequent (modal) food purchased was (shall we say) mini chocolate bars.

Sometimes there might be more than one mode in a set of data – it is possible that the most popular food items purchased were both lamingtons and mini chocolate bars.

The median is at the middle of an ordered (ranked) data set and is a useful measure for ordinal variables.

Strictly speaking, the mean only makes sense for interval and ratio scales of measurement. However, there is a tendency to calculate means for ordinal variables as well. The calculation of a mean for ordinal scales of measure assumes that the interval between the rankings is the same between each ranking [16]. But, can we be sure that my idea of the interval between 'agree' and 'strongly agree', for example, is the same as yours? An example of this type of scale is rating scales. These scales are divided into intervals and usually numbered similar to the method used in the example below.

However, although the scale implies that the intervals are equal, they are not intrinsically equal.

Use of mean and median compared

Sometimes the mean is thought of as an economic measure and the median as a social measure. For example, the mean income of a group of people might be of more interest to retailers and the tax office, but the median income might be of more interest to welfare organisations. In the calculation of the mean, these groups are interested in the total amount that is available (in this case it is money) but welfare groups are interested in the number of units that this amount is distributed amongst.

In real estate usage, the mean would be used to describe the average value of a portfolio of houses being offered for sale by a real estate agent. However, someone wanting to buy a home from that real estate agent would use the median or middle house value. This is because the median does not alter when there are extreme values (outliers) in a data set.

To illustrate:
Set A: 30 40 50 60 70 has mean = 50 and median = 50
Set B; 30 40 50 60 700 has mean = 176 and median = 50

The value 700 is an outlier because it is a long way from the nearest data value, 60. On the other hand, 70 is not a long way from 60.

 SCENARIO

Last year, a fast food outlet in a beachside city paid 3 kitchen hands \$16000 per year, 2 supervisors \$22000 and the owner \$85000.

The mean salary at this business was \$29500 and the median was \$19000 (the mean of the two centre observations in an ordered list). The mode was \$16000.

Explain why the mean is higher than the median.

### Question

The mean is higher than the median because theis affected by theof everyso a high income of \$85,000 affects the. The relative position of measurements affects theand \$85,000 has the same effect on theas \$22,000.