This unit begins with a question that comes before any graph: if you take a sample, how far can its mean be from the mean of the whole population? You will count every possible sample from a small population and see what a bigger sample and a stratified plan do to the spread of the answers.
Then the unit turns to two variables. You will build covariance from the deviations of each dot from the mean point, scale it into the correlation coefficient r, work r out by formula once, and find the regression line from the same sums. The last lesson asks the question that a high r does not answer: is a straight line the right model?
A graphic display calculator (GDC) is optional all the way through. Lessons 4 and 5 have a closed “With a GDC” section to open if you have one; everything else is done by hand.
The lessons, in order
- Lesson 1 · How far can a sample mean stray? Counting every possible sample, the range of sample means and how it narrows with sample size, and how a stratified sample in proportion centres the sample means. Choose a sample size and see the lowest and highest sample mean, then compare random and stratified samples of four. No calculator
- Lesson 2 · Covariance Deviations from the mean point, the product of a dot's two deviations and the quadrants where it is positive, covariance as the mean of the products, and why its size depends on the units. Step through five dots and their products, switch between rising, falling and scattered patterns, then change the unit of y. No calculator
- Lesson 3 · The correlation coefficient by formula The deviations dx and dy, the sum of their products, the sums of their squares, r as the first divided by the square root of the other two multiplied, and what r does and does not depend on. Build r for five stalls column by column, then flip the same y values to a falling pattern. Scientific calculator
- Lesson 4 · The regression line The mean point, the gap from a dot to a line, the total of the squared gaps, the formulas for the gradient and the intercept of the regression line, and finding it with a GDC. Tilt a line through the mean point and watch the total of the squared gaps, to find the gradient that makes it smallest. GDC optional
- Lesson 5 · Is a straight line appropriate? Residuals as the gaps from the regression line, their sum, reading a pattern in the residuals, why a high r does not prove a linear model, and the cost of predicting outside the data. Pick a dot and read its residual, then switch between a straight-looking and a curved set and compare r with the pattern of residuals. GDC optional
Assessing this unit
Your teacher chooses which of these tasks to set, and when: nobody does all of them. Each sheet says where it fits in the unit, how long it takes and how your work is judged. To print a sheet, open it first and then print the page.
| Task | Criteria | When | Time |
|---|---|---|---|
| A1: Quick check: samples, covariance and r | A | After lesson 3 | 30 minutes |
| B1: Changing the units: what happens to the line? | B | Mid-unit, after lesson 4 | 50 minutes |
| C1: A note for the stall owner | C | After lesson 5 | 35 minutes |
| A2: End-of-unit test | A | End of unit, after lesson 5 | 50 minutes |
| D1: Fertilizer and the rice harvest | D and C | End of unit, after lesson 5 | 50 minutes |
| D2: Counting life on a reef | D and C | End of unit, after lesson 5 | 50 minutes |
D1 and D2 are alternatives. Both assess criteria D and C in a real-life setting, one about crops and one about a reef: set one, not both.
Task A1: Quick check: samples, covariance and r (criterion A)
When: After lesson 3. Time: 30 minutes. Calculator: no calculator. Criterion: A, knowing and understanding.
Show every step of your working: a correct method earns credit even if the final answer slips. Questions 5 and 6 are meant to make you think, so try something rather than leaving them blank. No calculator is needed.
Part 1. Samples
- Five ferry crossings were late by 2, 4, 6, 10 and 18 minutes. (a) Work out the mean lateness of the five crossings. (b) A sample of two crossings is chosen. List every possible sample of two and give the mean of each. (c) Give the lowest and the highest sample mean, and say how many of the samples have a mean within 2 minutes of the mean of all five crossings. (d) Without listing them, would samples of size 3 have a wider or a narrower spread of sample means than samples of size 2? Give a reason. (7 marks)
- A ferry operator has sold 600 adult and 400 child tickets and wants a stratified sample of 50 tickets. How many of each kind should it take? Give one reason for stratifying rather than choosing 50 tickets at random. (2 marks)
Part 2. Two variables
- A bakery records, on four days, x, the number of staff on shift, and y, the minutes a customer waits: x = 2, 4, 6, 8 and y = 9, 5, 7, 3. (a) Find the mean of x and the mean of y. (b) Write down dx and dy for each day, and the product dx × dy. (c) Work out the covariance. (d) What does the sign of the covariance tell the bakery? (e) The sum of dx2 is 20 and the sum of dy2 is 20. Work out r. (8 marks)
- The waiting times in question 3 are re-measured in seconds (every y multiplied by 60). What are the covariance and r now? (2 marks)
Part 3. Think it through
- Four points have x = −2, −1, 1, 2 and y = 4, 1, 1, 4. Show that their covariance is 0. Does that mean there is no relationship between x and y? Explain. (3 marks)
- Priya says: “Data set P has covariance 120 and data set Q has covariance 0.8, so the link in P is stronger.” Explain why she may be wrong, and what she should compare instead. (2 marks)
How your work is judged
| Level | Criterion A in task A1: what it looks like |
|---|---|
| 1–2 | You answer some of the questions in Parts 1 and 2 correctly, for example the mean of the five crossings or the deviations in the bakery table. |
| 3–4 | You get most of questions 1 to 3 right: the list of samples, the stratified split, the deviations and the covariance. Finding r in question 3 or the effect of changing units in question 4 is missing or has errors. |
| 5–6 | Questions 1 to 4 are right, with working that shows your method, and you make a sound attempt at question 5 or 6 that uses the figures. |
| 7–8 | Questions 1 to 4 are right with working, and 5 and 6 are reasoned from the figures: you show the covariance is 0 and say what that does and does not show, and you explain why covariances cannot be compared across different units. |
Task B1: Changing the units: what happens to the line? (criterion B)
When: Mid-unit, after lesson 4. Time: 50 minutes. Calculator: gdc optional. Criterion: B, investigating patterns.
