How can 1,000 people speak for a country?
Tonight's idea
A poll asks a few people and uses their answers to speak for everyone. The people asked are the sample. If they are picked at random, the sample works like a spoonful of well-stirred soup: one spoonful tells you how the whole pot tastes. A bigger pot does not need a bigger spoon.
The sample will not match the country exactly. How far off? That distance is the margin, in percentage points. About 19 random polls in 20 land within it.
Rule of thumb: margin is about \(1 \div \sqrt{n}\), where \(n\) people were asked and \(\sqrt{n}\) is the number that times itself makes \(n\).
Ask 100: \(1 \div 10 = 0.10\), or 10 points. Ask 400: \(1 \div 20\), 5 points. Ask 1,600: \(1 \div 40\), 2.5 points. Ask 10,000: \(1 \div 100\), 1 point.
The country's size is nowhere in the rule. Only the sample's size counts. A poll of 1,000 sits between 400 and 1,600, so its margin is about 3 points.
Watch one: is Ana ahead?
A random poll of 400 voters gives Ana 54% and Ben 46%. \(\sqrt{400} = 20\), so the margin is \(1 \div 20\), or 5 points. Ana's true share is probably between 49% and 59%. That range dips below 50%, so Ben could still be ahead. Too close to call.
A poll of 1,600 gives the same 54%. Now the margin is 2.5 points: between 51.5% and 56.5%. All of that is above 50%, so Ana is very likely ahead.
Draw sample after sample from a town of 5,000 and watch the spread: mathdetours.com/demos/sample.html.
Your turn
- A random poll of 1,600 students finds that 38% cycle to school. (a) What is the margin? (b) Between which two percentages is the true figure likely to lie?
- How many times as many people must you ask to cut the margin (a) from 10 points to 5; (b) from 5 points to 2.5; (c) from 10 points to 1? (d) Finish the sentence: to halve the margin, ask ____ times as many people.
- A random poll of 400 people finds that 52% want a new park. (a) Can you be confident that more than half the town wants it? (b) A random poll of 10,000 also finds 52%. Can you be confident now?
- (a) Country A has 5 million people and country B has 300 million. Each asks 1,600 people at random. Which poll has the smaller margin? (b) A news website asks its own readers, "Do you read the news online?" and 10,000 reply. A random poll asks 400 people the same question. Which result do you trust more, and why?
Stretch: what does a point cost?
Each interview costs $5. What does a poll cost if you want a margin of 5 points? Of 2.5 points? Of 1 point? Most polls ask about 1,000 people. Use your three prices to explain why they stop there.
Answers
- (a) \(\sqrt{1600} = 40\) and \(1 \div 40 = 0.025\), so 2.5 points. (b) Between 35.5% and 40.5%.
- (a) From 100 people to 400: 4 times. (b) From 400 to 1,600: 4 times. (c) From 100 to 10,000: 100 times. (d) Four times as many.
- (a) No. The margin is 5 points, so the truth is likely between 47% and 57%, and part of that range is below 50%. (b) Yes. The margin is 1 point, so the truth is likely between 51% and 53%, all above 50%.
- (a) Neither. Both asked 1,600, so both have a margin of 2.5 points. The size of the country does not come into it. (b) The random 400. Everyone who answers a website's poll is already online, so that sample is lopsided, and no number of replies fixes that. The rule of thumb only works for a random sample. See Ask 50 people, learn about 5,000.
Stretch. 5 points needs 400 people: \(400 \times 5 = \$2{,}000\). 2.5 points needs 1,600: \(\$8{,}000\). 1 point needs 10,000: \(\$50{,}000\). The first halving costs \(\$6{,}000\) extra. The step from 2.5 points to 1 costs \(\$42{,}000\) extra for only 1.5 points. Each extra point of accuracy costs far more than the last, so around 1,000 people is where most pollsters stop.