Sampling and distributions: a week of ready-to-teach maths

Five lessons for Year 8 Statistics. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus 40 student cards to sample from, dot plots on numbered axes, a secondary data set to judge, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets, one copy per pair: the 40 student cards (Days 2 and 3) and the blank dot plots (any day).
4. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.

The idea in plain words

A distribution is the pattern of a variable’s values: which are common, which are rare, and how they spread. We describe it by its shape (symmetric or skewed, one peak or more), its centre (the mean, the median or the mode) and its spread (here, the range). Data come from primary sources, which we collect ourselves, or secondary sources, which someone else collected. Either way, how the sample was chosen matters: a random sample tends to look like the population, while a convenience or self-selected sample can shift the centre or change the shape, however large it is.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

The data in this pack

The 40 students and the other school’s survey are made up for this pack, so the class can see the whole population as well as the samples. Day 1 can use the class’s own data as well; ask first, and keep it anonymous.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 sets the words for describing a distribution; Days 2 and 3 choose samples in different ways and see what each does to the distribution; Day 4 adds secondary data and how to judge it; and Day 5 writes it all up. It builds on data collection earlier in Year 8 and on the mean, median, mode and range from Year 7, and prepares the effect of sample size in the next unit.

Materials for the week (one trip)

• From the classroom: pencils, coloured pencils, calculators, scissors, a paper bag or envelope per pair, this pack printed.
• Cut-out sheet 1, one per pair: 40 student cards (Days 2 and 3).
• Cut-out sheet 2, one per pair: blank dot plots on numbered axes (any day).

Words for the word wall

distribution, dot plot, shape, symmetric, skewed, peak, centre, mean, median, mode, spread, range, primary source, secondary source, random sample, systematic sample, convenience sample, self-selected sample, bias

Dear families

This week in maths, Year 8 is describing data by its shape, centre and spread, and learning why the way a sample is chosen can change what the data seem to say.

Try this at home

• Look at a graph in the news. Who collected the data, and how were the people chosen?
• Ask: if a gym asked its members how often people exercise, would the answer be fair for everyone? Why not?
• Count the people in the homes of five relatives or friends. What is the middle value?
• Notice a survey that anyone could answer online. Who is most likely to answer it?

Data I met this week

Fill one row a day. Name some data, say whether it is primary or secondary, and how the sample was chosen.

Printed from the free seegongsik Sampling and distributions teaching pack · seegongsik.com/au/y8/statistics/AC9M8ST02/pack

Distributions

A distribution shows how a variable’s values are spread. Students draw dot plots of two class data sets and describe each by its shape, its centre and its spread.

We are learning to

• draw a dot plot of one variable,
• describe its shape: symmetric or skewed, with one peak or more,
• find its centre (mean, median and mode) and its spread (range).

Success criteria

• I can draw a dot plot and say whether it is symmetric or skewed.
• I can find the mean, median, mode and range, and say which centre suits the data best.

You need

The worksheet and a calculator each; the class’s own count of people at home, if you collect it (anonymously).

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with the loop times.

Watch for these ideas

• Naming a skew by the pile instead of the tail: a long tail to the right is skewed right.
• Finding the median without putting the data in order first.
• Giving the range as the two end values instead of their difference.

Answers

• Part A: 2: 2, 3: 5, 4: 9, 5: 5, 6: 3, 7: 1, 8: 1; one peak at 4, with a short tail to the right, so slightly skewed right.
• Part B: mean 113 ÷ 26 ≈ 4.3; median 4; mode 4; range 8 − 2 = 6. The median or mode suits it; the mean is pulled up a little by the large homes.
• Part C: mean 17.9, median 17.5, mode 17, range 8; nearly symmetric with one peak.
• Part D: a long tail to the right: a few large values pull the mean above the median.

Distributions

A. People at home

A class of 26 counted the people living in each student’s home. Draw a dot plot, one dot per student.

Shape: ______________________ (symmetric or skewed? one peak or more?)

B. Centre and spread

Mean = ______   Median = ______   Mode = ______   Range = ______

Which centre would you use to describe a typical home here? Why?

C. Cross-country loop

Minutes taken by 20 students to walk the school’s loop: 14, 15, 15, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 19, 19, 20, 20, 21, 22, 22.

Mean = ______   Median = ______   Mode = ______   Range = ______   Shape: ______________

D. Stretch: mean above median

Sketch a dot plot where the mean is bigger than the median. What makes this happen?

Choosing a sample

How a sample is chosen decides what it can tell us. Students take four samples from 40 student cards: random (drawn from a bag), systematic (every fourth card), convenience (the students at sport training) and self-selected (those who handed back a survey).