A gardener measures the height of a bean seedling at the end of each of five weeks. In week 1, 2, 3, 4 and 5 the heights were 3, 4, 6, 7 and 10 cm. Write x for the week number and y for the height. You will be finding regression lines for this set of data and for changed versions of it, and looking for what the changes do to the line.
Part 1. The starting line
- Find the regression line of y on x, and r. Check that the line passes through the mean point.
Part 2. Four changed versions
- Find the regression line of y on x, and r, for each of these changed versions of the data, and record every line and r in a table you design yourself. (i) Every height written in millimetres instead of centimetres. (ii) Time in half-weeks instead of weeks, so week 1 becomes 2, week 2 becomes 4, and so on. (iii) Every height measured from a table top that is 5 cm above the soil, so every height is 5 cm more. (iv) Weeks counted from the start of week 1, so the first measurement is at 0 and the last at 4.
Part 3. Describe what you find
- Look across your five lines. Describe in words, as a rule another student could use, what the changes do to the gradient, to the intercept and to r.
Part 4. Test your rules
- Choose a change of your own to the data that is not one of the four above. Use your rules to predict the new line before you work it out, then work it out and compare.
Part 5. Predict, then check
- A report needs heights in millimetres, time in half-weeks, and heights measured from a table top that is 20 mm above the soil. Predict the regression line for this version of the data without working out any new sums. Then check your prediction using the new data values.
Part 6. Explain
- Explain why your rules must hold for any data set and not only this one.
How your work is judged
| Level | Criterion B in task B1: what it looks like |
|---|---|
| 1–2 | With help, you find the line for the original data and for at least one changed version, and you say something you notice about the numbers. |
| 3–4 | You find the line for all five versions and suggest a rule for what happens to the gradient or to the intercept for at least one kind of change, and it fits your results. |
| 5–6 | You find all five lines accurately, describe rules for both the gradient and the intercept for all four changes, and verify them on your own change in Part 4 and on the prediction in Part 5. |
| 7–8 | As the level before, and your rules are written in general form for any change of the type new value = multiplier times old value plus a shift, verified, and you justify from y = a + bx (or from the sums) why they must always hold. |
Task C1: A note for the stall owner (criterion C)
When: After lesson 5. Time: 35 minutes. Calculator: no calculator. Criterion: C, communicating.
Zoe runs a market stall. For five weeks she handed out leaflets, and counted visitors to the stall. Her friend Tomas worked out the link and the regression line, and gave her this working. It has mistakes in it.
Data. x is the number of leaflets handed out, in tens: 1, 2, 3, 4, 5. y is the number of visitors, in tens: 2, 4, 3, 6, 5.
Tomas’s working
Mean x = 3, mean y = 4. dx = −2, −1, 0, 1, 2 and dy = −2, 0, −1, 2, 1.
dx × dy = 4, 0, 0, 2, 2, which add to 8, so the covariance is 8.
The sum of dx2 is 10 and the sum of dy2 is 10, so r = 8 ÷ (10 + 10) = 0.4.
Gradient b = 8 ÷ 10 = 0.8. Intercept a = 3 − 0.8 × 4 = −0.2. Line: y = −0.2 + 0.8x.
For 12 tens of leaflets the line gives y = 9.4, so about 94 visitors, and r shows that leaflets cause the visitors.
Write a one-page note for Zoe. She has not studied statistics, and should be able to follow it on her own.
Part 1. Put it right
- Find every mistake in Tomas’s working. For each one, say what he did, why it is wrong, and write the correct working.
Part 2. Show the data
- Make a table that shows the deviations and products, and a small sketch of the five points with the corrected line. Mark the mean point on your sketch.
