Comparing data sets
Describe a data set by its shape (symmetrical or skewed, with clusters, gaps or outliers), its centre and its spread. Compare two sets on centre and spread, with numbers and in context, not by one value.
Two classes sit the same quiz. One has the top score; the other has nobody below 12. Which class did better? It depends on what you compare.
To describe a data set, look at three things:
- Shape: is it symmetrical (balanced about the middle) or skewed (bunched at one end with a tail at the other)? Are there clusters, gaps or outliers?
- Centre: the median (or mean): a typical value.
- Spread: the range: how varied the values are.
To compare two data sets, compare their centres and their spreads, using the numbers and saying what they mean in context.
- 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.
More consistent
Look at how far apart each class's lowest and highest scores are.
Change the centre
Did the range change when the median moved? Can you raise the median without changing the range?
Change Class A's scores (click or type) so that Class A's median is higher than Class B's. Was your prediction right about consistency?
Use the board. This ticks when you get there.
Describe, then compare
Here is how to describe and compare.
- Symmetrical, or skewed with a tail? Clusters, gaps, outliers?
- Close: a typical score is about the same.
- Very different: Class B's scores vary much more.
- "Similar centres, but A is more consistent (range 6, not 16)."
Now describe the shape of the books data.
Read Mia's reasoning. What does one score tell you about a class?
Mia compares the two classes. Click the line where her reasoning goes wrong.
One change at a time
Each task changes one thing.
Need a hint?
Team A's median is the middle value, 10: keep it.
Another hint
Bring the lowest and highest values closer to 10.
Change Team A so its median stays 10 but its range is smaller than Team B's.
Use the board. This ticks when you get there.
Class A's median is 15 and Class B's is 16. By how much is B's median higher?
Compare in context
Review: a stem-and-leaf plot reads and . What is the median?
A surprising comparison
Write two headlines about your sets: one where A wins, one where B wins.
Need a hint?
Set B's median is 15 and its mean 15.6.
Another hint
Give Set A a middle value above 15, but some very low values to drag its mean down.
Change Set A (5 values) so it has a higher median than Set B but a lower mean.
Use the board. This ticks when you get there.
For teachers
Students describe data sets by shape (symmetrical or skewed, clusters, gaps, outliers), centre and spread, and compare two sets on both centre and spread, with numbers and in context (AC9M7ST02, nodes 2 and 3). About 45 minutes.
Open the teacher guide: answers, solutions, slides and a worksheet.
Students will be able to
- I can describe the shape of a data set (symmetrical, skewed, clusters, gaps, outliers).
- I can compare two data sets using a measure of centre and a measure of spread.
- I can write a comparison with numbers and in context.
Before class
Do the Run and Make boards first. Students need Mean, median, mode and range, and Stem-and-leaf plots.
The one idea: a data set has a shape, a centre and a spread; comparing two needs both centre and spread. The common mistake is comparing single values (the highest score); it is planted in Mia's reasoning and again in Practise.
The board draws the second set above the first, each with its own median line and range arrow; only the first set can be changed.
Running the lesson
- Predict: take a show of hands; ask students who chose B why (often the 22).
- Run: ask whether raising the median changed the range.
- Investigate: write a sentence frame on the board: 'On average … (median …), but … is more spread (range …).'
- Modify: for the same-mean edge case, ask students to invent a third set with mean 10 and range 100.
- Make: read out pairs of headlines.
When students get stuck
- A student compares one value from each set (Mia's mistake). Ask: Does one student's score describe the whole class? Fix: Compare a centre (median) and a spread (range) for each.
- A student gives numbers without meaning. Ask: What does a bigger range mean for these students? Fix: Say it in context: 'less consistent', 'more varied'.
- A student confuses the direction of skew. Ask: Where are the few values that trail off? Fix: The tail points to the few values; that is the direction of the skew.
Stretch and support
Stretch: Find two data sets with the same median and range but clearly different shapes.
Support: Use the sentence frame and the board's measures for every comparison.
Where it leads
Next: the Chance unit (sample spaces and probability). A statistical investigation (AC9M7ST03) is a class task: pose a question, collect data, display and compare it with these tools (see the unit primer).
Where it leads: sampling and variation (Year 8), the five-number summary and box plots (Years 9 and 10).
What each step is for
- more consistent: Committing to which class is more consistent before the board shows the ranges separates centre from spread.
- change the centre: Changing one class's scores and watching both medians and ranges shows that centre and spread are separate: one can change without the other.
- describe, then compare: Shape, centre and spread describe one set; comparing two needs both centre and spread, with numbers and context. Mia's mistake compares the single highest values.
- one change at a time: One change at a time: same centre with a different spread on the board, a symmetrical shape, an outlier, a difference in medians, then the edge case of equal means hiding different spreads.
- compare in context: Describing shape and comparing centres and spreads in new contexts, mixed; the single-value slip planted again; the review keeps stem-and-leaf reading fresh.
- a surprising comparison: Making one set have the higher median but the lower mean shows that the measures can disagree, so a comparison should say which it uses.
- Describe the shape of a data set (symmetrical, skewed, clusters, gaps, outliers).
- Compare two data sets using a measure of centre and a measure of spread.
- Write a comparison with numbers and in context.