--- title: "How samples vary" subtitle: "Data and chance · Samples · Lesson 19" --- # How samples vary **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** Two polls of the same election, taken the same week, give 52% and 48% for the same party. Is one of them wrong? :::teacher-note **In one sentence:** Students take many random samples of a simulated school, see their results vary, and find that bigger samples vary less, for proportions and means (AC9M8ST03, nodes 1 and 2). **Students should be able to say:** I can explain why two fair samples give different results. I can describe how the spread of sample results changes with sample size. I can choose a sample size that is big enough for a job. **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 :::keep-together **Two random samples of 20 students are taken from the same school. Will they give the same proportion who play sport?** ☐ Probably not: they'll be a bit different ☐ They will be very different ☐ Yes, exactly the same My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** Probably not exactly. Each sample happens to hold different students, so the proportions differ a little, scattered around the school's value. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Variation is normal :::keep-together ### Range of results **Five samples of 20 gave these proportions who play sport: 0.45, 0.6, 0.55, 0.7, 0.5. What is the range of the results?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 0.25 0.7 − 0.45 = 0.25. ::: :::keep-together ### Find the mistake **Mia and Jack compare their samples. Click the line where the reasoning goes wrong.** ``` 1 Mia and Jack each take a random sample of 20 students from the same school. 2 In Mia's sample, 12 of 20 play sport (0.6). In Jack's, 9 of 20 do (0.45). 3 The results are different, so one of them must have made a mistake. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 3 Line 3: random samples from the same school differ because they hold different students. Neither made a mistake: this is sampling variation. ::: :::keep-together ### Least variation **Which samples will give results that vary the least?** ☐ Samples of 20 ☐ They all vary the same ☐ Samples of 10 ☐ Samples of 100 ::: :::teacher-note **Answer:** Samples of 100 Bigger samples average over more people, so one unusual person changes the result less. ::: :::typical ## Change the size **Bigger samples.** Here is how to describe sampling variation. 1. $\text{10 samples of } 20:\ 0.40 \text{ to } 0.75$ — The spread of results. 2. $\text{10 samples of } 100:\ 0.52 \text{ to } 0.63$ — Bigger samples: a narrower spread. 3. $\text{Both centred near } 0.57$ — Fair samples scatter around the population's value. 4. $\text{Bigger sample} \Rightarrow \text{more reliable estimate}$ — The pattern. **Means vary too.** **Weather forecasts.** **Polls.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Variation :::keep-together ### Small samples **Two classes each take a random sample of 10 students from the same school. One finds 0.3 own a dog; the other finds 0.6. What is the best explanation?** ☐ Sampling variation: samples of 10 vary a lot ☐ The second class made a mistake ☐ The first class made a mistake ☐ Dogs changed ::: :::teacher-note **Answer:** Sampling variation: samples of 10 vary a lot Small samples vary a lot. Larger samples would agree more closely. ::: :::keep-together ### Bigger samples **Ten samples of 25 give means from 8.4 to 9.2 hours. Ten samples of 100 are taken. What will most likely happen to the spread of their means?** ☐ It stays the same ☐ It disappears completely ☐ It gets narrower ☐ It gets wider ::: :::teacher-note **Answer:** It gets narrower Bigger samples give means closer together. ::: :::keep-together ### A census **A census of the whole school is taken twice on the same day. How do the two results compare?** ☐ They are very different ☐ You can't take a census twice ☐ They vary like samples ☐ They are the same: a census has no sampling variation ::: :::teacher-note **Answer:** They are the same: a census has no sampling variation Sampling variation comes from who happens to be in a sample. A census includes everyone. ::: :::keep-together ### Review: a claim **Review: which is the best claim from a random sample of 40 students where the median sleep was 9 hours?** ☐ Students should sleep 9 hours ☐ The school's median is exactly 9 hours ☐ Every student sleeps 9 hours ☐ Students at the school sleep about 9 hours a night ::: :::teacher-note **Answer:** Students at the school sleep about 9 hours a night A sample gives an estimate: 'about'. ::: ## Make: how big is big enough? Write the smallest size that worked for you, and why you might not always choose a bigger one. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Two random samples of 10 students give 0.4 and 0.7 for the proportion who walk to school. Is one wrong? What would you do?** {{lines-3}} ::: :::teacher-note **Answer:** Not necessarily: small samples vary a lot. Take bigger samples (or more of them) for a more reliable estimate. :::