We are learning to

• take a simple random sample and a systematic sample,
• recognise convenience and self-selected samples,
• find each sample’s mean and compare it with the whole group’s.

Success criteria

• I can explain why drawing cards from a bag gives every student an equal chance.
• I can say who is missing from the sport-training sample.

You need

Cut-out sheet 1 (the student cards), scissors and a paper bag or envelope per pair; a calculator and the worksheet each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with the systematic sample.

Watch for these ideas

• Choosing cards by looking at them, which is not random.
• Thinking a self-selected sample is random because people chose freely.
• Expecting every random sample to give the same mean.

Answers

• Part A: samples vary; for example cards 4, 5, 7, 8, 11, 15, 28, 35, 39, 40 give a mean of 45.0 minutes.
• Part B: starting at card 1, 2, 3 or 4 gives means of 45.5, 56.5, 48.0, 43.0 minutes.
• Part C: training (12 students) 89.6 minutes; survey returners (10) 84.0 minutes. The whole group: 1930 ÷ 40 = 48.3 minutes. Both non-random samples favour active students.
• Part D: 12 of 40 play a team sport, so a sample of 10 should have 3 of them, chosen at random from each group.

Choosing a sample

Each card is one of 40 students: their number, their minutes of exercise yesterday, whether they play a team sport, and whether they handed back a voluntary survey.

A. A random sample

Put all 40 cards in the bag, shake it, and draw 10 without looking. Record each card.

B. A systematic sample

Draw one card numbered 1 to 4 at random to start. Take that card and every fourth card after it, in number order.

C. Two non-random samples, and a summary

Convenience: the students at sport training (team sport: yes). Self-selected: the students who handed back the survey (returned: yes). Find each mean, then compare all four.

D. Stretch: a fair share

12 of the 40 students play a team sport. How many team-sport students should a sample of 10 include to match the group? How would you choose them fairly?

When a sample misleads

A sample can mislead without anyone telling a lie. Students plot the Day 2 samples beside the whole group of 40 and see which ways of choosing shifted the centre and changed the shape.

We are learning to

• compare the distribution of a sample with the population’s,
• explain how a non-random sample shifts the centre,
• explain why a bigger non-random sample still misleads.

Success criteria

• I can show on dot plots how the training sample sits to the right of the whole group.
• I can explain why asking more volunteers would not fix the bias.

You need

The student cards and Day 2 results; the worksheet and a calculator each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with the table.

Watch for these ideas

• Comparing counts when the groups are different sizes.
• Thinking a large sample must be accurate.
• Blaming chance for a gap that the method caused.

Answers

• Part A: the whole group runs from 0 to 110 minutes; the training sample sits from 65 up, at the right; the survey sample is mostly at the right too.
• Part B: All 40: median 50, range 110; Random: median 50, range 85; Training: median 90, range 45; Survey: median 80, range 55. The random sample is spread like the whole group; the other two are shifted right.
• Part C: no: active students are still the ones most likely to answer; more of them repeats the same lean.
• Part D: only readers who choose to vote are counted, and people who care about the topic are more likely to vote.

When a sample misleads

A. Four dot plots, one scale

The whole group and one random sample are drawn. Plot the training sample and the survey sample from the cards.

B. Centre and spread

C. Bigger is not fairer

Jo says, “Only 10 students returned the survey. If 30 had, the mean would be right.” Do you agree? Explain.

D. Stretch: a big online vote

A news site asks readers to vote on whether teenagers exercise enough, and 5000 people vote. Why might the result still mislead?

Primary and secondary data

Data we collect ourselves are primary; data someone else collected are secondary. Students compare our 40 students with another school’s published survey, describe both distributions, and judge how far the secondary source can be trusted.

We are learning to

• tell primary data from secondary data,
• compare two distributions by shape, centre and spread,
• judge a secondary source by who collected it, how, when and how many.

Success criteria

• I can compare our group with the other school using the median and the range.
• I can name two questions to ask before trusting a secondary source.

You need

The worksheet and a calculator each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with the comparison.

Watch for these ideas

• Comparing only the means.
• Trusting a secondary source without asking how it was collected.
• Treating a difference between two samples as certain proof.

Answers

• Part A: their 30 values run from 30 to 90 minutes, bunched in the middle and upper values.
• Part B: ours mean 48.3, median 50, range 110; theirs mean 61.3, median 65, range 60. Their centre is higher and their spread is smaller (no one at 0 and no one above 90).
• Part C: the day before a Monday is a Sunday, when weekend sport is common; we do not know which day our “yesterday” was, who chose the sample, or exactly how the question was asked. The comparison is not fair until the days match.
• Part D: yes: data are primary for whoever collected them and secondary for anyone else who uses them.