Part 3. Tell Zoe what it means
- In plain words, say what the corrected r and the corrected line tell Zoe, and one thing neither of them tells her. Use the words deviation, correlation and gradient in a way she could follow.
- Give a prediction for 3 tens of leaflets and say whether Zoe can trust a prediction for 12 tens, and why.
How your work is judged
| Level | Criterion C in task C1: what it looks like |
|---|---|
| 1–2 | Some of your note can be followed, but it is hard to tell which of Tomas’s steps you are talking about, the table or sketch is missing or confusing, and key words are missing. |
| 3–4 | You explain several of Tomas’s mistakes with some correct words and notation, and your steps are in a sensible order, so Zoe could follow most of it, with gaps. |
| 5–6 | You deal with every step in order, with reasons; your table and sketch match your words; you usually use the right notation (for example dx, dy, r, b and a) and the key words; and your plain-words paragraph says what the results mean and do not mean. |
| 7–8 | Each correction sits beside the step it corrects, and your table, sketch and words agree with each other; the notation is correct throughout, the key words are used as Zoe would need them, and the plain-words part says in a few precise sentences what can and cannot be concluded. |
Task A2: End-of-unit test (criterion A)
When: End of unit, after lesson 5. Time: 50 minutes. Calculator: gdc optional. Criterion: A, knowing and understanding.
Three sections, from familiar questions to ones you have not met in this form. Show your working everywhere: it earns credit even when an answer slips. A GDC is optional; every number can be found by hand.
Section 1. Samples
- A population has four values: 1, 3, 5 and 7. List every possible sample of size 2 and the mean of each. (2 marks)
- Give the range of the sample means for samples of size 2. (1 mark)
- List every sample of size 3 and its mean. Give the range of these sample means. (2 marks)
- What happens to the range as the sample size increases? What is the range for samples of size 4, and why? (1 mark)
- A survey of 600 residents has 240 in the north, 160 in the centre and 200 in the south. A stratified sample of 30 is taken. How many come from each area? (2 marks)
Section 2. A linear model
- A kiosk records the outside temperature, x in degrees Celsius, and the number of hot drinks sold, y, on five days: x = 3, 5, 7, 9, 11 and y = 14, 10, 4, 6, 6. Write down the mean point. (1 mark)
- Work out dx, dy and dx × dy for each day, and the covariance. (3 marks)
- Work out r to two decimal places and describe the link in words. (2 marks)
- Find the regression line of y on x. (3 marks)
- Say what the gradient and the intercept mean in this context, and whether the intercept makes sense. (2 marks)
- Use the line to predict the number of drinks sold at 8 degrees and at 20 degrees. Comment on each. (2 marks)
Section 3. Judging the model
- Work out the five residuals for the line in Section 2. Check that they add to 0. (2 marks)
- Use the residuals and r to say whether a straight line is a suitable model for this kiosk. (2 marks)
- A sixth day is added at 7 degrees with 8 drinks sold. Without redoing the whole calculation, say what happens to the covariance, to r and to the regression line. Explain using the sums from Section 2. (4 marks)
- Samira says: “A stratified sample of 30 is always better than a simple random sample of 30.” Describe a situation in which stratifying would make almost no difference, and explain why. (2 marks)
How your work is judged
| Level | Criterion A in task A2: what it looks like |
|---|---|
| 1–2 | You answer some of the questions in Sections 1 and 2 correctly, for example the sample means, the mean point or the deviations. |
| 3–4 | You answer most of Sections 1 and 2 correctly, with a method visible, but finding r, the regression line or the residuals has errors or is missing. |
| 5–6 | You answer Sections 1 and 2 correctly with working, find and use the residuals, and make a sound attempt at one question in Section 3 that goes beyond the lessons. |
| 7–8 | You answer Sections 1 and 2 correctly with working, and in Section 3 you reason correctly from the sums and the residuals in the unfamiliar questions, including the new point. |
Task D1: Fertilizer and the rice harvest (criteria D and C)
When: End of unit, after lesson 5. Time: 50 minutes. Calculator: gdc optional. Criteria: D, applying mathematics in real-life contexts; C, communicating.
A farmers’ co-operative in a rice-growing region tried fertilizer on five equal trial plots. The amount of fertilizer, in kilograms per hectare, and the yield, in tonnes per hectare, were:
| Fertilizer, x (kg per hectare) | 20 | 40 | 60 | 80 | 100 |
|---|---|---|---|---|---|
| Yield, y (tonnes per hectare) | 3.0 | 3.6 | 4.0 | 4.2 | 4.2 |
Part 1. The model
- Find the regression line of yield on fertilizer, and r. What does the gradient say, in words with its units?
Part 2. Is a straight line the right model?
- Work out the residuals and say what they show about how the yield responds to more fertilizer.
Part 3. Using the line
- Predict the yield for 70 kg per hectare. How accurate should you say it is?
