Sampling - Estimating a Population: Edexcel GCSE Foundation maths revision notes

Samples and the exam

When you want to find something out about a big group of people, such as all 1500 students at a school, it takes too long to ask everyone. Instead you ask some of them. The people you ask are a sample.

Sampling questions are not common on Edexcel Foundation papers. When one appears, it is one of three kinds.

Use the results from a sample to work out an estimate for the whole group, then write down an assumption you made. This is the most common kind, usually worth 2 or 3 marks.
Decide whose results are more reliable, then use both sets of results together. These are often about throwing a coin many times, with 1 mark for each part.
Compare the results of two surveys that asked different numbers of people.

This page works through all three kinds.

Fractions and percentages of a sample

Results from a sample are written as so many out of the number asked. If 7 out of 20 people chose football, that is \frac{7}{20} of the sample.

To write it as a percentage, make the bottom of the fraction 100: \frac{7}{20}=\frac{35}{100}=35\%

Worked example 1: A sample as a fraction and a percentage

Leah asks a sample of 24 people which season they like best. 18 of them say summer. (a) Write 18 out of 24 as a fraction in its simplest form. (b) Write 18 out of 24 as a percentage.

(a) 18 out of 24 is \frac{18}{24}
6 goes into both 18 and 24, so divide the top and the bottom by 6: \frac{18}{24}=\frac{3}{4}
Your turn. (b) Write \frac{3}{4} as a percentage.

Your answer

\frac{3}{4}=\frac{75}{100}=75\%
Answer: (a) \frac{3}{4} (b) 75\%

Comparing two surveys

When two surveys ask different numbers of people, do not compare the numbers who gave an answer. Compare the proportions: the fraction or percentage of each survey who gave that answer.

Kim asks 50 people and 20 of them like jazz. Sam asks 20 people and 12 of them like jazz. More people like jazz in Kim's survey, but \frac{20}{50}=40\% and \frac{12}{20}=60\%, so a greater proportion like jazz in Sam's survey.

In a pie chart the whole circle, 360^\circ, stands for everyone asked. A sector with an angle of 90^\circ is \frac{90}{360}=\frac{1}{4} of the people, however many were asked.

Worked example 2: Comparing a table and a pie chart

Zara asks 50 people which hot drink they like best. Her results are tea 16, coffee 24 and hot chocolate 10. Leo asks 20 people the same question and shows his results in a pie chart. In whose survey did a greater proportion of people choose tea? Show how you decide.

Zara: 16 out of 50 chose tea, and \frac{16}{50}=\frac{32}{100}=32\%
Leo: the tea sector has an angle of 144^\circ, so tea is \frac{144}{360} of his survey.
Your turn. Write \frac{144}{360} as a percentage.

Your answer

144\div360=0.4, which is 40\%
40\% is more than 32\%. Fewer people chose tea in Leo's survey (40\% of 20 is 8, against Zara's 16), but a greater proportion did.
Answer: Leo's survey, since 40\% of his survey chose tea, against 32\% of Zara's.

More results are more reliable

Results from more people, or from more throws of a coin or a dice, are more reliable: they give a better estimate. A few unusual results make less difference when there are lots of results.

Jo throws a coin 20 times and Mo throws the same coin 180 times. Mo's results are more reliable. Name the person and compare: "Mo's, because he threw the coin more times than Jo." Writing only "Mo threw it 180 times" does not compare, so it does not get the mark.

To use both sets of results together, add the counts and add the totals. Jo gets 9 heads and Mo gets 111 heads, so together that is 9+111=120 heads in 20+180=200 throws. The estimate for the probability of heads is \frac{120}{200}=0.6

Do not take the mean of the two results. Mo's 180 throws must count for more than Jo's 20.

Worked example 3: Using two sets of results

Tom and Asha want to find out what fraction of the students at a college cycle to college. Tom asks 20 students chosen at random, and 7 of them cycle. Asha asks 80 students chosen at random, and 21 of them cycle. (a) Whose results are likely to be more reliable? Give a reason for your answer. (b) Use both sets of results to work out an estimate for the fraction of the students at the college who cycle. Give your answer as a decimal.

