--- title: "Which measure?" subtitle: "Data and chance · Data · Lesson 3" --- # Which measure? **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** One student gets 40 dollars a week; the rest get 5 to 8. 'The average is 12 dollars,' says the newspaper. Is that fair? :::teacher-note **In one sentence:** Students see an outlier drag the mean but not the median on a dot plot, and choose and justify the most useful measure (mean, median or mode) for different data and questions (AC9M7ST01, node 3). **Students should be able to say:** I can explain how an outlier affects the mean and the median. I can choose the most useful measure of centre for a data set and a question, and justify it. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict How many students get more than the mean? :::keep-together **Weekly pocket money (\$): $5, 6, 6, 7, 8, 40$. Which best describes what a typical student gets?** ☐ The mean, 12 dollars ☐ The median, 6.5 dollars ☐ The range, 35 dollars My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** The median, 6.5 dollars. The mean is 12 dollars, but five of the six students get less than that: the one 40 drags it up. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Choosing a measure :::keep-together ### Look for outliers first Here is how to choose. 1. $5, 6, 6, 7, 8, 40$ — Look first: 40 is far from the rest, an outlier. 2. $\text{mean} = \tfrac{72}{6} = 12$ — The mean uses every value, so the outlier pulls it up: above 5 of the 6. 3. $\text{median} = \tfrac{6 + 7}{2} = 6.5$ — The median only uses the order: a typical value. 4. $\text{no outlier: } 5, 6, 6, 7, 8$ — Mean 6.4 and median 6 now agree: the mean is a fair summary. 5. $\text{most common?} \to \text{mode}$ — For "which is most popular?", use the mode. Now choose a measure for the house prices. **Five house prices (\$ thousands): $600, 620, 640, 650, 3200$. Which measure best describes a typical house?** ☐ The mean, 1142 ☐ The mode ☐ The median, 640 ::: :::teacher-note **Answer:** The median, 640 One mansion is an outlier. The median, 640, is a typical price; the mean, 1142, describes no house in the street. ::: :::keep-together ### Find the mistake **Sam describes the street's house prices. Click the line where his reasoning goes wrong.** ``` 1 House prices in a street ($ thousands): 600, 620, 640, 650, 3200. 2 The mean uses every price, so it is always the best measure. 3 The mean is 1142, so a typical house costs about $1 142 000. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 2 Line 2: using every value is the mean's weakness here: one outlier (3200) pulls it above four of the five prices. With an outlier, the median (640) is the better description. ::: :::keep-together ### The most common **A shoe shop records the sizes it sells: $7, 8, 8, 8, 9, 9, 10, 11$. Which measure tells it which size to stock most of?** ☐ The mean, 8.75 ☐ The median, 8.5 ☐ The mode, 8 ::: :::teacher-note **Answer:** The mode, 8 The mode, 8, is the size most people bought: the shop's real question. ::: :::typical ## One change at a time **Make them agree.** Each task changes one thing. **No outliers.** **Add the mean.** **A useless mode.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Choose and justify :::keep-together ### Outlier **Times (s) to run 100 m: $13.2, 13.5, 13.8, 14.0, 22.4$ (one runner fell). Which describes a typical time best?** ☐ The median, 13.8 s ☐ The mean, 15.38 s ::: :::teacher-note **Answer:** The median, 13.8 s The outlier pulls the mean up; the median 13.8 s is typical. ::: :::keep-together ### How much it adds **Find the mean of $2, 3, 3, 4, 28$, then the mean without the $28$. How much does the $28$ add to the mean? (Type the difference.)** Answer: {{field-15}} ::: :::teacher-note **Answer:** 5 With it: 40 ÷ 5 = 8. Without: 12 ÷ 4 = 3. The outlier adds 5. ::: :::keep-together ### Most popular **A café wants to know its most popular cake. Which measure?** ☐ The median ☐ The mode ☐ The mean ::: :::teacher-note **Answer:** The mode The most popular choice is the mode. ::: :::keep-together ### Review: median **Review: find the median of $9, 4, 11, 6$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 7.5 Sorted: 4, 6, 9, 11. Halfway between 6 and 9: 7.5. ::: ## Make: a misleading mean Then write a sentence someone could use to mislead with your mean. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Salaries in a small company ($ thousands): 50, 52, 55, 58, 400. Which measure would you report as typical, and why?** {{lines-3}} ::: :::teacher-note **Answer:** The median, 55: the 400 is an outlier that pulls the mean up to 123, more than four of the five salaries. :::