Mean, median, mode and range, and comparing rates

Last time

Before we start: last time, in Mean and standard deviation of grouped data, and combining groups, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

One number for the data

An average sums up a set of data in one number, and the range shows how spread out it is. This lesson covers the mean, median and mode, the range, and finding missing values, and how to compare rates fairly.

The three averages

The mean is the total of the values divided by how many there are. The median is the middle value when the data are in order; the mode is the most common value. The range is the largest value minus the smallest.

A worked set

The total is 42 and there are 6 values, so the mean is 7. With 6 values, the median is halfway between the 3rd and the 4th: 5 and 8 give 6.5. The mode is 5, and the range is 12 minus 3, which is 9.

Which average?

The mean uses every value, but one extreme value can pull it a long way. The median is not affected by extreme values, so it suits skewed data such as incomes. The mode is the only average for qualitative data, but there may be more than one mode, or none.

The range

The range is quick to find, but it depends only on the two most extreme values. A single unusual value makes the range large, so the interquartile range is often a better measure of spread. A smaller range means the values are more consistent.

Missing values

If you know the mean, you know the total: the mean times the number of values. The four values 6, 9, 11 and x have a mean of 10. The total must be 40, so x is 40 minus 26, which is 14. If the range is given, a missing value is the smallest value plus the range, or the largest minus it.

Mean from a frequency table

For a frequency table, the mean is the sum of f times x, divided by the sum of f. Values 0, 1, 2 and 3 with frequencies 4, 6, 7 and 3: Σf is 20, and Σfx is 0 plus 6 plus 14 plus 9, which is 29. So the mean is 29 over 20, which is 1.45.

Comparing rates

Totals mislead when the amount of opportunity differs, so compare rates instead. A scoring rate per 1,000 minutes is the number of goals divided by the minutes played, times 1,000. 6 goals in 1,500 minutes is a rate of 4 per 1,000 minutes: better than 7 goals in 2,000 minutes, a rate of 3.5.

Mistakes that lose marks

Mistake one: finding the median without first putting the values in order. Mistake two: dividing Σfx by the number of different values instead of by Σf. Mistake three: comparing totals when the time or the population differs. The fix: order, Σf, and rates.

The full-marks tip

The full-marks tip: when choosing an average, say why it suits this data, such as an extreme value. When comparing two sets, compare one average and one measure of spread, in context.

On one card

So, on one card: averages and range. Keep it for revision.