Standard Error Examples in Statistics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Standard Error.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Statistics.

Concept Recap

The standard deviation of a sampling distribution, measuring how much a sample statistic typically varies from the true population parameter.

Standard error tells you how much your sample estimate might be 'off' from the true value. Larger samples have smaller SE because they're more precise - like asking 1000 people vs 10.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Standard error measures how much a sample statistic varies from sample to sample. It decreases as sample size increases, so larger samples give more precise estimates.

Common stuck point: Students confuse standard error with standard deviation. SD measures spread of individual data values; SE measures precision of a sample statistic.

Worked Examples

Example 1

easy
A population has a standard deviation of \sigma = 20. If you take a random sample of n = 100, what is the standard error of the sample mean?

Solution

  1. 1
    Step 1: The standard error (SE) of the sample mean is given by SE = \frac{\sigma}{\sqrt{n}}.
  2. 2
    Step 2: SE = \frac{20}{\sqrt{100}} = \frac{20}{10} = 2.
  3. 3
    Step 3: This means the sample mean is expected to vary by about 2 units from the true population mean across different samples of size 100.

Answer

SE = 2.
The standard error measures the precision of the sample mean as an estimate of the population mean. It decreases as sample size increases, meaning larger samples give more precise estimates. The SE is the standard deviation of the sampling distribution of the mean.

Example 2

medium
How does the standard error change when you quadruple the sample size from n = 25 to n = 100? Assume \sigma = 30.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
A researcher measures the reaction time of 64 participants and finds a sample standard deviation of s = 40 ms. Calculate the standard error and construct an approximate 95% confidence interval if the sample mean is 250 ms.

Example 2

hard
A polling company wants the standard error of a proportion to be no more than 0.02 (2%). If a preliminary estimate suggests \hat{p} \approx 0.5, what minimum sample size is needed? Use SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

standard deviation introsampling distribution