Building Bridges
Hey Diddle Diddle the median’s the middle.
You add and divide for the mean.
The mode is the one that you see the most,
And the range is the difference between.
Moving from the process skills of calculating these summary statistics to deeper contextual understanding of which to use, and when, and why, is often tricky for pupils. The problem in part is one of memory: quite often you will hear a pupil say “medium” instead of “median” and then muddle the other two. This alliterative nightmare continues right up to the GCSE exam for some. One response is to teach each element separately and as far apart in the curriculum as you can manage, to try to prevent pupils from muddling the meaning of mean, median and mode. This strategy is explored in more detail in this blog by Bruno Reddy.
But keeping the topics apart in the curriculum and teaching the processes is not, of course, “job done”. When pupils can calculate mean, median and mode, do they have the full understanding of which one to use and how to interpret the value(s) they calculate? Could they make a decision about which is the best statistic to apply, given the scenario, and why? This diagnostic question about the use of the mean average could easily be adapted to cover the other three summary statistics:
Ten pupils take part in some races on Sports Day, and the following times are recorded. For each race, do you think that the mean average of the times would give a useful summary of the ten individual times? Explain your decision.
 Time to run 100m (seconds): 23, 21, 21, 20, 21, 22, 24, 23, 22, 20.
 Time to run 100m holding an egg and spoon (seconds): 45, 47, 49, 43, 44, 46, 78, 46, 44, 48.
 Time to run 100m in a threelegged race (seconds): 50, 83, 79, 48, 53, 52, 85, 81, 49, 84.
In Eight Effective Principles of Mathematics Teaching, Malcolm Swan demonstrates the importance, and impact, of questioning. A powerful technique is to turn the questions around, from simple closed question to “Show me” open questions: rather than ask “What is the mean, median and mode of 5, 4, 5, 6, 5?”, ask instead “If the mean of a set of numbers is 5 what might the numbers be?” and then “If the mode is 4 and the mean is still 5 what might the numbers be?”. Or, “If the mean, median and mode of a set of six numbers is 6, what might the numbers be?”
“Always, sometimes, never” statements challenge and deepen pupils’ thinking and reasoning. The examples here are from NRICH:
 The mean, median and mode of a set of numbers can't all be the same.
 The mean cannot be less than both the median and the mode.
 Half of the students taking a test will score less than the average mark.
 Nobody scores higher than the average mark in a test.
 In a game where you can only score an even number of points (0, 2, 10 or 50), the average score over a series of games must be an even number.
Alongside ensuring that pupils can calculate summary statistics fluently, you can deepen their understanding using some of the following resources:
 the Standards Unit Mostly Statistics materials S4 Understanding Mean, Median, Mode and Range
 the followon unit is Interpreting Bar charts, Pie charts, Box and whisker plots, to help pupils develop further the connections between raw data and its visual representation and analysis;
 Learn and WinAtSchool offers a range of resources as well as an annual competition. The activity The Averages is an enjoyable challenge;
 within the Durham Maths Mysteries there is the excellent Ratio and Proportion Mystery, which draws on knowledge of the mean average;
 and as the end of term approaches…everyone loves a game of Top Trumps!
Image credits
Page header by Alan Levine (adapted), some rights reserved
