Math Detours · Grade 6 · Sharpen · 8

Lying with averages

A billionaire walks into a bar and everyone in it becomes a millionaire, on average. Learn which average to trust, how to measure spread, and how to design numbers that mislead.

Name ____________________Date ____________

Tonight's idea

There are three common "averages", and they answer different questions. The mean shares the total out equally: add everything, divide by how many. The median is the middle value when the numbers are lined up in order (or halfway between the two middle ones). The mode is the most common value.

Five friends earn 30, 35, 40, 45 and 50 thousand dollars: mean 40, median 40. A billionaire earning 1 000 000 thousand joins them. The mean jumps to about 166 700; the median creeps to 42.5. The median barely noticed; the mean was dragged away by one value. That is the rule: the mean follows the total, the median follows the crowd. House prices are reported as medians for exactly this reason.

An average alone hides how spread out the numbers are: 50, 50, 50 and 20, 50, 80 both have mean 50. The mean absolute deviation (MAD) measures spread: find how far each number is from the mean (ignoring sign), then average those distances. The first set has MAD 0; the second has MAD \((30 + 0 + 30)/3 = 20\).

Watch one

Data: 2, 4, 4, 4, 5, 5, 7, 9. Mean \(= 40 \div 8 = 5\). Median: the 4th and 5th values are 4 and 5, so 4.5. Mode 4.

MAD: distances from 5 are 3, 1, 1, 1, 0, 0, 2, 4. Sum 12. MAD \(= 12 \div 8 = 1.5\). On average, a value sits 1.5 away from the mean.

Your turn

  1. Test scores: 72, 85, 91, 68, 85, 99. Find the mean, median and mode. Which would the student rather have on the report card?
  2. The mean of five numbers is 12. Four of them are 10, 15, 8 and 14. Find the fifth. (Work with the total.)
  3. Which average would you want for each, and why: the typical house price in a city; the shoe size a shop should stock most of; the amount of rain to expect in a month?
  4. Find the mean and MAD of 3, 7, 8, 10, 12.
  5. Class A has 20 students with mean score 70. Class B has 10 students with mean 80. Find the mean of all 30 students. Why is it not 75?
  6. Alice and Bob practise free throws for two weeks. (a) Who has the higher rate in week 1? In week 2? (b) Predict who has the higher rate over both weeks together, and only then fill in the last column. (c) Count the circles and check. Point at the picture: what changed? (d) Redo it with Alice 6 of 10 and 10 of 10, Bob 5 of 10 and 9 of 10. Who wins now?
    PlayerWeek 1Week 2Both weeks
    Alice6 of 101 of 1____ of ____
    Bob1 of 29 of 10____ of ____
    Week 1Week 2 AliceBob 11 shots12 shots
    One circle per attempt, filled for a hit: count circles for shots, share filled for rate. Orange marks each big pile.

Stretch: building liars

(a) Write down seven whole numbers with mean 10 and median 4. (The total must be 70. Where does the "extra" have to go?)

(b) Invent a two-week table where one player has the higher rate in both weeks but the lower rate pooled. Keep the "of how many" showing. Where must each player's big pile of attempts land?

Answers

  1. Mean \(500 \div 6 = 83.3\). In order: 68, 72, 85, 85, 91, 99; median 85. Mode 85. The student prefers the median or mode (85) to the mean (83.3); the low 68 pulls the mean down.
  2. Total must be \(5 \times 12 = 60\). The four given sum to 47, so the fifth is 13.
  3. House prices: median (a few mansions would wreck the mean). Shoe sizes: mode (you need the size most people actually buy). Rain: the mean, if you care about the total water over a year, or the median if you want a typical month. Accept either with a reason.
  4. Mean \(40 \div 5 = 8\). Distances 5, 1, 0, 2, 4, total 12, MAD \(= 2.4\).
  5. Totals: \(20 \times 70 + 10 \times 80 = 1400 + 800 = 2200\), over 30 students gives 73.3. It is not 75 because class A has twice as many students, so its mean counts twice as much. This is a weighted mean.
  6. (a) Week 1: Alice 60%, Bob 50%. Week 2: Alice 100%, Bob 90%. Alice, both times. (b) Most people predict Alice. (c) Both weeks: Alice 7 of 11 ≈ 64%, Bob 10 of 12 ≈ 83%. Bob. Nothing strange happened to the arithmetic: a pooled rate is a weighted mean of the week rates (the same idea as question 5), weighted by attempts. Alice: \(\tfrac{10}{11} \times 60\% + \tfrac{1}{11} \times 100\% \approx 64\%\), almost all her weak week. Bob: \(\tfrac{2}{12} \times 50\% + \tfrac{10}{12} \times 90\% \approx 83\%\), almost all his strong week. In the picture, Alice's big pile sits in her weak week and Bob's in his strong week. (d) Alice 16 of 20 = 80%, Bob 14 of 20 = 70%, Alice. With equal attempts each week, the pooled rate is the plain average of the two week rates, so the week-by-week winner stays the winner.

Stretch. (a) One answer: 1, 2, 3, 4, 5, 6, 49. The median (4th value) is 4, so three values must be at most 4 and three at least 4; to reach a total of 70 the big ones have to be huge. (b) The recipe: the player who wins each week takes most attempts in their weak week, and the other player takes most in their strong week. The rates must also be close enough, so always check by pooling. One example: Carol 3 of 4 and 10 of 20 (75%, 50%); Dan 14 of 20 and 2 of 5 (70%, 40%). Carol wins both weeks, but pooled she has 13 of 24 ≈ 54% and Dan has 16 of 25 = 64%.

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