Standard Error Formula
Calculate the standard error of the mean with SE = s / sqrt(n).
Understand sampling variability and confidence intervals with examples.
The Formula
The standard error measures how much a sample mean is likely to differ from the true population mean. Larger samples produce smaller standard errors, meaning more precise estimates.
Variables
| Symbol | Meaning |
|---|---|
| SE | Standard error of the mean |
| s | Sample standard deviation |
| n | Sample size (number of observations) |
Confidence Interval
The standard error is used to build confidence intervals around the sample mean. Common z-values: 1.645 (90%), 1.96 (95%), 2.576 (99%).
Example 1
A sample of 36 students has a mean test score of 78 with a standard deviation of 12. What is the standard error?
SE = s / √n = 12 / √36
SE = 12 / 6
SE = 2.0 (the sample mean is likely within about 2 points of the true mean)
Example 2
Using the same data (mean = 78, SE = 2), find the 95% confidence interval.
CI = x̄ ± z × SE = 78 ± 1.96 × 2
CI = 78 ± 3.92
95% CI = [74.08, 81.92] — we are 95% confident the true mean falls in this range
When to Use It
Use the standard error formula when working with sample data:
- Estimating how close a sample mean is to the population mean
- Building confidence intervals for survey results and experiments
- Determining the sample size needed for a desired precision
- Comparing means from different samples or groups
Key Notes
- Formula: SE = σ/√n (or s/√n): The standard error of the mean (SEM) measures how precisely the sample mean estimates the true population mean — not how spread out the individual data points are. Use σ when population standard deviation is known; use sample s otherwise.
- SE vs standard deviation: Standard deviation describes the spread of individual values. Standard error describes the precision of the mean. A large dataset can have large SD and very small SE simultaneously. They answer different questions.
- Square-root diminishing returns: SE decreases as 1/√n. To halve the SE, you must quadruple the sample size. Going from n=100 to n=400 halves the SE — each additional sample contributes less precision than the previous one.
- Confidence interval: x̄ ± z × SE: For a 95% CI, use z = 1.96. The margin of error in opinion polls ("±3 percentage points") is z × SE for a proportion. A smaller SE directly narrows the confidence interval.
- Applications: Standard error is reported in scientific studies to convey precision of measurements, in polling to compute margins of error, in A/B testing to determine if an observed difference is statistically significant, and in meta-analysis to weight study results.