standard error formula
The standard error (SE) measures how much a sample statistic (usually the mean) varies from the true population parameter.
Standard Error of the Mean
$$SE = \frac{s}{\sqrt{n}}$$
Where:
- s = sample standard deviation
- n = sample size
If the population standard deviation σ is known:
$$SE = \frac{\sigma}{\sqrt{n}}$$
Key Points
- As n increases, SE decreases - larger samples give more precise estimates.
- SE is not the same as standard deviation: SD measures spread within a sample; SE measures precision of the sample mean as an estimate of the population mean.
- SE is used to construct confidence intervals: Mean ± (z × SE) or Mean ± (t × SE)
Other Common Standard Errors
| Statistic | Formula |
|---|---|
| Proportion | $SE = \sqrt{\dfrac{p(1-p)}{n}}$ |
| Difference of two means | $SE = \sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}$ |
| Regression coefficient | $SE_{\hat{\beta}} = \sqrt{\dfrac{MSE}{\sum(x_i - \bar{x})^2}}$ |
Example
Sample: n = 25, s = 10
$$SE = \frac{10}{\sqrt{25}} = \frac{10}{5} = 2$$
This means the sample mean is expected to be within about 2 units of the true population mean (on average).
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