A news report says 54% of a town’s voters support a new library, with a margin of error of 3 percentage points. Nobody asked every voter, so 54% is a guess built from a sample. The margin of error says how far that guess could reasonably be off because of which voters happened to be picked. SAT questions on this topic ask you to read that sentence correctly without claiming more than it says.
Key terms
Estimate
Definition An estimate is a value computed from a sample and used in place of an unknown population value. A sample proportion estimates the population proportion, and a sample mean estimates the population mean. The population value is a fixed number that is usually unknown, while the estimate changes from one sample to another. This variation from sample to sample is called sampling variability, and the difference between an estimate and the true value that it causes is called sampling error.
In plain words An estimate is your best single guess about the whole group, based on the part you actually measured. A different random sample would almost always give a slightly different guess.
Example A random sample of 200 of the 12,000 students at a university finds that 70 of them, or , walk to class. The estimate of the proportion of all students who walk is , and the estimated number of walkers is
This is not a count of walkers. It is a reasonable guess for a number nobody counted.
Margin of error
Definition A margin of error is a positive number reported with an estimate from a random sample. It describes how far the estimate might plausibly be from the population value because of sampling variability alone, at a stated confidence level. For a sample proportion from a large simple random sample of size , a common approximate 95% margin of error is
SAT questions usually state the margin of error, so you interpret it rather than compute it. A margin of error accounts only for random sampling variability. It does not cover bias from how the sample was chosen, from questions that are worded unfairly or from people who refuse to answer.
In plain words The margin of error is the wiggle room around the estimate. It is honest about luck in the sample, and silent about every other kind of mistake.
Example Suppose the library poll asked a random sample of 900 voters. Then and , and the approximate 95% margin of error is
which is about 3 percentage points, the margin in the report.
Uncertainty interval
Definition For an estimate with margin of error , the interval of plausible values is
When the interval comes from a random sample, the standard interpretation is that it is plausible the population value lies in this interval. The interval is not a guarantee. A particular reported interval may miss the population value, and nothing in the data tells you whether it did. Values outside the interval are not impossible, only less plausible.
In plain words Put the estimate on a number line and step to the left and to the right. The population value is probably somewhere in that stretch, but you never know for sure.
Example The library poll gives to . The right claim is that it is plausible that between 51% and 57% of all the town’s voters support the library. Saying that exactly 54% support it, or that the support is certainly above 50%, claims more than the poll shows.
The same reading works for means. A random sample of students at a school sleeps an average of hours a night, with a margin of error of hours. The plausible values for the mean of all students at the school run from to hours. The interval says nothing about how long any single student sleeps.
Percentage point
Definition A percentage point is the unit for the difference between two percentages. If a value changes from to , the change is percentage points. The relative percent change is
A margin of error for a percentage is given in percentage points, so it is added to and subtracted from the percentage itself.
In plain words Percentage points measure the gap on the percent scale. Percent change measures the gap compared with where you started.
Example Support rises from 40% to 46%. That is an increase of 6 percentage points, but a relative increase of
For the library poll, a margin of 3 percentage points means 51% to 57%. It does not mean 3% of 54%, which would only be percentage points.
Sample size
Definition The sample size is the number of individuals in the sample. When samples are drawn at random and the method, population and confidence level stay the same, a larger sample generally gives a smaller margin of error. In the approximate formula above, with about the same, is proportional to , so multiplying the sample size by 4 roughly halves the margin of error. A larger sample does not fix a biased sampling method. A huge sample of volunteers still describes only the kind of people who volunteer.
In plain words More randomly chosen people means less luck in the result, so the interval gets narrower. More people chosen badly just gives you a very precise answer to the wrong question.
Example With at 95% confidence, the approximate margins are these.
| Sample size | Margin of error |
|---|---|
| about percentage points | |
| about percentage points | |
| about percentage points |
Each time the sample is four times as large, the margin is cut roughly in half. Compare this with an online poll that gets 20,000 answers from anyone who clicks. No margin of error can describe its bias, because the respondents were not chosen at random.
Confidence level
Definition The confidence level, such as 95%, describes the method that produced the interval, not one particular interval. A 95% confidence level means that if many random samples were drawn in the same way and an interval were built from each, about 95% of those intervals would contain the population value. For the same sample, a higher confidence level gives a wider interval. When you compare margins of error for different sample sizes, the confidence level and the sampling method must be the same, or the comparison is not fair.
In plain words The confidence level is how often the method works in the long run. To be right more often, you have to give a wider range.
Example For and , the 95% margin is about percentage points. The 99% margin uses in place of ,
about percentage points. Same data, wider interval. It is still wrong to say there is a 99% chance that the population value lies in this one interval. The 99% describes how the method behaves over many samples.
Check yourself
- A random sample of 1,200 adults in a city found that 62% favor building a new park. The margin of error is 3 percentage points at the 95% confidence level. Which conclusion is most appropriate? (A) Exactly 62% of adults in the city favor the park. (B) It is plausible that between 59% and 65% of adults in the city favor the park. (C) It is plausible that between 59% and 65% of adults in the whole country favor the park. (D) The percentage of adults in the city who favor the park is certain to be between 59% and 65%.
Show the answer
The correct choice is (B). The interval is to percent, and it describes the population that was sampled, the adults of the city. Choice (A) treats the estimate as exact. Choice (C) extends the result to people who were never sampled. Choice (D) turns a plausible range into a guarantee.
- In a random survey last year, 35% of residents said they ride a bike to work. This year a survey with the same method found 42%. By what percent did the proportion of bike riders increase relative to last year?
Show the answer
The answer is . The change is percentage points, and relative to last year that is
The tempting answer is the change in percentage points, not the relative percent change.
- Two researchers estimate the mean commute time of workers in the same city, both using random samples and both reporting margins of error at the 95% confidence level. Researcher A samples 300 workers and researcher B samples 1,200 workers. Which statement is most likely true? (A) The margin of error of A is smaller, because a small sample is easier to measure carefully. (B) The margin of error of B is smaller, because B used a larger random sample. (C) The two margins of error are equal, because the confidence levels are equal. (D) An online poll of 10,000 workers who choose to answer would have a smaller margin of error than both, so it would be more reliable.
Show the answer
The correct choice is (B). With the same population, method and confidence level, a larger random sample generally gives a smaller margin of error. Here B has four times as many workers, so the margin is roughly half as large. Choice (A) gets the effect of sample size backward, since a smaller sample gives a larger margin. Choice (C) ignores sample size. Choice (D) fails because volunteers are not a random sample, and a larger sample does not repair that bias.