Exercise: Illustrating the mean, the median and the mode

Goal: to understand different types of averages exist, and know which is most appropriate to use 
Materials needed: pen and paper.
Estimated time: 5 minutes.
Average is a word that gets thrown about a lot. But what does it mean, really?  Does it mean the most representative figure? The most “middling” one? The most common one?  Median, mean, and mode are all types of average, but which is the “right” one to use in a given situation?
Pass around a piece of paper around the room and ask everyone to write down their age.  Once complete, run a quick calculation to work out the mean age.
"Is anyone exactly this age? If not, is it a fair summary?  For most of the students it will be a good approximation, but any lecturers present might disagree.  Explain how, if the older lecturer were to leave the room, the mean may come down markedly, but the same is not true for one of the students leaving the room  Thus we see that extreme values have a disproportionate affect on the mean."
Have the students gather into groups of the same age.  Explain that the largest group is the modal average, the most frequently occurring value.
"Is this an accurate summary? Ask how many new students would need to be removed or added to the room to shift this average.  What would happen if the groups were made more precise – e.g. age in months?  Most of the time, working out the mode will rely on putting continuous data into discrete bands, so how should we decide how wide those bands should be?"
Finally, have the students line up from youngest to oldest.  Select the person at the half way point to find the median age.
"Is this an accurate summary? Note how the oldest person in the room has no more effect on the average age than someone near the middle.  Thus the median is robust against small groups of extreme values, like we might see in income distribution."
Ask which average students would use.  What about other situations?  Suggest different distributions – continuous but skewed such as income, discontinuous such as eye colour or number of children, or very diverse distributions, such as “size of fruit”.
The take home point is that averages are a way of summing up a collection of  different values.  Without identifying and communicating that distribution, the average may not tell us very much at all.
Image CC Sharon Mollerus, Flickr

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