Math

Margin of Error

2 min readEasy5-question drill

When a poll says '52% support the measure, plus or minus 3 points,' that little 'plus or minus' is the whole point — and the test loves to check whether you actually understand what it means.

Where this lesson fits

Margin of Error

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→Evaluating Statistical Claims: Observational Studies and Experiments →One-Variable Data: Distributions and Measures of Center and Spread →Percentages

When researchers want to know what a whole population thinks, they can't ask everyone. So they take a sample — a smaller group — and measure that. But a sample is never a perfect copy of the population, so the result comes with a margin of error: a cushion of uncertainty around the sample's number.

Here's the key idea. If a survey says 52% with a margin of error of ±3 percentage points, you build an interval:

Low end: 52% - 3% = 49%
High end: 52% + 3% = 55%

This is called a confidence interval. The correct interpretation is: the true value for the whole population is plausibly between 49% and 55%. That range is your answer to 'what does the population really think?'

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A sample of 52% with ±3 points gives a confidence interval of 49% to 55% for the population.

Three things the test checks over and over:

1. The interval is about the POPULATION, not the sample. We already know exactly what the sample said (52%). The margin of error tells us how far the population's true value might be from that sample number. Wrong answers often say 'between 49% and 55% of the sample' — that's a trap.

2. Bigger sample → smaller margin of error. The more people you ask, the closer your sample gets to the truth, so the cushion shrinks. Specifically, the margin of error is inversely proportional to the square root of the sample size (√n). To cut the margin in half, you must make the sample 4 times bigger (because √4 = 2).

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Sample size vs. margin of error

Sample size	Effect on margin of error	Why
Small (e.g. 250)	Larger margin	Less representative of population
Large (e.g. 1000)	Smaller margin	More representative; scales with 1/√n
×4 the size	Margin cut in half	√4 = 2

Bigger samples shrink the margin of error.

3. Random sampling matters. The margin of error formula only works if the sample was chosen randomly from the population. If the survey only asked people leaving a gym, you can't generalize to everyone — no margin of error fixes a biased sample.

You almost never have to calculate a margin of error on the test. Instead you interpret it (build the interval, describe what it means) or reason about it (what happens if the sample gets bigger or smaller). Get those two skills down and this topic is easy points.

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Quick check

A biologist tags 40 fish in a lake and releases them. A week later, she catches a sample of 100 fish and finds that 8 are tagged. Using the capture-recapture method, what is the estimated total number of fish in the lake?

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800

Worked examples

Example 1

A poll of 600 registered voters found that 47% favor a new transportation bill, with a margin of error of ±4 percentage points. Based on this poll, the proportion of ALL registered voters who favor the bill is plausibly between what two values?

Example 2

A survey of 250 randomly selected students found that 70% attend at least one school event per year, with a margin of error of ±6 percentage points at a 95% confidence level. If the researchers had instead surveyed 1,000 randomly selected students, what would most likely happen to the margin of error?

Example 3

A researcher reports that 38% of a random sample of 900 commuters use public transit, with a margin of error of ±3 percentage points. Which statement is the BEST interpretation of this result?

Common pitfalls

Applying the interval to the sample instead of the population

We already know the sample's exact percentage — the margin of error is about how far the whole population's true value might be. Any answer that says 'X% of the sample' is wrong.

Thinking bigger sample = bigger margin of error

It feels like 'more people = more spread,' but it's the opposite. A larger sample is more representative, so the margin of error shrinks. More data = more precision.

Forgetting the √n relationship

To halve the margin of error you need 4× the sample, not 2×, because margin of error scales with 1/√n. Doubling the sample only shrinks it by a factor of √2 ≈ 1.4.

Ignoring whether the sample was random

Margin of error only lets you generalize to the population if the sample was randomly selected. A convenience sample (e.g. only people at a gym) can't be fixed by any margin of error.

Key takeaways

Confidence interval = sample percent ± margin of error (subtract and add).
The interval describes the true POPULATION value, not the sample.
Larger sample size → smaller margin of error (inversely proportional to √n).
To halve the margin of error, multiply the sample size by 4.
Valid generalization requires a random sample.

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Try it yourself

5 practice questions on Margin of Error, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

Margin of Error

You are here

→Evaluating Statistical Claims: Observational Studies and Experiments →One-Variable Data: Distributions and Measures of Center and Spread →Percentages