1. Definition

  • The Standard Error (SE) measures how much a sample statistic (like a mean or proportion) would vary if you took many random samples from the same population.
  • It reflects the uncertainty of the estimate.
  • In other words: it shows how far the sample mean is likely to be from the true population mean (μ).

2. Formulas

(a) For the sample mean

$SE_{\bar{x}} = \frac{σ}{\sqrt{n}}$

  • $σ$ = population standard deviation
  • $n$ = sample size

If σ is unknown, use the sample standard deviation (s):

$SE_{\bar{x}} = \frac{s}{\sqrt{n}}$

(b) For a proportion

$SE_{\hat{p}} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

  • $\hat{p}$​ = sample proportion
  • $n$ = sample size

3. Key Properties

  • Larger samples (n ↑) → SE decreases. Bigger samples give more precise estimates.
  • More variability (σ ↑) → SE increases. More spread in data → more uncertainty.
  • SE is tied to the Central Limit Theorem: with large n, sample means follow a normal distribution with SE as their spread.

4. Examples

Example 1 – Mean

  • Population mean μ = 100, σ = 20, n = 25.

$SE = \frac{20}{\sqrt{25}} = \frac{20}{5} = 4$

The sample mean will typically vary about ±4 around the true mean.

Example 2 – Proportion

  • 1,000 people surveyed; 520 said “yes.” $\hat{p} = 0.52$.

$SE = \sqrt{\frac{0.52 \times 0.48}{1000}} \approx 0.016$

The estimated proportion (52%) has a margin of error of about ±1.6%.

5. Uses

  • Confidence intervals (CI): $\text{Estimate} \pm Z_{\alpha/2} \times SE$
  • Hypothesis testing: test statistics use SE in the denominator.
  • A/B testing: SE is used when comparing conversion rates.

6. SE vs Standard Deviation (SD)

  • Standard deviation (σ or s): