Sample size and variation: 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 a bag of 20 slips to sample from, a grid of 100 students, two random digit tables, a computer simulation to read, twelve poll cards, 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 20 slips for the bag (Day 1) and the twelve poll cards (Day 5).
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

Random samples of the same size from one population give different results: different per cents and different distributions. That scatter is sampling variation, and it is not a mistake. Larger samples scatter less, so their results bunch closer to the population’s true value, but the gain slows as they grow: four times the size halves the spread. A bigger random sample is more reliable, never certain, and a big sample that was not chosen at random stays biased however big it gets.

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 bag, the 100 students, the digit tables, the simulation and the polls are made up for this pack from fixed random draws, so the class knows the true value and can see exactly how far each sample strays from it.

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

Days 1 and 2 show that random samples of the same size vary, first as per cents and then as whole distributions; Day 3 changes the size and watches the spread shrink; Day 4 measures how fast it shrinks; and Day 5 uses all of it to read polls. It builds on random and non-random samples from the previous unit and on the median and range from Year 7, and prepares planning a full investigation in the next unit.

Materials for the week (one trip)

• From the classroom: pencils, calculators, scissors, a paper bag or envelope per pair, this pack printed.
• Cut-out sheet 1, one per pair: the 20 slips for the bag (Day 1).
• Cut-out sheet 2, one per pair: the twelve poll cards (Day 5).

Words for the word wall

population, sample, random sample, sample size, proportion, per cent, true value, sampling variation, distribution, median, range, spread, random digits, simulation, poll, reliable, bias, census

Dear families

This week in maths, Year 8 is learning why two random samples of the same size give different results, and why bigger samples give steadier answers.

Try this at home

• Flip a coin 10 times and count the heads. Do it again. Did you get the same number?
• Find a poll in the news. How many people did it ask, and how were they chosen?
• Ask: would you trust a survey of 5 people or of 500 people more? Why?
• Notice “plus or minus 3%” in a poll report. What do you think it tells you?

Samples I met this week

Fill one row a day. Name a sample you met, how many it included, and how it was chosen.

Printed from the free seegongsik Sample size and variation teaching pack · seegongsik.com/au/y8/statistics/AC9M8ST03/pack

Same size, different results

Two random samples of the same size rarely agree, and neither is wrong. Pairs draw four samples of 10 from a bag of 20 slips, the class pools its results, and everyone sees how far same-size samples stray from the bag’s true 30%.

We are learning to

• take a random sample by drawing blind and putting each slip back,
• turn a count in a sample into a per cent,
• describe how samples of the same size vary around the true value.

Success criteria

• I can turn 3 out of 10 into 30% and plot it on the class dot plot.
• I can explain why a sample that gives 60% does not mean the bag has changed.

You need

Cut-out sheet 1 (the slips), scissors and a paper bag or envelope per pair; the worksheet each; a dot plot from 0% to 100% drawn on the board.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the class dot plot; start the second with the reveal.

Watch for these ideas

• Thinking a result far from 30% means someone drew wrongly.
• Feeling for slips instead of drawing blind.
• Expecting a fair sample to give exactly the true value.

Answers

• Part A: results vary; with 6 walk slips in 20, most samples of 10 give 20% to 40%.
• Part B: 10%: 4, 20%: 7, 30%: 15, 40%: 7, 50%: 5, 60%: 1, 70%: 1. Most common 30%; range 70 − 10 = 60 points; 29 of 40 within 10 points of 30%.
• Part C: no. A small sample can give 6 out of 10 by chance (the other class got 60% or more 2 times), and the bag still holds 6 of 20. Mia’s sample is one of the far ones, not a wrong one.
• Part D: a sample of 10 counts whole people, so it can only give 0%, 10%, 20% and so on; 35% would be 3.5 people. A sample of 8: 0%, 12.5%, 25%, 37.5%, 50%, 62.5%, 75%, 87.5%, 100%.

Same size, different results

The bag holds 20 folded slips: some say “walks or rides”, the rest say “car or bus”. Draw one without looking, write it down, put it back and shake. Ten draws make one sample of 10.

