Choose the measure that tells the truth about the data. An outlier drags the mean, so the median is better for data with extreme values; the mean uses every value when there are none; the mode suits the most common choice.
One student gets 40 dollars a week; the rest get 5 to 8. 'The average is 12 dollars,' says the newspaper. Is that fair?
Mean, median and mode all describe a "typical" value, but they can disagree. Which is most useful depends on the data and the question:
An outlier (a value far from the rest) pulls the mean towards it; the median hardly moves.
With no outliers, the mean uses every value, so it is often the best summary.
The mode answers "which is most common?": the size a shop should stock most.
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
A fair average
How many students get more than the mean?
TryTry it on the board and compare the result with your prediction.
Remove the outlier
Then put a very large value back (type it in). Which measure jumps?
Try it yourselfYour edits are saved in this browser.
Click the dot at 40 to remove it. How much does the mean change? The median? Was your prediction right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Choosing a measure
Here is how to choose.
Worked exampleBefore each step, say what you think comes next.
5,6,6,7,8,40Look first: 40 is far from the rest, an outlier.
mean=672=12The mean uses every value, so the outlier pulls it up: above 5 of the 6.
median=26+7=6.5The median only uses the order: a typical value.
no outlier: 5,6,6,7,8Mean 6.4 and median 6 now agree: the mean is a fair summary.
most common?→modeFor "which is most popular?", use the mode.
Now choose a measure for the house prices.
Read Sam's reasoning. When is using every value a weakness?
Sam describes the street's house prices. Click the line where his reasoning goes wrong.
ModifyChange the problem on purpose and see what happens.
One change at a time
Each task changes one thing.
Need a hint?
16 is the outlier. Remove it and add a value nearer the rest.
Another hint
Try making the data symmetric around the middle.
Try it yourselfYour edits are saved in this browser.
Change the data (keep 5 values) so that the mean and the median are equal.
Use the board. This ticks when you get there.
PractiseA few on your own, to check you have it.
Choose and justify
Find the mean of 2,3,3,4,28, then the mean without the 28. How much does the 28 add to the mean? (Type the difference.)
Review: find the median of 9,4,11,6.
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
A misleading mean
Then write a sentence someone could use to mislead with your mean.
Try it yourselfYour edits are saved in this browser.
Need a hint?
Put four values close together.
Another hint
Make the fifth much bigger.
Make 5 values where the mean is at least 3 more than the median.
Use the board. This ticks when you get there.
For teachers
Students see an outlier drag the mean but not the median on a dot plot, and choose and justify the most useful measure (mean, median or mode) for different data and questions (AC9M7ST01, node 3). About 40 minutes.
I can explain how an outlier affects the mean and the median.
I can choose the most useful measure of centre for a data set and a question, and justify it.
Before class
Do the Run board first. Students need Mean, median, mode and range.
The one idea: look for outliers first. With one, the median describes a typical value better; without, the mean uses every value; for "most common", the mode. The common mistake is trusting the mean whatever the data; it is planted in Sam's reasoning about house prices.
There is often more than one defensible answer: reward the justification, not just the choice.
Running the lesson
Predict: ask how many students get more than the mean of 12 dollars.
Run: ask students to predict the new mean before removing the 40.
Investigate: link house prices to real median prices in the news.
Modify: collect different data sets that make the mean and median equal.
Make: read out some misleading sentences and ask how to correct them.
When students get stuck
A student always picks the mean (Sam's mistake). Ask: How many values are bigger than the mean? Fix: If most values are on one side of it, an outlier is pulling it: use the median.
A student cannot spot an outlier. Ask: Which dot is far from the others on the plot? Fix: Draw the dot plot; look for gaps.
Stretch and support
Stretch: Find a data set where the mean, median and mode are all different and each could be argued as 'typical'.
Support: Use the board for every question: draw the data, look at where the triangle and the dashed line sit.
Where it leads
Next: Stem-and-leaf plots (AC9M7ST02), then describing and comparing distributions.
Where it leads: skewed distributions and the five-number summary (Years 9 and 10).
What each step is for
a fair average: Committing to the best description of pocket money with one extreme value shows whether they default to the mean.
remove the outlier: Removing the outlier and watching the mean fall from 12 to 6.4 while the median goes from 6.5 to 6 shows which measure the outlier controls.
choosing a measure: Look for outliers first: with one, prefer the median; without, the mean; for the most common choice, the mode. Sam's mistake trusts the mean whatever the data.
one change at a time: One change at a time: make the mean and median agree on the board, scores with no outlier, adding a value equal to the mean, then the edge case of a mode that tells nothing.
choose and justify: Choosing a measure for new contexts and calculating the effect of an outlier, mixed; the review keeps calculating the median fresh.
a misleading mean: Building data where the mean is well above the median shows what an outlier (or a long tail) does, from the other side.
Now you can
Explain how an outlier affects the mean and the median.
Choose the most useful measure of centre for a data set and a question, and justify it.