1.6 Different types of averages

Statistics makes use of different types of averages. The most commonly used average value is the arithmetic mean, which is obtained by dividing the sum of all measurements by the number of observations.

The mean height of this group is:

However, the arithmetic mean cannot be used with all kinds of distributions, but only with a ratio scale (e.g. money, weight, height) or an interval scale (e.g. temperature, index).

If the subject of interest can only be measured using an ordinal scale (e.g. level of education), it is necessary to use the median instead. The median indicates the middle point of the data set so that one-half of the observations fall on either side of that point.

The median of this group is 130 cm. If there were an even number of individuals in the group, the median would be midway through the height difference between the two middle individuals in the group.

The median is also a useful average value for ratio scale distributions because it is less sensitive to extreme deviations than the arithmetic mean. This has practical implications in the case of income statistics, for example: the relatively small number of people with a high income push up the arithmetic mean so that the majority of income earners remain below the mean. Changes in income at the very highest level of the scale also have a major impact on the arithmetic mean. The median, on the other hand, lying as it does at the middle point of all observations, is unaffected by changes in the incomes of people in the highest income bracket.

But even the median cannot be used when the distribution reflects differences in quality. In this case we need to use the mode, i.e. the value that occurs with the greatest frequency. The mode can also be used as the average value in ratio scales and ordinal scales. The mode value for the observations in the material below is 130 cm.

Average values can also be computed from categorical data. To count the mean of a categorical data set, the values of the observations in each category are replaced by the class midpoint, i.e. the observations in each category get the exact same value. The class midpoint is then multiplied by the number of observations in the category, or by the class frequency,and the products are then added together. The sum is divided by the total number of observations. This yields the arithmetic mean of the classified material.