A. Four samples of 10

B. Another class’s results

Another class pooled 40 samples of 10 from the same bag. Plot them, one dot per sample.

50%, 40%, 30%, 40%, 50%, 30%, 30%, 10%, 20%, 10%, 30%, 20%, 40%, 10%, 20%, 70%, 30%, 30%, 20%, 50%, 30%, 20%, 20%, 40%, 10%, 30%, 60%, 30%, 40%, 30%, 30%, 50%, 30%, 20%, 30%, 40%, 40%, 30%, 30%, 50%

Most common result: ______   Range: ______ points   Within 10 points of the bag’s true value: ______ of 40

C. Mia’s sample

Mia draws a sample of 10, gets 6 walk slips and says, “The bag must be 60% walk slips.” Is Mia right? Use the class results.

D. Stretch: possible results

Why can a sample of 10 never give 35%? List every per cent a sample of 8 could give.

Two samples, two distributions

A sample gives a whole distribution, and that varies too. Pairs use random numbers to take two samples of 10 from 100 students and compare their dot plots with each other and with the whole group.

We are learning to

• choose a random sample from a numbered list with a random number table,
• compare two samples’ distributions by centre (median) and spread (range),
• explain why two random samples of the same size differ.

Success criteria

• I can take a random sample of 10, skipping repeats, and draw its dot plot.
• I can say how my two samples differ, and why neither is wrong.

You need

The worksheet and a calculator each; give each pair a different line of the random number table (with more pairs, a pair reads a line from the right).

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Keeping a repeated student, so the sample has fewer than 10 people.
• Comparing one number only, not the centre and the spread.
• Thinking the sample closest to the whole group was the one done properly.

Answers

• Part A (line 1, skipping repeats of 76, 67, 26, 21, 10): A = students 01 67 19 45 76 26 71 21 33 99, minutes 41, 14, 10, 5, 38, 31, 12, 47, 40, 25; B = students 10 61 43 34 96 05 30 74 59 39, minutes 18, 4, 11, 15, 25, 9, 11, 24, 16, 10.
• Part B: A median 28, range 42; B median 13, range 21. A is spread widely with several long trips; B is bunched at the short end.
• Part C: the whole group has median 16, range 51. A sits above it and B below; both are fair random samples, and the gap is sampling variation. Across all ten lines the 20 sample medians run from 11.5 to 28 minutes.
• Part D: line 1: A has 0 of 10 who walk or ride (0%), B has 4 (40%); the whole group 30 of 100 (30%).

Two samples, two distributions

The grid is the 100 Year 8 students at a made-up school: minutes to get to school, and W for a student who walks or rides. Student 47 is in row 4, column 7.

Random number table: use your line. Read two digits at a time and skip any student you already have, in either sample.

A. Two samples of 10

B. Two dot plots, one scale

C. Against the whole group

The whole group of 100 has a median of 16 minutes and a range of 51 minutes. Compare each sample with the whole group and with each other. Is either sample wrong?

D. Stretch: who walks?

What per cent of each sample walks or rides? In the whole group it is 30 of 100. Why might your samples differ?

Bigger samples, smaller spread

Random digits can stand for people: 0, 1 and 2 are a student who walks or rides, so 30% of digits. Pairs read the same 60 digits as twelve samples of 5 and then as three samples of 20, and the class sees the larger samples bunch closer to 30%.

We are learning to

• simulate random samples with random digits,
• compare the spread of results for two sample sizes,
• explain why larger samples vary less.

Success criteria

• I can find the per cent for a sample of 5 and for a sample of 20 from my line.
• I can compare the ranges for the two sizes and say which size I would trust.

You need

The worksheet each; a different line of the digit table for each pair; two tallies on the board (samples of 5 and samples of 20).

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with the class tallies.

Watch for these ideas

• Treating the samples of 20 as new people rather than the same 60 digits.
• Trusting the small samples more because there are more of them.
• Reading a wider spread as more accurate.

Answers

• Part A (line 1): 20%, 0%, 40%, 40%, 20%, 20%, 60%, 40%, 20%, 20%, 40%, 40%; range 60 points.
• Part B (line 1): 25%, 35%, 30%; range 10 points. Each is the mean of its four small samples.
• Part C (all 12 lines): samples of 5: 0%: 22, 20%: 63, 40%: 41, 60%: 16, 80%: 2, range 80; samples of 20: 10%: 1, 15%: 4, 20%: 5, 25%: 6, 30%: 8, 35%: 8, 40%: 3, 45%: 1, range 35. Missed by 20 points or more: 40 of 144 samples of 5, 1 of 36 samples of 20.
• Part D: four groups put together rarely all lean the same way, so high and low groups cancel and their mean stays nearer the middle.

Bigger samples, smaller spread

Each digit is one student: 0, 1 or 2 walks or rides; 3 to 9 comes by car or bus. Use your line.

A. Twelve samples of 5

Count the 0s, 1s and 2s in each group of 5, then write the per cent.

B. The same digits as three samples of 20

C. Compare the spreads

Range of my samples of 5: ______ points   of my samples of 20: ______ points

Class: missed 30% by 20 points or more: samples of 5 ______ of ______   samples of 20 ______ of ______

D. Stretch: why combining helps

Your first sample of 20 is groups 1 to 4 put together. Why does it land nearer 30% than most of its four groups?

How much bigger?

Bigger samples vary less, but by how much? Students read a computer’s 30 random samples at each of three sizes, 10, 40, 160, find each size’s spread, and discover that four times the size about halves it.

We are learning to

• read and plot the results of a simulation,
• measure spread by the range and by how many results land near the true value,
• predict the spread for a larger sample size.

Success criteria

• I can find the range of the results for each sample size.
• I can use “four times the size, half the spread” to predict.

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 table.

Watch for these ideas

• Thinking twice the size halves the spread.
• Expecting a big enough sample to have no spread at all.
• Reading one lucky small sample as proof that small samples are fine.

Answers

• Part A: samples of 10 run from 0% to 60%; samples of 40 run from 13% to 43%; samples of 160 run from 22% to 37%.
• Part B: ranges 60, 30, 15 points; within 5 points of 30% (25% to 35%): 11, 19, 27 of 30.
• Part C: each size is four times the one before, and the range halves: 60 → 30 → 15. Samples of 640 would have a range of about 7.5 points.
• Part D: about 2560 (four times 640) for a range near 4 points. Twice the size only cuts the spread by about 3 in 10, as the curve shows.

How much bigger?

A computer took 30 random samples of each size from a school where 30% walk or ride. Each number is one sample’s per cent, to the nearest whole number.

Samples of 10: 20, 40, 50, 30, 50, 10, 30, 40, 30, 50, 30, 50, 40, 30, 50, 10, 30, 30, 60, 40, 30, 50, 30, 30, 20, 40, 20, 50, 30, 0

Samples of 40: 28, 30, 28, 18, 33, 33, 40, 33, 18, 30, 43, 25, 28, 28, 25, 38, 13, 33, 25, 25, 25, 33, 35, 23, 20, 20, 35, 23, 30, 20

Samples of 160: 33, 31, 33, 32, 29, 30, 28, 31, 27, 29, 31, 32, 35, 34, 31, 22, 26, 24, 29, 29, 30, 37, 28, 25, 30, 35, 32, 31, 31, 34

A. Three dot plots, one scale

The samples of 10 are drawn. Plot the other two sizes.

B. How spread out?

C. Four times the size

Each size is four times the one before. What happens to the range each time? Predict the range for samples of 640.

D. Stretch: how big is enough?

About how big would the samples need to be for a range of about 4 points? Why does doubling the size not halve the range?

Reading polls

A poll is a sample, so its result varies. Pairs sort twelve poll cards about the canteen by method and size, check them against the whole-school vote, report which to trust, and play the quiz as a class game.

We are learning to

• tell random polls from polls that were not random,
• compare how far small and large random polls land from the true value,
• explain why a large random sample is more reliable, but not certain.

Success criteria

• I can sort the polls and say which pile landed closest to the vote.
• I can explain why a big poll that was not random still missed.

