Year 11 Standard · Unit 4 Sampling and relationships in data · Lesson 2 of 6

How do you pick a sample so that nobody has a better chance than anybody else?

Random sampling gives every member the same chance. Systematic sampling takes every kth member from a random start. Both work from a list, and each can fail in its own way.

Statistics and ProbabilityGDC optional

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Lesson 1: a sample of 30 pupils from a year group of 240 has 6 left-handed. Estimate the left-handed pupils in the year group.
    Answer48. 6 ÷ 30 = 0.2, and 0.2 × 240 = 48.
  2. From Year 7: list the multiples of 4 from 4 up to 24.
    Answer4, 8, 12, 16, 20, 24.
  3. From Year 8: 84 ÷ 12 = ?
    Answer7, because 12 × 7 = 84.
Words used in this lesson
sampling frame
the numbered list of all the members the sample is taken from. For example: a class register.
simple random sample
a sample in which every member has the same chance and each choice is by chance. For example: names drawn from a hat.
systematic sample
a sample taken by choosing a random start then every interval-th member. For example: every 6th seat in a hall.
interval
the gap between one chosen member and the next. For example: population size divided by sample size.
random start
a starting number chosen by chance between 1 and the interval. For example: a number drawn from a bag of slips.
bias
a lean in the method that pushes results the same way every time. For example: asking only the first people through the door.

The fifth nozzle

A drinks factory fills bottles from a machine with five nozzles, used in turn: bottle 1 from nozzle 1, bottle 2 from nozzle 2, and so on, then back to nozzle 1. An inspector will check every few bottles and use the mean fill to judge the whole line. One nozzle is set to overfill.

Predict first: if the inspector checks every 5th bottle, does it matter which bottle she starts at? Try other intervals too. Then use the second figure to see where the interval comes from.

3
1
20bottles checked
501mean of the checked bottles in ml

Every 3rd bottle from bottle 1: 20 bottles from 5 of the 5 nozzles, mean 501 ml against 501 ml for all 60.

Show me the steps
  1. Every 3rd bottle from bottle 1: 20 bottles, spread over all 5 nozzles equally, mean 501 ml, the same as the whole line.
  2. Every 8th bottle from bottle 1: 8 bottles from all 5 nozzles, but unevenly, so the mean is 501.6 ml, a little off the 501 ml of the whole line.
  3. Every 5th bottle from bottle 1: 12 bottles, all from nozzle 1. The mean is 508 ml, which is the nozzle's fill and says nothing about the other four.
  4. Every 5th bottle from bottle 2: all from nozzle 2, mean 499 ml. The interval matches the cycle, so the start decides everything.

Two ways to avoid choosing by hand.

  • A sampling frame is the list the sample is taken from. Each member needs a number.
  • Simple random sample: every member has the same chance. Use random numbers, ignore any that repeat or lie outside the list, and stop when you have enough.
  • Systematic sample: the interval is the population size ÷ the sample size. Pick a random start from 1 up to the interval, then take every interval-th member.
  • A systematic sample is quick and spreads evenly, but it can be biased if the list repeats a pattern whose length matches the interval or shares a factor greater than 1 with it.
10
1
12interval between chosen positions
25third position chosen

Population 120, sample 10, start 1: interval 120 ÷ 10 = 12, chosen positions 1, 13, 25, and so on up to 109.

Show me the steps
  1. Sample 10, start 1: the interval is 120 ÷ 10 = 12. The chosen positions are 1, 13, 25 and so on, ending at 109.
  2. Sample 20, start 1: the interval halves to 6. Twice as many positions are chosen, and the third is 13.
  3. Sample 40, start 3: the interval is 3. The third position chosen is 3 + 2 × 3 = 9 and the last is 120.
  4. In every case the start is a random number from 1 up to the interval, so every member has the same chance of being the one that starts the pattern.

“Random” is not the same as “without a plan”. Picking whoever looks friendly, or whoever is nearest, feels random but is not: some members had no chance at all. Also, the random start belongs to the method. A systematic sample that always starts at member 1 gives some members no chance of ever being chosen.

Worked examples

A register has 200 names. Take a systematic sample of 25, starting from a random start of 3.

  1. The interval is 200 ÷ 25 = 8, so the start must be a number from 1 to 8. The start 3 is allowed.
  2. The chosen positions are 3, 11, 19, and so on, adding 8 each time.
  3. The 25th position is 3 + 24 × 8 = 195, still inside the list.

A class of 40 is numbered 01 to 40. A list of random two-digit numbers reads 52, 17, 08, 17, 39, 71, 24. Choose a sample of 4.

  1. 52 is above 40, so skip it. 17 is valid and is chosen.
  2. 08 is valid and is chosen. The second 17 repeats, so skip it.
  3. 39 is chosen. 71 is above 40, so skip it. 24 is chosen. The sample is 17, 08, 39, 24.

Your go. The first step is given. A market has 150 stalls numbered 1 to 150. A systematic sample of 10 stalls begins with a random start of 4. List the first three stalls chosen and the last.

  1. The interval is 150 ÷ 10 = 15.
  2. Add the interval to the start each time.
Check the answer

The first three are 4, 19 and 34. The last is 4 + 9 × 15 = 139. A common slip is to count the start as one of the intervals and work out 4 + 10 × 15 = 154, which is outside the list.

With a GDC (optional)

Most GDCs have a random integer command (often written randInt) in a probability or maths menu. Give it the lowest number, the highest number and how many numbers you want, and it lists them. Use it to pick members from a numbered register, or to pick the random start for a systematic sample.

It can repeat a number, so cross out repeats and ask for a few more, exactly as with a printed list. Everything here can be done by hand with a printed list of random digits or numbered slips drawn from a hat.

Check before you go on

1. A school has a register of all 900 students. Which of these is a simple random sample of 30 students?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A list has 360 members and a systematic sample of 45 is wanted. What is the interval?

3. A systematic sample has a start of 7 and an interval of 12. What is the position of the 5th member chosen?

Level 2 · Building

4. A list has 230 members. A systematic sample starts at 6 and takes every 10th member (6, 16, 26, and so on) as far as the list goes. How many members are chosen?

5. Inspectors study the number of passengers on a bus route for 12 weeks. To save time they use the journeys of every 7th day. What is the problem with this?

Level 3 · Stretch

6. A class register is numbered 01 to 60. A list of random two-digit numbers reads 47, 83, 12, 47, 96, 05, 31. Skip any number above 60 and any repeat. What is the 4th student chosen?

7. A systematic sample of 30 members uses a start of 5 and an interval of 8. What is the position of the last member chosen?

Something to think about

A café owner wants to know how long customers wait. She times every 10th customer from opening. The café serves a coach party of exactly 10 people at the start of every hour. What could go wrong, and how would you fix it?

One way to think about it

Ten customers in a row always include exactly one timed customer, so the party is never missed. The risk is lock-step: if the number of other customers between parties is a whole number of tens, she times the same place in the party every hour, perhaps always the first served, whose wait is shortest. A new random start each hour, or a simple random sample of customer numbers, breaks the lock-step.

Exit ticket

8. A register has 90 names numbered 1 to 90. A systematic sample of 10 is taken with a random start of 3. Which list is the sample?