Primary and secondary data

A. Their data

From another school’s newsletter (made up for this pack): “We asked a random sample of 30 of our Year 8 students, on a Monday morning, how many minutes they exercised the day before.”

Their minutes: 65, 75, 80, 55, 65, 45, 40, 85, 70, 40, 75, 35, 80, 40, 70, 65, 50, 85, 75, 65, 40, 65, 65, 30, 65, 40, 45, 70, 90, 70. Our 40 students are drawn; plot theirs on the same scale.

B. Compare

Their centre is ________ than ours. Their spread is ________ than ours. The shapes differ because ________________

C. Can we trust it?

What does the newsletter tell us about how the data were collected? What does it leave out? Is the comparison with our group fair?

D. Stretch: whose data?

Our 40 students are primary data for us. If the other school used our data, would it be primary or secondary for them? Why?

Reporting a distribution

A report says what the data show and how far to trust them. Students compare the random sample with the survey volunteers by shape, centre and spread, decide which better represents the 40 students, and play the quiz as a class game.

We are learning to

• write a report comparing two distributions by shape, centre and spread,
• say which sample better represents the population, and why,
• state the limits of a conclusion.

Success criteria

• I can write three sentences comparing two samples, one for each feature.
• I can say which sample I would trust, and why.

You need

The worksheet and a calculator each; the self-check quiz at the bottom of the unit page for the game.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with Part C and the quiz.

Watch for these ideas

• Reporting one number and calling it the result.
• Drawing a conclusion about all students from a self-selected sample.
• Leaving out how each sample was chosen.

Answers

• Part A: random sample mean 45.0, median 50, range 85; volunteers mean 84.0, median 80, range 55.
• Part B, for example: “The volunteers’ distribution sits higher: median 80 against 50. The random sample is spread more widely, from 0 to 85, while the volunteers start at 55.”
• Part C: the random sample: every student had an equal chance, and its mean (45.0) is close to the whole group’s (48.3); the volunteers chose themselves and lean active.
• Part D: survey a random sample and follow up anyone who does not hand the survey back.

Reporting a distribution

A. Two samples

Random sample of 10: 50, 25, 50, 60, 55, 0, 15, 85, 35, 75. Survey volunteers (10): 60, 110, 105, 60, 110, 55, 110, 85, 70, 75. Minutes of exercise yesterday.

B. Write the report

Use one sentence for each: how the samples were chosen; their centres; their spreads; their shapes.

C. Which would you trust?

Which sample better describes all 40 students? Give two reasons, one about how it was chosen and one about its numbers.

D. Stretch: a fairer survey

The voluntary survey gave a biased picture. How would you change the way it is collected so the results can be trusted?

Forty student cards

For Days 2 and 3, one set per pair

Cut out the cards. Each card is one student: minutes of exercise yesterday, team sport, and whether they handed back the survey.

The 40 students are made up for this pack, so the class can compare every sample with the whole group.

Blank dot plots

For any day, one sheet per pair

One dot for each student, stacked over their value. Write the name of each sample at the left.

What we know: Sampling and distributions

Work on your own. Show your working.

1. A team’s goals in 15 games: 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 6, 8. Describe the shape of the distribution. ______________________
2. Find the mean, median, mode and range of the goals. ______________________
3. Primary or secondary: (a) our class measures its own heights; (b) we use a table of heights from a government website? ______________________
4. A survey takes every fifth name on the roll, starting from a random name. What kind of sample is this? ______________________
5. A radio station asks listeners to text in their opinion. What kind of sample is this, and why might it be biased? ______________________
6. Two random samples from the same population have means of 52 and 46. The population mean is 49. Why do the samples differ? ______________________
7. Two classes both have a mean test score of 70. Class A’s range is 20 and class B’s is 55. Compare the two classes. ______________________
8. Which tells you more about all students: a random sample of 50 or a convenience sample of 500? Why? ______________________

Answers and marking guide

Answers with working

1. One peak at 1, with a long tail to the right: skewed right.
2. Mean 34 ÷ 15 ≈ 2.3; median 2; mode 1; range 8.
3. (a) primary: we collected it; (b) secondary: someone else collected it.
4. A systematic sample.
5. Self-selected: only people who choose to text are counted, often those with strong views.
6. Random samples vary by chance; each lands near the population mean but rarely on it.
7. The same centre, but class B’s scores are far more spread out; class A’s are bunched together.
8. The random sample of 50: a convenience sample leans towards whoever is easy to reach, however many there are.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions and reports shape, centre and spread together.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: distributions, choosing a sample, misleading samples, primary and secondary sources, and reporting.