Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- (Lesson 2) Find the mean of 7, 9, 13 and 11.
Answer
10. The total is 40 and 40 ÷ 4 = 10. - (Lesson 1) Find the median of 21, 5, 13, 8, 30, 9.
Answer
11. In order: 5, 8, 9, 13, 21, 30, and (9 + 13) ÷ 2 = 11. - (Year 7, Unit 6, lesson 5) A pie chart sector is a quarter of the circle. What angle is it?
Answer
90°. A quarter of 360°.
Words used in this lesson
- outlier
- a value that is far from the rest of the data. For example: 3 in 40, 42, 3, 45, 41.
- mean
- the total shared out equally. For example: mean of 2, 4, 9 is 5.
- median
- the middle value when the data is in order. For example: median of 2, 5, 9 is 5.
- mode
- the value that appears most often. For example: mode of 2, 5, 5, 9 is 5.
- typical
- the sort of value that most of the data are like. For example: most journeys took about 18 minutes.
- pulled
- moved towards something, as if by a rope. For example: in a tug of war, the stronger team pulls the rope towards itself.
A value that does not fit
A bus company in Lagos says its mean journey is 18 minutes. Nine journeys are recorded, and eight of them took between 14 and 22 minutes. The ninth is the day of the roadworks, and it could be anything. Predict first: as that one journey gets longer, which will move more, the mean or the median? Slide it and watch the two markers.
The ninth journey is 18 minutes, like the others: the mean is 18, the median is 18 and the mode is 18.
Show me the steps
- Ninth journey 18 minutes: the total is 144 + 18 = 162, and 162 ÷ 9 = 18. The median (5th of 9) is 18 and the mode is 18, so all three agree.
- Ninth journey 45 minutes: the total is 189 and the mean is 21. The median is still 18 and the mode is still 18. The mean is 3 above the median.
- Ninth journey 90 minutes: the total is 234 and the mean is 26, higher than eight of the nine journeys. The median and the mode are both still 18.
An outlier is a value that sits far from the rest of the data. It might be a mistake in recording or a real extreme, so look before you act.
The mean uses every value, so an outlier pulls it towards itself. The median depends only on the middle value or values, so changing an outlier to an even more extreme value on the same side does not change it at all. The mode depends only on which value is most frequent.
Choosing: the mode for the most popular choice, such as a shoe size; the median for a typical value when there are outliers or the data is lopsided, such as wages or house prices; the mean when every value should count and the values are fairly even, or when the total matters.
A common mistake: deleting an outlier because it “spoils” the average. First ask whether it is an error, such as a typed 58 that should be 18, or a real value, such as the roadworks day. A real value that is left out changes the story the data tells. If you do leave one out, say so and say why.
Worked examples
Five workers in a workshop earn, in thousands of naira a month: 60, 65, 70, 75 and 480. The owner wants to say what a worker typically earns. Compare the three averages.
- Mean: 60 + 65 + 70 + 75 + 480 = 750, and 750 ÷ 5 = 150.
- Median: the values are already in order, and the 3rd is 70.
- Mode: every value appears once, so there is no mode.
- Four of the five workers earn at most 75, so 150 is not a typical worker's pay. The 480 pulls the mean up. The median, 70, is the fair figure for a worker.
What happens if the 480 is left out?
- Mean of 60, 65, 70 and 75: 270 ÷ 4 = 67.5.
- Median of the four: (65 + 70) ÷ 2 = 67.5.
- The mean fell from 150 to 67.5, a change of 82.5. The median fell from 70 to 67.5, a change of only 2.5.
Your go. The first step is given. The daily number of visitors, in hundreds, to a small museum over six days: 12, 14, 13, 15, 14, 52. Find the mean, median and mode, then say which you would quote and why.
- In order: 12, 13, 14, 14, 15, 52.
- Now find the three averages and decide which is fair, then check.
Check the answer
Mean: 120 ÷ 6 = 20. Median: (14 + 14) ÷ 2 = 14. Mode: 14. Five of the six days had 15 or fewer, so the median (or the mode) says what a usual day is like. The mean, 20, is pulled up by the one very busy day.
Check before you go on
1. The journey times are 10, 12, 11, 13 and 64 minutes. The 64 is replaced by 640. Which measure does NOT change?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. Which value is an outlier in this list? 14, 16, 15, 17, 58, 13.
3. Find the median of 3, 4, 4, 5, 5, 90.
Level 2 · Building
4. The mean of 3, 4, 4, 5, 5 and 90 is 18.5. What is the mean of the same list with the 90 removed?
5. Monthly rents of five flats, in thousands of pesos: 9, 9.5, 10, 10.5 and 51. A new tenant wants a fair idea of what a flat costs. Which is the best summary?
6. Five readings are 10, 11, 12, 22 and 58. The 58 was a typing slip and should have been 25. What is the mean after the correction?
Level 3 · Stretch
7. Five values 10, 12, 13, 14 and 16 have a mean of 13. A sixth value is added and the mean of the six becomes 18. What is the sixth value?
8. Six values are 4, 6, 9, 10, 15 and x, where x is larger than 15. What is the median? (Your answer must work for every such x.)
Something to think about
A shop owner says “the average sale today was 640 rupees” and a customer says “I never spend near that”. Both are telling the truth. How can that be, and which average would the customer want to see?
One way to think about it
A few very large sales pull the mean up, so most customers spend much less than the mean. The customer would want the median: half of all sales are at or below it, so it describes what a typical customer pays.
Exit ticket
9. A very large value is added to a data set. Which statement is true?