At a glance. One lesson of about 60 minutes. It assumes the median and the idea of quartiles from Year 8 and cumulative frequency from Lesson 2. Students find quartiles from halves, apply the fence rule, draw a box plot with a whisker and outliers, and read counts from a box plot.
Calculator: GDC optional. A GDC gives the five-number summary of a list and can draw a box plot, but the quartile rule varies between models for short lists, so the by-hand halves come first and the GDC is a check.
What students will learn
Know
- Quartiles are the medians of the lower and upper halves of an ordered list.
- Fences are Q1 − 1.5 × IQR and Q3 + 1.5 × IQR.
- Each quarter of a box plot holds about a quarter of the values.
Understand
- An outlier is defined by the spread of the data, not by how far it looks from the rest.
- The whisker stops at the most extreme value that is not an outlier.
- A short section means crowded values, a long section means spread-out values.
Do
- Find the five-number summary of a short list.
- Test values against the fences and draw the box plot.
- Read a box plot to answer questions about counts and spread.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: quartiles from halves, the IQR, the fences (questions 1 to 8).
- Criterion B. Investigating patterns: the first figure. Ask students to predict at which value the point becomes an outlier.
- Criterion C. Communicating: setting out the halves, the quartiles, the IQR and the fence in order, and saying where the whisker ends (the worked example, the faded example and questions 6 and 7).
- Criterion D. Applying mathematics in real-life contexts: race times, seedling heights, deciding whether a value is an error or a real extreme.
The one thing to do with the figure
Make students commit before they slide. In the first figure, ask “At which largest value will the point become an outlier?” and take a number from each pair before moving the slider.
In the second figure, ask which section holds the most values. Most expect the longest one. Slide to show that every section holds a quarter.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: the slow runner. Collect predictions about how far is far enough. |
| 10 min | First figure and key idea: halves, quartiles and the fence. |
| 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, rebuild the lower half of four values on the board. |
| 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 | 45; 7; 18. |
|---|---|
| Hook | It depends on the spread of the other values: the fence is 1.5 times the IQR beyond the quartile, so a more spread-out set tolerates larger gaps. For the first figure the fence is at 32.5, so a largest value of 32 is not an outlier but 34 is. |
| Worked example | Q1 = 45, median 50, Q3 = 58, IQR 13, fences 25.5 and 77.5; 90 is an outlier; five-number summary 41, 45, 50, 58, 90 with the upper whisker ending at 61. |
| Faded example | Q1 = 23.5, Q3 = 31.5, IQR 8, upper fence 43.5; 52 is an outlier. |
| 1. Quartile of an even-sized list | A. B picks the second value; C takes five values as the lower half; D is the middle of the first two values only. |
| 2. Lower quartile of an odd-sized list | 6.5. (5 + 8) ÷ 2. |
| 3. Interquartile range of a list | 12. 18.5 − 6.5. |
| 4. The upper fence | 59. 38 + 1.5 × 14. |
| 5. Which value is an outlier | C. A is inside the fence; B is on it; D is outside the box but inside the lower fence. |
| 6. The largest value that is not an outlier | 68. 41 + 27. |
| 7. Median after removing an outlier | 14. The 6th of 11 values. |
| 8. Reading a box plot | B. A confuses median and mean; C counts two quarters; D counts the wrong fraction above the upper quartile. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Picks the second value, takes five values as the lower half, or averages only the first two values. | Write the four values of the lower half on their own line. |
| 5, 8 or 9 in question 2 | Picks the second value, keeps the median in the half, or uses the value before the median. | Cross out the median before looking for the quartile. |
| 20, 9 or 25 in question 3 | Gives the range, uses quartiles that include the median, or adds the quartiles. | Ask: which two numbers is the middle half between? |
| 21, 52 or 57 in question 4 | Gives only the extra distance, adds the IQR once, or multiplies the quartile. | Write the fence as a three-step line: IQR, times 1.5, plus Q3. |
| A, B or D in question 5 | Misses the fence, treats a value on the fence as beyond it, or treats any value outside the box as an outlier. | Mark the fence on a number line and plot each option. |
| 69, 59 or 27 in question 6 | Gives the smallest outlier, adds the IQR once, or forgets to add to Q3. | Ask: which end of the fence is being asked for? |
| 14.5, 15 or 13 in question 7 | Keeps all 12 values, takes the wrong middle, or averages 12 and 14. | Count the values before and after removing the outlier. |
| A, C or D in question 8 | Confuses median with mean, counts two quarters, or counts a half above the upper quartile. | Shade quarters of the box plot and label each with a count. |
Questions worth asking
- Why do we leave the median out when finding quartiles of an odd-sized list? It belongs to neither half, so each half has the same number of values.
- Why does the fence depend on the IQR? It judges a value against the usual spread of the data.
- Why can the box not move when the largest value changes? The quartiles depend on the middle of the list, not on the extremes.
- Does an outlier have to be removed? No: it should be investigated, and removed only if it is an error.
Differentiation
A way in for anyone stuck
- Give the list with the median and the two halves already bracketed.
- Use a checklist: ordered list? median? halves? Q1 and Q3? IQR? fences?
- Provide a box-plot template with the scale printed and the box ready to draw.
Stretch
- Find a list of seven values with an upper quartile of 20 and an IQR of 10 in which the largest value is exactly on the fence.
- Explain why one outlier changes the mean more than the median, using a short list.
Students learning in English as an additional language
“Quartile” comes from “quarter”. “Fence” here means a boundary, as around a field. “Outlier” is a value that lies outside the usual range. “Whisker” is the line extending from the box.
| Word | Plain meaning | Example |
|---|---|---|
| five-number summary | the minimum, lower quartile, median, upper quartile and maximum | the five values that a box plot is drawn from |
| quartile | a value that divides ordered data into quarters | the median of the upper half of a list |
| interquartile range | the upper quartile minus the lower quartile | the width of the box |
| outlier | a value beyond a fence, drawn as a separate point | a finishing time far from the rest |
| fence | a boundary at 1.5 times the IQR beyond a quartile | the line Q3 + 1.5 × IQR |
| whisker | the line from the box to the most extreme value that is not an outlier | the line on each end of a box plot |
Sentence starters for explaining aloud or in writing:
- The lower half of the list is … so Q1 is ….
- The fence is … because the IQR is ….
- The value is / is not an outlier because ….
Two things not to say
“An outlier is a value that looks wrong.” It is a value beyond the fence, and it might be correct.
“The long part of the box has the most values.” Each part holds a quarter of the values.
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.
- QuartilesFind Q1, the median and Q3 of 4, 6, 7, 9, 12, 13, 15.
Q1 = 6, median 9, Q3 = 13. - FenceQ1 = 10 and Q3 = 18. Find the lower fence.
10 − 1.5 × 8 = −2.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextThe daily sales of a market stall in Accra have Q1 = 50 and Q3 = 90. A day of 160 sales is recorded. Is it an outlier?
Fence 90 + 60 = 150, so 160 is an outlier. - Earlier work (Lesson 2)A curve is drawn for 200 values. At what count is the median read?
100.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA data set's largest value is an outlier. It is replaced by a value just inside the fence. What happens to the quartiles and to the whisker?
The quartiles do not change; the whisker now reaches that value. - Working backwardsQ1 = 30 and the lower fence is 12. Find Q3.
30 − 1.5 × IQR = 12 gives IQR = 12, so Q3 = 42.
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 each pair 15 numbered cards, have them find the quartiles by folding the row, and draw the box on a strip of paper.