Four numbers that sum up a data set: the mean (add, then share equally), the median (the middle value once sorted), the mode (the most common) and the range (highest minus lowest).
A team scored 3, 5, 5, 6 and 11 goals. What is a typical game? There are at least three good answers, and they are all useful.
A data set can have many values. To describe it in a few numbers, statisticians use measures of centre (what is typical) and of spread (how varied):
the mean: add all the values, then divide by how many there are;
the median: sort the values; it is the middle one (or halfway between the two middle ones);
the mode: the value that appears most often;
the range: highest minus lowest, a simple measure of spread.
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 typical game
The mean shares the total equally. What is the total?
TryTry it on the board and compare the result with your prediction.
Move the mean
Watch the triangle (mean) and the dashed line (median) as you add values. Which moves more when you add a big score?
Try it yourselfYour edits are saved in this browser.
Add one more game so that the mean becomes exactly 7. Was your prediction right about the first mean?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
The four measures
Here is each measure for the goals data.
Worked exampleBefore each step, say what you think comes next.
3,5,5,6,11The data, sorted (the median and range need it sorted).
mean=53+5+5+6+11=530=6Add, then divide by how many.
median=5The middle (3rd of 5) value of the sorted list.
mode=5The most common value (it appears twice).
range=11−3=8Highest minus lowest.
8,3,9,4,6,7→3,4,6,7,8,9An even count: sort, then the median is halfway between the two middle values, 26+7=6.5.
Now find the median of 8,3,9,4,6,7.
Read Leo's working. What must you do before finding the middle?
Find the median of 8,3,9,4,6,7.
Leo finds a median. Click the line where his working goes wrong.
Find the mean of 4,8,9,3.
ModifyChange the problem on purpose and see what happens.
One change at a time
Each task changes one thing.
Need a hint?
The median of 2, 4, 6, 8, 10 is 6.
Another hint
Which value could you add so 6 is still in the middle?
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Temperatures over four mornings were −3,2,5,−1 °C. Find the range.
Edge case: add one value so the median is still 6.
Use the board. This ticks when you get there.
PractiseA few on your own, to check you have it.
The four measures
Find the mean of 12,15,9,14.
Find the median of 21,17,30,25,19.
Find the range of 4.5,7.2,3.9,6.1.
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
Build a data set
Then make a different set with the same mean and median.
Try it yourselfYour edits are saved in this browser.
Need a hint?
The median of 5 values is the 3rd once sorted: make it 5.
Another hint
A mean of 6 with 5 values needs a total of 30.
Make 5 values (from 0 to 12) with a mean of 6 and a median of 5.
Use the board. This ticks when you get there.
For teachers
Students calculate the mean, median, mode and range of small data sets (odd and even counts, two modes, no mode, negative values), watching each measure move on a dot plot as values are added (AC9M7ST01, node 2). About 45 minutes.
I can calculate the mean, median, mode and range of a data set.
I can find the median of an even number of values.
I can handle data with two modes or no mode.
Before class
Do the Run and Make boards yourself first. Students need Counting and measuring and division with decimals.
This is a hinge: the next lesson asks which measure is most useful, which is meaningless if the values are wrong. The common mistake is taking the middle of the unsorted list for the median; it is planted in Leo's working and again in Practise.
The board marks the mean with a triangle, the median with a dashed line and the range with an arrow; students can add values by clicking the line or typing them.
Running the lesson
Predict: take a show of hands; students who chose 5 found the median or mode.
Run: ask why one big score moves the mean but not the median much.
Investigate: model the even-count median with the sorted list on the board.
Modify: for the keep-the-median task, ask which other values would work (only 6).
Make: collect different answers and check them on the board.
When students get stuck
A student takes the middle of the unsorted list (Leo's mistake). Ask: Is the list in order? Fix: Sort first, then count to the middle.
A student gives the total as the mean. Ask: How many values share that total? Fix: Divide the total by the count.
A student picks one of two middle values for an even count. Ask: Which value is exactly halfway between them? Fix: Add the two middle values and halve.
Stretch and support
Stretch: Find five whole numbers with mean 10, median 8 and mode 7.
Support: Write the data sorted on a strip of paper; fold to find the middle.
Where it leads
Next: Which measure? (node 3), choosing the most useful measure for a data set.
Where it leads: comparing data sets (this unit) and samples (Year 8).
What each step is for
a typical game: Committing to the mean of 3, 5, 5, 6, 11 before the board shows it brings out whether they share equally or pick a middle or common value.
move the mean: Adding a value and watching the mean, median and mode move shows how each responds: the mean feels every value, the median only the order.
the four measures: Mean = total ÷ count; median = middle of the sorted list (average the two middle values for an even count); mode = most common; range = highest − lowest. Leo's mistake takes the middle of the unsorted list.
one change at a time: One change at a time: two modes, no mode, a range with negatives, then the edge case of a value that leaves the median unchanged.
the four measures: Each measure from a fresh data set, mixed; sorting before the median is planted again; the review keeps discrete and continuous fresh.
build a data set: Building five values with a given mean and median reverses the lesson: the median fixes the middle value, the mean fixes the total.
Now you can
Calculate the mean, median, mode and range of a data set.