At a glance. One lesson of about 60 minutes. It assumes the summaries from Lessons 1 to 3. Students compare two data sets on centre and spread, choose measures for skewed data, and write comparisons in context.
Calculator: GDC optional. A GDC can give both lists' summaries at once, which saves time, but the point is the sentence, and a quartile from a short list should be checked by hand.
What students will learn
Know
- The median and IQR are steady when extreme values are present.
- The mean and range follow extreme values.
- A comparison needs a centre and a spread.
Understand
- Centre and spread are independent: one can change without the other.
- A higher average is not automatically better; it depends on what the measure means.
- A comparison is written in the words of the situation, with numbers.
Do
- Compare medians and IQRs from lists or box plots.
- Choose between the mean and median for a given data set.
- Write a two-part comparison in context.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the effect of extreme values on each summary (questions 1 to 8).
- Criterion B. Investigating patterns: the second figure. Ask students to predict which of mean, median, range and IQR will move.
- Criterion C. Communicating: a comparison that names a centre and a spread with numbers (the worked example and question 8).
- Criterion D. Applying mathematics in real-life contexts: bus waits, crop yields, bakery sales; deciding what better means.
The one thing to do with the figure
Make students commit before they slide. In the second figure, ask “As the largest value grows, which of the mean, median, range and IQR will change?” and have them write down their four answers first.
In the first figure, ask for a sentence about B before and after each move, using both centre and spread.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: the two bus routes. Collect what students would want to know. |
| 10 min | First figure and key idea: a centre and a spread. |
| 10 min | Second figure, the warning card and the worked example; pairs finish the faded example. |
| 5 min | The check. If many choose B or C, return to the first figure and move only the spread. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | 16; 9; 10. |
|---|---|
| Hook | At least a typical wait and how consistent it is: a median and an IQR for each route. Better is not one number. For the first figure, B's median is 7 higher than A's while the spreads match. |
| Worked example | X: median 9, IQR 7. Y: median 10, IQR 5. Y's typical wait is 1 minute longer, its middle half is tighter, and its largest wait, 25, is the longest of either route. |
| Faded example | Same typical yield (62); Q's IQR 24 is more than twice P's 10, so Q is less reliable. |
| 1. What a larger IQR says | A. B and C confuse spread with centre; D confuses spread with number of values. |
| 2. Difference of two IQRs | 6. 18 − 12. |
| 3. Choosing measures for skewed data | B. A uses two measures that follow the extreme value; C and D each keep one that does. |
| 4. Comparing two data sets by IQR | 3. 10 − 7. |
| 5. Mean against median | 4. 8 − 4. |
| 6. Adding a value to a mean | 35. 315 ÷ 9. |
| 7. Effect of one value on the mean | 5. 40 ÷ 8. |
| 8. Writing a comparison | C. A and B each get one of the two comparisons wrong; D ignores both differences. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Confuses spread with centre or with the number of values. | Ask: what does the IQR measure? Is it a position or a width? |
| 5, 30 or 1.5 in question 2 | Used the medians, added the IQRs, or divided. | Underline “IQRs” and “how much larger” in the question. |
| A, C or D in question 3 | Chose a measure that follows the extreme value. | Replace the flood reading with a typical one and ask what moves. |
| 2, 1.5 or 1 in question 4 | Compared ranges, medians or means. | Ask which summary the question names. |
| 29, 8 or 5 in question 5 | Gave the extreme value minus the median, the mean alone, or used the mode. | Write mean and median on separate lines. |
| 31.5, 39.375 or 9 in question 6 | Averaged two numbers, divided by the old count, or gave a difference. | Total, add, count, divide: say the steps aloud. |
| 40, 0 or 4.4 in question 7 | Gave the change in the value, the change in the median, or divided by 9. | Ask how many values share the extra 40. |
| A, B or D in question 8 | Got one comparison wrong, or ignored the differences. | Ask for the centre comparison and the spread comparison separately. |
Questions worth asking
- Can two groups have the same median and different IQRs? Yes: the middle values can match while the middle halves are spread differently.
- When would you prefer the mean? When the data are roughly symmetric and every value matters, such as a total.
- Why mention the situation in a comparison? Because a number alone does not say whether larger is good or bad.
- How many values do you need before an average is reliable? There is no fixed number, but few values make any average unreliable.
Differentiation
A way in for anyone stuck
- Give a sentence frame: “The typical value is higher for …. The values are more spread out for ….”
- Use a two-row table of median and IQR for the two groups, to be filled in before writing.
- Provide the quartile halves already marked for the lists.
Stretch
- Find two lists of 7 values that have the same mean and the same median but different IQRs.
- A data set has a mean much higher than its median. Describe its shape and say which centre you would quote.
Students learning in English as an additional language
“Consistent” means not changing much. “Typical” means usual. “Spread” is how far apart values are. “Reliable” means you can depend on it.
| Word | Plain meaning | Example |
|---|---|---|
| centre | a single value that represents the typical value of a data set | the median or the mean |
| spread | how far apart the values in a data set are | the IQR or the range |
| consistent | not varying much from one value to the next | bus waits that are always close to the same |
| skewed | having more values on one side, with a long tail on the other | a few very long waits |
| extreme value | a value far from the rest of the data | one house price ten times the others on its street |
| compare | say how two things are alike and how they differ, with numbers | “the typical mass is greater for plot P” |
Sentence starters for explaining aloud or in writing:
- On average, group A is … because the median is ….
- Group B is more / less consistent because its IQR is ….
- I used the median because ….
Two things not to say
“The bigger number wins.” Which is better depends on what is being measured.
“Use the mean, it is the real average.” The median is also an average, and is better for skewed data.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- DifferenceMedian of A is 22 and median of B is 17. How much higher is A's?
5. - ChooseDelivery times with one very late parcel. Mean or median for the typical time?
The median.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextTwo phone batteries last a median of 18 hours with IQRs of 2 and 6 hours. Which is more predictable?
The one with IQR 2. - Earlier work (Lesson 4)A class 10 wide holds 45 values. What is its frequency density?
4.5.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningGroup A has mean 50, median 50. Group B has mean 60, median 50. What does that suggest about B?
B has a few high values pulling its mean up: it is skewed to the right. - Working backwardsFive values have a mean of 20. One is removed and the mean of the rest is 18. What was the removed value?
100 − 72 = 28.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Give pairs two sets of 9 cards and have them find the median and the IQR of each and write the comparison.