Standard Error
In short: Standard error estimates how far a sample statistic, like a mean or proportion, typically lands from the true population value, and it shrinks as the sample size grows.
AP StatisticsStatisticsAP Statistics Unit 2
What is Standard Error?
The standard error is the standard deviation of a statistic’s sampling distribution, estimated from the data at hand. It answers: “if I drew another sample of this size, how much would my answer typically move?”
For a sample mean, \(SE = \frac{s}{\sqrt{n}}\). For a sample proportion, \(SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\). Because \(n\) sits under a square root, cutting the standard error in half requires four times as many observations. Standard error is the building block of both confidence intervals and significance tests.
How to spot it
- A formula has s divided by the square root of n, or p-hat times (1 – p-hat) over n under a root.
- The question asks how precise an estimate is, or how much it varies from sample to sample.
- A confidence interval or test statistic is being built, and you need the denominator.
- A computer output lists “SE Mean” or “SE Coef” beside an estimate.
The method
- Identify the statistic: a mean (use \(s\) and \(n\)) or a proportion (use \(\hat{p}\) and \(n\)).
- Substitute into \(\frac{s}{\sqrt{n}}\) or \(\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\).
- Compute with care: take the square root of \(n\) (or of the entire fraction) before dividing.
- Interpret in context: the estimate typically varies about one standard error from the true value.
Standard Error: solved example
Problem
A random sample of 36 students reports an average of 4.2 hours of daily screen time with \(s = 1.8\) hours. Another survey finds that 120 of 400 sampled adults own a smartwatch. Find the standard error of the mean and of the proportion.
Solution
Mean: \(SE = \frac{1.8}{\sqrt{36}} = \frac{1.8}{6} =\) 0.3 hour.
Proportion: \(\hat{p} = \frac{120}{400} = 0.30\), so \(SE = \sqrt{\frac{0.30 \times 0.70}{400}} = \sqrt{0.000525} \approx\) 0.0229, about 2.3 percentage points.
Try it
Question 1
If \(s = 12\) and the sample size increases from 16 to 64, how does the standard error of \(\bar{x}\) change?
Show the answer
From \(\frac{12}{4} = 3\) to \(\frac{12}{8} = 1.5\): it is cut in half when \(n\) is multiplied by 4.
Question 2
A poll of 100 people has \(\hat{p} = 0.50\). Find the standard error.
Show the answer
\(\sqrt{\frac{0.5 \times 0.5}{100}} = \sqrt{0.0025} = 0.05\).
Standard Error vs. Standard Deviation
The standard deviation measures how spread out individual values are in one data set. The standard error measures how spread out a statistic, such as the mean, would be from sample to sample. The second is always smaller by the factor \(\sqrt{n}\), because averages are steadier than single values.
Keep going
Lessons
Worksheets and quizzes
- Sampling Distributions and the Central Limit Theorem – Worksheet
- Sampling Distributions and the Central Limit Theorem – Quiz
- t Intervals and t Tests for a Mean – Worksheet
- t Intervals and t Tests for a Mean – Quiz
Related words
Whole dictionary: A–Z · AP Statistics · Statistics