You need

Cut-out sheet 2 (the poll cards) and scissors per pair; the worksheet each; the self-check quiz at the bottom of the unit page.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Trusting the biggest poll, whatever its method.
• Expecting a large random poll to land exactly on the vote.
• Reading a small poll’s big miss as a mistake.

Answers

• Part A: not random: D, E, H; random, fewer than 50: B, C, F, G, J; random, 100 or more: A, I, K, L.
• Part B: small random polls 45% to 80%, range 35, farthest F (22 points off); large random polls 53% to 63%, range 10, all within 5 points; not random 74% to 90%, 16 to 32 points too high.
• Part C, for example: “The large random polls landed within a few points of the vote; the small ones scattered from 45% to 80%; the polls that were not random said far more yes, however many they asked.”
• Part D: the random 400: its result is likely within a few points. The online 4000 chose themselves, and more answers keep the same lean.

Reading polls

Cut out the twelve poll cards. Every poll asked students the same question: should the canteen sell hot lunches?

A. Sort the polls

B. The vote

Later the whole school voted: 58% said yes. How close did each pile come?

C. A report for the principal

Three sentences: which polls came closest, which missed, and why.

D. Stretch: 400 or 4000?

A poll of 400 students chosen at random says 61% yes. An online poll answered by 4000 people says 75%. Which result would you report, and why?

Twenty slips for the bag

For Day 1, one set per pair

Cut out the 20 slips, fold each one in half with the words inside, and put them all in a paper bag or envelope. Draw one without looking, write it down, put it back and shake before the next draw.

Keep the bag shut until Day 1’s reveal: the class finds the true share from its samples first.

Twelve poll cards

For Day 5, one set per pair

Every poll asked students at the same school: should the canteen sell hot lunches? Cut out the cards.

The school, the polls and the vote are made up for this pack.

What we know: Sample size and variation

Work on your own. Show your working.

1. Two random samples of 20 students from one school found that 35% and 50% walk to school. Was one of the samples done wrongly? Explain. ______________________
2. A random sample has 8 students. List every per cent of the sample that could walk to school. ______________________
3. Plot P shows 25 sample results from 10% to 65%. Plot Q shows 25 results from 30% to 45%. One used samples of 10, the other samples of 100. Which is which? ______________________
4. Random samples of 30 gave results with a range of 36 points. About what range would samples of 120 give? Samples of 480? ______________________
5. In a random sample of 50 students, 17 walk to school. What per cent is that? A second random sample of 50 has 12. Find its per cent and say why the two differ. ______________________
6. One poll asked 1500 people chosen at random; another asked 30 people chosen at random. Which would you trust more? Is it certain to be right? ______________________
7. A website poll had 6000 replies. A poll of 400 people chosen at random asked the same question. Which is more likely to be close to the truth? Why? ______________________
8. Heights (cm) of two random samples of 8 students: A: 152, 160, 147, 165, 158, 150, 171, 155; B: 149, 163, 156, 158, 146, 161, 153, 166. Find each median and range. Is the difference a problem? ______________________

Answers and marking guide

Answers with working

1. No. Random samples of the same size vary; 35% and 50% are both ordinary results for samples of 20.
2. 0%, 12.5%, 25%, 37.5%, 50%, 62.5%, 75%, 87.5%, 100%: each student is 100 ÷ 8 = 12.5% of the sample.
3. P is the samples of 10: small samples spread more widely. Q is the samples of 100.
4. About 18 points for 120 (four times the size halves the range) and about 9 points for 480.
5. 17 ÷ 50 = 34%; 12 ÷ 50 = 24%. Different random samples give different results: sampling variation, not a mistake.
6. The 1500: larger random samples vary less. It is not certain: it can still be a few points out.
7. The random 400: the website poll counts only people who chose to reply, and more replies keep the same lean.
8. A: median 156.5, range 24; B: median 157, range 20. A small difference between two random samples is ordinary sampling variation, not a problem.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions and links a smaller spread to a larger random sample.

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: same size, different results; two distributions; size and spread; four times the size; reading polls.