(a) Asha asked 80 students and Tom asked only 20
More students asked means more reliable results, so name Asha and compare: Asha's, because she asked more students than Tom.
Your turn. (b) Use both sets of results together. What fraction of all the students asked cycle? Give your answer as a decimal.

Your answer

7+21=28 cycle, out of 20+80=100 asked, and \frac{28}{100}=0.28
Taking the mean of Tom's \frac{7}{20}=0.35 and Asha's \frac{21}{80}=0.2625 would be wrong: Asha's 80 students must count for more than Tom's 20
Answer: (a) Asha's, because she asked more students than Tom (b) 0.28

Using a sample to estimate

A sample can be used to estimate for the whole group. Find the fraction of the sample who gave the answer, then find that fraction of the whole group.

\text{estimate}=\frac{\text{number in sample with the answer}}{\text{number in the sample}}\times\text{number in the whole group}

So if 9 people out of a sample of 50 say yes, and the whole group is 400 people, the estimate is \frac{9}{50}\times400=72

A quicker way: divide the number in the whole group by the number in the sample. Here 400\div50=8, so each person in the sample stands for 8 people, and 9\times8=72

Check that your answer is sensible. The estimate cannot be more than the number in the whole group. If it is, you have multiplied or divided the wrong way.

Worked example 4: Estimating from a table of results

A cinema expects 780 people at a film festival. The manager asks a sample of 36 of these people which snack they will buy. Her results are sweets 15, popcorn 12 and nachos 9. Work out an estimate for the number of people at the festival who will buy popcorn.

12 out of the 36 people in the sample chose popcorn.
Divide the top and the bottom by 12: \frac{12}{36}=\frac{1}{3}
780\div36 is not a whole number, so find \frac{1}{3} of 780 instead.
Your turn. Work out an estimate for the number of people at the festival who will buy popcorn.

Your answer

\frac{1}{3} of 780 is 780\div3=260
Check: 260 is less than 780, so the answer is sensible.
Answer: 260 people

Writing down an assumption

When you use a sample to estimate, you are assuming the sample is representative: that the whole group would answer in the same way as the people in the sample.

So a good answer is "I assumed the sample is representative of all the students", or "I assumed the other students want the same as the students in the sample".

If you are asked how this could affect your answer, say which way: if the sample is not representative, the estimate could be too high or too low.

An assumption is not a description of your working, and it is not a complaint about the sample. "I divided 1800 by 60" and "The sample is too small" are not assumptions, so they get no marks.

Worked example 5: Estimating and writing down an assumption

A theatre has sold 1800 tickets for a show. In a random sample of 60 ticket holders, 22 say they will buy an ice cream at the interval. (a) Work out an estimate for the number of ice creams that will be sold. (b) Write down one assumption you have made. Explain how it could affect your answer.

(a) In the sample, 22 out of 60 will buy an ice cream, which is \frac{22}{60} of the sample.
1800\div60=30, so each person in the sample stands for 30 ticket holders.
Your turn. 22 people in the sample said yes. Work out the estimate for all 1800 ticket holders.

Your answer

22\times30=660 ice creams
(b) Assumption: the sample is representative, so the same fraction of all 1800 ticket holders will buy an ice cream.
Effect: if the other ticket holders are less keen on ice cream than the sample, fewer will be sold, and 660 is too high.
Answer: (a) 660 (b) the sample is representative of all the ticket holders; if it is not, the estimate could be too high or too low.

Common mistakes

Describing your working, or criticising the sample, instead of giving an assumption. An assumption is something you took to be true, such as "the sample is representative of everyone".
Subtracting, or dividing the wrong way, when scaling up. Divide the whole group by the number in the sample, then multiply by the number who gave the answer, and do not stop after dividing; an estimate bigger than the whole group must be wrong.
Rounding a fraction too early. Using 0.3 or 0.33 for one third makes the estimate wrong, so keep the fraction.
Choosing the results with more heads, or with fewer throws, as more reliable. More throws, or more people asked, give more reliable results; name the person and compare.
Taking the mean of two results instead of using them together. Add the two counts and add the two totals, then write the answer as a fraction or a decimal.