Sample Statistics and Margin of Error — SAT Math

Introduction

When researchers want to learn something about a large population — say, the percentage of a city's residents who use public transit — they can't survey everyone. Instead, they draw a sample, compute a sample proportion, and use it to estimate the true population proportion (or a population total). This skill falls under Problem-Solving and Data Analysis, which makes up about 15% of SAT Math questions. You'll be asked to read an estimate alongside its margin of error, understand what that interval does and does not mean, and recognize how sample size affects precision.

Note scope: This note covers estimating a population total from a sample proportion, reading an estimate ± its margin of error as a range of plausible values, what a margin of error does and does not claim, and how sample size affects it. For the underlying ideas of mean, median, and range, see the Mean, Median, and Range note.

Core Concept

From Sample Proportion to Population Estimate

If a sample of $n$ individuals contains $k$ with some characteristic, the sample proportion is:

\hat{p} = \frac{k}{n}

To estimate the number of individuals in a population of size $N$ who have that characteristic, multiply:

\text{Estimated population count} = \hat{p} \times N

Example: 60 out of 400 sampled commuters bring a packed lunch. $\hat{p} = 60/400 = 0.15$ . In a city of 20,000 commuters, the estimated count is $0.15 \times 20{,}000 = 3{,}000$ .

Margin of Error: A Range, Not a Guarantee

Because a sample is not the whole population, the sample proportion will rarely equal the true population proportion exactly. The margin of error (MoE) captures the typical size of that gap.

When a survey reports $\hat{p} \pm \text{MoE}$ , the plausible range for the true population proportion is:

(\hat{p} - \text{MoE}) \quad \text{to} \quad (\hat{p} + \text{MoE})

Example: A survey finds $35\%$ of registered voters plan to vote, with a margin of error of $\pm 5$ percentage points. The plausible range is $30\%$ to $40\%$ .

What the MoE does NOT mean

It does not mean exactly $35\%$ is correct.
It does not mean $5\%$ of respondents were undecided.
It does not guarantee the true proportion is in that interval — but the interval is your best, mathematically grounded guess.

Sample Size and Margin of Error

A rough but useful benchmark: for a proportion estimate at a standard confidence level,

\text{MoE} \approx \frac{1}{\sqrt{n}}

The key takeaway: doubling the sample size does not halve the MoE — you need to quadruple the sample size to halve it. The relationship is:

\text{MoE} \propto \frac{1}{\sqrt{n}}

Sample size $n$	Approximate MoE
100	$\approx 10\%$
400	$\approx 5\%$
900	$\approx 3.3\%$
1,600	$\approx 2.5\%$

The SAT will not ask you to compute MoE from scratch. It will ask you to compare MoEs across different sample sizes, or to apply a given MoE correctly.

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