Key takeaway: The standard error measures how far a sample statistic typically lands from the quantity it estimates, and for a mean it equals the sample standard deviation divided by the square root of the sample size.
Grade 12 – AP StatisticsAP Statistics Unit 2Statistics and Data
Definition of Standard Error
The standard error (SE) is the standard deviation of a statistic’s sampling distribution. It tells how much a sample mean or proportion would typically bounce around from sample to sample, so it measures the precision of an estimate.
It is not the spread of the data itself. Individual values may vary a lot while the average of many of them varies little. Because it contains the square root of the sample size, collecting more data shrinks the standard error, but with diminishing returns.
Formula and notation
Using sample values to estimate the spread:
\[ SE_{\,\overline{x}}=\frac{s}{\sqrt{n}},\qquad SE_{\hat{p}}=\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}. \]
Here \(s\) is the sample standard deviation and \(n\) the sample size. Margins of error are built from multiples of the SE.
Key facts
- A smaller SE means a more precise estimate.
- To halve the SE, the sample size must be multiplied by four.
- The SE of a sample mean is smaller than the standard deviation of the individual values.
- Confidence intervals and test statistics both rely on the SE.
Standard Error example
A sample of 49 customers at a food truck has a mean wait of 11.6 minutes with standard deviation 21 minutes. Find the standard error of the mean, then the value if the sample had 196 customers.
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For 49 customers: \(SE=\frac{21}{\sqrt{49}}=\frac{21}{7}=3\).
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For 196 customers: \(SE=\frac{21}{\sqrt{196}}=\frac{21}{14}=1.5\).
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The sample is four times larger, and the SE is half as big.
The standard errors are 3 minutes for \(n=49\) and 1.5 minutes for \(n=196\).
Common mistake
Watch out
A student reports the standard error as 21 minutes, the sample standard deviation, and calls the mean wait “11.6 plus or minus 21”.
Why it is wrong
The 21 describes how individual waits spread out. The sample mean varies far less, by \(\frac{21}{\sqrt{49}}=3\) minutes. Dividing by the square root of the sample size is the step that turns a standard deviation into a standard error.
Related terms and lessons
- AP Statistics · Sampling DistributionsLessonWorksheetQuiz
- AP Statistics · Inference for MeansLessonWorksheetQuiz
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