At a glance. One lesson of about 60 minutes. Students find the effect of an outlier on the mean, median and mode, decide what to do with one, and choose the fairest average for a context.
Calculator: no calculator. The totals are chosen to divide cleanly, so the thinking is about which average, not the arithmetic.
What students will learn
Know
- An outlier is a value far from the rest of the data.
- The mean uses every value; the median only on the middle value or values; the mode on frequency.
- Which average is fair depends on the question being asked.
Understand
- An outlier pulls the mean towards it but hardly moves the median.
- An outlier may be an error or a real extreme, and the decision to leave one out must be explained.
- No average is best in itself: it has to suit the data and the question.
Do
- Identify an outlier and describe its effect on each average.
- Choose and justify an average for a context.
- Recalculate an average after correcting or removing a value.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: calculating the averages with and without an outlier (questions 3, 4, 6 and 7).
- Criterion B. Investigating patterns: the journey figure. Ask what the mean does for each step of 9 minutes, and why.
- Criterion C. Communicating: explaining which average is fair and why, in a sentence (question 5 and the faded example).
- Criterion D. Applying mathematics in real-life contexts: pay and rent, and the difference between an error and a real value.
The one thing to do with the figure
Ask for the prediction first: mean or median, which moves more? Then slide the journey one step at a time and ask what the mean does each step. It rises by exactly 1, because 9 more minutes shared among 9 journeys is 1 minute each.
Then stop at 90 and ask the whole room: is 26 minutes a fair thing for the company to put on its sign? Take the median as the answer, and let someone say what the company would have to say about the day of the roadworks.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and journey figure on the board; predictions first. |
| 10 min | Key card, the workshop pay example, then removing the 480. |
| 10 min | Common-mistake card: error or real value? Ask for one example of each from students' own lives. |
| 5 min | Question 1, the check. Hands up for each letter. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 10; 11; 90°. |
|---|---|
| Worked example 1 | Mean 150, median 70, no mode; the median is fair. |
| Worked example 2 | Without 480: mean 67.5, median 67.5. |
| Faded example | Mean 20, median 14, mode 14; quote the median or the mode. |
| 1. Which average an outlier cannot move | B. The median is 12 before and after. The mean changes from 22 to 137.2. |
| 2. Spotting an outlier | C, 58. A and B are the ends of the cluster. |
| 3. Median with an outlier | 4.5. (4 + 5) ÷ 2. |
| 4. Mean without the outlier | 4.2. 21 ÷ 5. |
| 5. Choosing the fair average | B. A is dragged by the 51; D is spread, not a typical value. |
| 6. Correcting an outlier | 16. 80 ÷ 5. |
| 7. The value that sets a new mean | 43. 108 − 65. |
| 8. Median does not depend on the outlier | 9.5. (9 + 10) ÷ 2. |
| 9. What a large value does | B. A treats the two as moving together; C confuses mode with the maximum. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A or C in question 1 | Believes the mean and median move together. | Replace the largest value by a much larger one in the figure and watch both markers. |
| 18.5 in question 3 | Gave the mean when asked for the median. | Ask which measure depends on the size of the 90 and which on its position. |
| 18.5 in question 4 | Did not remove the 90. | Ask what list the question is about. |
| 22.6 or 12 in question 6 | Used the uncorrected list, or the median. | Rewrite the corrected list on the board first. |
| 18 or 5 in question 7 | Gave the new mean, or the rise. | Ask for the total before and after. |
| 10 or 9 in question 8 | Took one of the two middle values. | Write the six values in a row and mark the middle pair. |
Questions worth asking
- When is it right to leave an outlier out? When it is a known error, and you say so. A real value should normally stay in.
- Which average would you use for the price of a house in a city with a few mansions? The median: it is not dragged by the mansions.
- Why can the mode be a poor choice? When most values appear once, the mode may be an accident of the data, not a typical value.
- Who might want the mean rather than the median, even with an outlier? Someone who cares about the total, such as a council planning a budget.
Differentiation
A way in for anyone stuck
- Give the five values for the workshop example on cards, so students can remove the 480 physically.
- Provide a sentence frame: “I would use the … because …”.
- Let students use the figure's steps list to read the three states before they predict.
Stretch
- Take the five values of question 7 and add a sixth value v. What are the smallest and largest the median of the six can be, whatever v is? (12.5 and 13.5: the median can only move half a step either way.)
- Ask students to write a data set of 7 values where one outlier makes the mean larger than the median by more than 10.
Students learning in English as an additional language
Say each word while pointing to the dot plot. “Outlier” is built from out and lie, a value that lies outside the rest.
| Word | Plain meaning | Example |
|---|---|---|
| outlier | a value that is far from the rest of the data | 3 in 40, 42, 3, 45, 41 |
| mean | the total shared out equally | mean of 2, 4, 9 is 5 |
| median | the middle value when the data is in order | median of 2, 5, 9 is 5 |
| mode | the value that appears most often | mode of 2, 5, 5, 9 is 5 |
| typical | the sort of value that most of the data are like | most journeys took about 18 minutes |
| pulled | moved towards something, as if by a rope | in a tug of war, the stronger team pulls the rope towards itself |
Sentence starters for explaining aloud or in writing:
- The outlier is … because it is far from …
- I would use the … because the … is pulled by …
- Without the outlier the mean changes from … to …, but the median changes from … to …
Two things not to say
“Always use the median.” The median ignores most of the values' sizes, and sometimes the total is what matters.
“Throw the outlier away.” Real values are data; say why a value is removed.
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.
- OutlierWhich value is an outlier: 31, 29, 33, 5, 30, 28?
5. The rest lie between 28 and 33. - MeanFind the mean of 20, 22, 24, 26.
23. The total is 92 and 92 ÷ 4 = 23.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- EffectSeven values: 6, 8, 9, 10, 11, 12, 14. Replace the 14 by 84. Find the median and the mean before and after.
The median is 10 both times. The mean is 70 ÷ 7 = 10 before and 140 ÷ 7 = 20 after. - ChoosingWhich average should a school use to describe the typical time students spend travelling, if a few students travel over two hours?
The median: it is not pulled up by the few very long journeys.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningEight values have mean 20, and one of them is 76. What is the mean of the other seven?
12. The total is 160, so the other seven add to 84 and 84 ÷ 7 = 12. - From lesson 2Four numbers have mean 12. A fifth number is added and the mean stays 12. What is the fifth?
12. It takes exactly its fair share.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The slider works with the keyboard (tab, then the arrow keys) and by touch. Under the figure, “Show me the steps” gives the three states in words.
No device? Write the nine times on the board and replace the last one three times.