C9.3 Averages and range

Learning objectives

C9.3.1Calculate the mean, median, mode and range for individual data and• Calculate the mean, median, mode and range for individual data and distinguish between the purposes for which these are used. Data may be in a list or frequency table, but will not be grouped.

Mean, median and mode describe a typical or central value; range describes spread. The calculation and the purpose of each measure are different.

Measure	How to find it	When it is useful
Mean	add all values, then divide by how many values	uses every numerical value; sensitive to extreme values
Median	order the data and find the middle; average the two middle values if needed	gives a central value that is less affected by extremes
Mode	identify the most frequent value or category	shows what occurs most often; suitable for categorical data
Range	maximum minus minimum	measures the total spread, not a typical value

For the ordered data $3,4,4,6,8$ , the total is $25$ . Therefore the mean is $25\div5=5$ , the median is $4$ , the mode is $4$ , and the range is $8-3=5$ . Always order a list before locating its median.

Value $x$	2	3	5
Frequency $f$	1	3	2
Product $fx$	2	9	10

For this ungrouped frequency table, $\sum f=6$ and $\sum fx=21$ , so the mean is $21\div6=3.5$ . The ordered positions are $2,3,3,3,5,5$ , so the median and mode are both $3$ , and the range is $5-2=3$ . Frequencies count repeated observations; do not average the headings alone.

Choose the measure to match the question: use the median when an extreme value would distort the mean; use the mode for the most common category; use the range only to describe spread. A data set can have no mode or more than one mode. This objective covers lists and ungrouped frequency tables, not grouped-data estimates.

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