Surveys and the margin of error.
Every poll headline ("45%, ±3%") is one descriptive number, one inferential leap, and one confidence claim. The ± tells us how big the leap is.
A poll headline scrolls past: "45% of likely voters favor the candidate, ±3%." We saw this same headline back in Lesson 1, when we used it to name the difference between descriptive and inferential statistics. The 45% is descriptive — the proportion the pollsters measured in their sample of interviewed voters. The ±3% is inferential — an honest admission that the 1,000 people they called might not perfectly represent the 150 million voters they're trying to predict.
This lesson unpacks the ±. The margin of error comes from two ideas we've already met: the standard deviation (Lesson 3) and the empirical rule (Lesson 5). Sample proportions are themselves normally distributed, and the standard deviation of that distribution shrinks as the sample size grows. The headline's ±3% is just two standard deviations of the sample proportion, scaled into percent.
By the end of this lesson, we can read any poll, compute a rough margin of error from the sample size alone, and explain to a friend what "95% confidence" means — which closes the loop on every Topic 5 objective and finishes the data half of the course.
The poll headline, anatomized.
For most introductory work, the margin of error for a poll at 95% confidence is approximated by the rule of thumb
MOE ≈ 1 / √n (as a proportion)
or equivalently MOE ≈ 100 / √n in percentage points. The crucial feature is the square root: doubling the sample size doesn't halve the MOE, it only shrinks it by a factor of √2 ≈ 1.41. To cut the MOE in half, the sample size has to quadruple.
The confidence interval is just the sample value with the MOE attached on each side: sample value ± MOE. When a poll reports "45% ± 3%," the confidence interval is 42% to 48%, and the claim is: "with 95% confidence, the true population proportion is somewhere in that range."
Left: a poll's sample proportion (45%) plus its 95% confidence interval (42-48%), the way a statistician reads a headline. Right: the rule of thumb — MOE shrinks like 1/√n, so 100 interviews give about ±10%, 400 give about ±5%, and 1,000 give about ±3%. To cut the MOE in half, we have to quadruple the sample size.
The margin of error (MOE) is the ± attached to every inferential claim — the honest acknowledgment that a sample is a stand-in for the population, and the stand-in may be off by some amount. For a 95% sample-proportion headline, MOE shrinks like 1 / √n.
Words you'll see on every poll story
- Sample size (n) The number of people (or items) actually interviewed / measured. Larger n is more expensive to collect but produces a tighter MOE. National political polls typically use n between 600 and 1,500.
- Sample proportion (p̂) Pronounced "p-hat." The proportion of the sample that gave a particular answer (e.g., 0.45 = 45% favoring the candidate). Our best estimate of the population proportion p, but not the same number.
- Margin of error (MOE) Half the width of the confidence interval. Rule of thumb at 95% confidence: MOE ≈ 1 / √n, or as percentage points ≈ 100 / √n %. Reported alongside the sample proportion in any reputable poll.
- Confidence interval (CI) The range sample proportion ± MOE. For "45% ± 3%," the CI is 42% to 48%. Always reported with a confidence level (95% is the convention).
- Confidence level How often the true population value would land inside the confidence interval if we repeated the poll many times. 95% is standard; higher confidence (99%) gives a wider interval, lower confidence (90%) gives a narrower one.
Read the poll: 45%, n = 1,000, 95% confidence.
"A polling firm interviews 1,000 likely voters and reports that 45% favor the candidate. The firm reports the result at 95% confidence. What is the margin of error, and what does the headline actually claim?"
Identify the sample size and sample proportion.
Read straight from the report. n = 1,000 is the number of people interviewed; p̂ = 45% = 0.45 is the proportion of the sample favoring the candidate.
Compute the margin of error.
Use the rule of thumb at 95% confidence: MOE ≈ 1/√n. With n = 1,000:
Build the confidence interval.
Take the sample proportion and add/subtract the MOE on each side.
Translate to plain English.
The poll's claim, said carefully: "We are 95% confident that the true fraction of likely voters favoring the candidate is somewhere between about 42% and 48%." Most journalism rounds the MOE to a single digit ("±3%") and the headline reads "45% support, ±3%."
Sanity check.
If a tighter MOE were needed (say, ±1%), the sample size would have to grow to about n = 1/(0.01)² = 10,000 voters. Most polls don't bother because the marginal cost of going from 1,000 to 10,000 interviews is huge and the additional precision rarely changes any decision based on the result.
Three problems. The rule of thumb is enough.
A campus club surveys n = 100 students. Using the rule of thumb, what is the margin of error at 95% confidence, in percentage points?
A national poll interviews n = 1,225 likely voters. Using the rule of thumb, what is the approximate margin of error at 95% confidence, in percentage points? Round to one decimal.
A poll of n = 400 voters finds 60% support for a ballot measure. Using the rule of thumb, what is the upper bound of the 95% confidence interval, in percent?
Three fast questions before you move on.
Q1. A poll reports "52% support, ±4%" at 95% confidence. The honest claim being made is...
Q2. If you wanted to cut the margin of error in half, you would need to...
Q3. Which two earlier Topic 5 ideas combine to produce the margin of error?
Topic 5, in five sentences.
That closes the chapter. The course's data half is now in our hands, and it fits in five sentences. Statistics comes in two flavors: descriptive describes the data we have, inferential reaches beyond it. Center (mean, median, mode) and spread (range, SD) together summarize any data set in just two kinds of number. Charts make those two numbers visible at a glance. The normal distribution — the bell curve — is the shape most real-world measurements tend toward, and the 68-95-99.7 rule reads percentages off it instantly. Polls and surveys use exactly that bell-curve machinery to turn a sample of 1,000 voters into a defensible claim about 150 million, with the honest size of the leap reported as a margin of error.
Next: Topic 5 review brings together the six ALEKS questions on median, mode, all-three computations, range, the empirical rule, and the mean-as-balance picture. After that, Topic 6 picks up probability and expected value — the other half of the data-and-chance branch of the course.
Back to Topic 5Different angle? Need another rep? These are optional — tap any that look helpful.
Sampling Methods and Bias with Surveys — Crash Course Statistics #10
How polls actually work: sample, population, sources of bias. Doesn't compute margin of error itself — pair with the confidence-intervals video below.
Confidence Intervals — Crash Course Statistics #20
The '47% support, ±3%' polling framing is directly demonstrated using the election-polling example. Note: this episode crosses slightly into z-score territory after ~7 minutes — for a 100-level audience, watch the first portion (polling + margin of error) and stop before the z-score formulas.