Averages

Mean, median and mode

Mean

The mean is the most commonly used ‘average’ and the one often referred to in everyday conversation, eg the average wage. The mean is usually used when the data involved is fairly evenly spread, and there are no exceptional cases that are much higher or lower than the rest. If a few exceptional results exist, the mean may give a misleading impression, because it takes account of all the data given. It is found by adding together all the data values in a set of data and then dividing this total by the number of values in the set.

Example

In a test a group of 11 pupils scored the following marks:

5, 10, 3, 4, 4, 8, 4, 3, 11, 9, 5

The mean mark is found by adding together all the marks and dividing the total by the number of pupils.

(5+10+3+4+4+8+4+3+11+9+5)÷11

= 66 ÷ 11

= 6

Therefore the mean mark is 6.

Median

The median is the middle value of a set of data when placed in order, it can be found with little or no calculation. The median is particularly useful when the data has a wide spread, as the middle value is not affected by exceptional cases. To find the median age of a group of 11 children it is only necessary to arrange their ages in order and the age of the middle child (the sixth child) is the median. This means that for the median there are as many values in the data set above the median as there are below it. The median for any data can also be found by drawing a cumulative frequency graph. See the page on cumulative frequency for more information.

Example

The median mark is the middle value in the group of marks when arranged in order of size.

In order of size the marks are:

There are 11 numbers in this set of data. The sixth number is the middle value or median. In this example the median is 5.

Note: when there is an even number of values, the median is found half way between the two middle values.

For example, if the test results for a particular pupil are: 12, 13, 16, 17, 21 and 25, then 16 and 17 are the middle values. The median is the mean of the two values, or half way between the two values. That is (16 + 17)/2. The median in this example is 16.5.

Mode

The mode is the most frequently occurring value in a set of data. For example, shoe manufacturers are not interested in the mean or median value of shoe sizes but they may want to know the size most frequently sold.

Example

The mode is the value which occurs most often in the set of marks.

For the set of test marks:

5, 10, 3, 4, 4, 8, 4, 3, 11, 9, 5

The score which occurs most often is 4 with three pupils scoring 4 marks, so the modal mark, or mode, is 4.

Worked examples

Example one

The table below records the amount of time, rounded to the nearest quarter of an hour, each member of a class spends on homework in hours.

Time spent on homework during the week ending 24/1/2003

What is the mean amount of time pupils spend on homework?

To find the mean, first add all the values together. The answer is 30.

Then divide by the number of items, in this case, 15.

So the mean is 30/15 = 2.

Therefore the mean amount of time spent on homework during the week is two hours.

The teacher may also be interested in the most common amount of time spent by pupils, the mode, ie:

The amount of time pupils spent varies a good deal. However, four pupils spent 1½ hours.

Therefore the mode is 1 ½ hours.

Example two

A teacher sets a test for 20 pupils that is marked out of a possible total of 20 marks. The teacher wants to know the median mark for the test.

Ordering the test scores gives:

As there are 20 marks there will be two middle marks, the tenth and eleventh. These are: 15 marks and 14 marks.

The median is half way between them or the mean of the two marks.

So the median is 14.5 marks.

Example three

The set of data below is the GCSE business studies grade results for a group of 70 pupils.

Business studies GCSE results 1999

The teacher wants to know what the average performance is and chooses the mode as the most appropriate average, ie. the grade that the most pupils achieve.

To find the mode the data needs to be sorted. The number of times each grade occurs is counted and entered as below:

The grade that occurs most frequently is C. So C is the modal grade.

The teacher might also be interested in the average (mean) points score. If points are attributed to grades as follows:

The mean can be calculated by multiplying the points score by the number of pupils at each grade, finding the total and dividing by the number of pupils (70), ie:

(3x8 + 3x7 + 8x6 + 15x5 + 13x4 + 8x3 + 9x2 + 7x1 + 4 x 0) ÷ 70

= 269 ÷ 70

= 3.8 to 1 d.p.

The average (mean) points score is 3.8.

The nearest grade to 3.8 points is D.

This example shows that the mean and the mode do not necessarily give the same result. Both can be useful in this case, but are giving the teacher different information.

One tells the teacher the most common grade, while the other gives an average (mean) points score which might be useful for comparison with other subjects. The fact that the mean is less than the mode indicates that the result as a whole are likely to be spread out amongst the lower grades as shown in the second table above.

Avoiding common errors

Whenever the word ‘average’ is used, make sure you understand which average is being referred to in the context of the set of data: is it the mean, median or mode?

When finding the median, make sure that the data is arranged in ascending or descending order.

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