- An adviser says: “The line gives 5.3 tonnes for 160 kg, so tell farmers to use 160 kg.” Reply.
Part 4. Testing the soil
- The co-operative has 1000 farms in three districts, with 450, 300 and 250 farms. It will soil-test 40 farms. How many from each district should be chosen if the sample is stratified? Say why it is better than asking for volunteers.
Part 5. A note
- Write a short note (about 100 words) to the co-operative: how much fertilizer to try next season, and what the trial results can and cannot tell them.
How your work is judged
| Level | Criterion D in task D1: what it looks like |
|---|---|
| 1–2 | You identify some of the quantities involved and find a regression line with help, but your answers are not related to the farms, so it is unclear whether they make sense. |
| 3–4 | You choose a regression line, residuals and a stratified allocation as the methods, apply them with some errors, and comment on whether the results suit the situation when prompted. |
| 5–6 | You choose suitable methods, apply them correctly, and justify the accuracy of your prediction and whether 160 kg is sensible, using the residuals and the range of the data. |
| 7–8 | As the level before, and you also use the shape of the residuals to criticize the linear model, say what an alternative should show, and justify your recommendation in the context of the harvest. |
| Level | Criterion C in task D1: what it looks like |
|---|---|
| 1–2 | Your working can sometimes be followed, but the layout is hard to read and the notation or words are often missing. |
| 3–4 | Your working is mostly in order, with some correct notation and words, so a reader could follow most of your solution. |
| 5–6 | Your solution is organized by part, uses correct notation and key words, and your note is clear and complete enough for a reader outside statistics. |
| 7–8 | Your solution and note are concise and logical, move between table, graph and words, and every claim in the note is backed by a figure from your working. |
Task D2: Counting life on a reef (criteria D and C)
When: End of unit, after lesson 5. Time: 50 minutes. Calculator: gdc optional. Criteria: D, applying mathematics in real-life contexts; C, communicating.
A marine survey team studies a section of coral reef. It has eight equal quadrats (square marked areas): four in the shallow zone and four in the deep zone. A diver can survey only two quadrats on one trip. The number of sea urchins in each quadrat is:
| Zone | Quadrat 1 | Quadrat 2 | Quadrat 3 | Quadrat 4 |
|---|---|---|---|---|
| Shallow | 8 | 10 | 12 | 14 |
| Deep | 2 | 4 | 4 | 6 |
Separately, the team counted the number of coral species at five depths: at 2, 4, 6, 8 and 10 metres they found 14, 11, 9, 5 and 6 species.
Part 1. Planning the dive
- Work out the mean number of sea urchins per quadrat for the whole reef section.
- A trip surveys two quadrats chosen at random. How many different pairs are possible? Find the lowest and the highest mean number of sea urchins that a trip could report, and how many pairs have both quadrats in the same zone.
- A second plan takes one quadrat from each zone. Find the lowest and highest mean it could report. Which plan would you use, and why?
Part 2. Species and depth
- Find the regression line of species on depth and r. Say what the gradient means.
- Work out the residuals. Is a straight line a reasonable model?
Part 3. Using the line
- Predict the number of species at 7 metres. A volunteer wants to use the line to predict the number of species at 15 metres. Explain what goes wrong.
Part 4. A reply
- The survey leader asks whether the team should use the one-quadrat-from-each-zone plan and the line in its report. Write a short reply (about 100 words), with figures from your working.
How your work is judged
| Level | Criterion D in task D2: what it looks like |
|---|---|
| 1–2 | You pick out some of the quantities in the reef survey and carry out a calculation or two with help, but you do not check your answers against the situation. |
| 3–4 | You choose a method for each part (the mean, listing pairs, the regression line), apply them mostly correctly, and say something about whether the results are sensible when prompted. |
| 5–6 | You choose suitable methods, apply them correctly, and justify the plan you prefer and the limits of the line using your figures, including why 15 metres is a problem. |
| 7–8 | As the level before, and you compare the two sampling plans using the full list of results, judge the line using the residuals, and your recommendation to the leader follows from both and says how far each can be trusted. |
| Level | Criterion C in task D2: what it looks like |
|---|---|
| 1–2 | Parts of your reply can be followed, but the order is hard to follow and key figures or words are missing. |
| 3–4 | Your solution is mostly in order, with some correct notation and words, and your reply gives a view with some figures behind it. |
| 5–6 | Your solution is set out by part with correct notation and key words, and your reply is clear, complete and uses figures from the working. |
| 7–8 | Your solution and reply are concise and logically ordered, move between table, graph and words, and every claim in the reply is tied to a specific figure. |
Teachers: the assessment teacher page has the markschemes, exemplar answers, the MYP descriptors for this year and support for each task.
Every lesson has a teacher page with the answers, the misconceptions each wrong answer reveals, discussion questions, differentiation and support for students learning in English, and extra questions in three tiers.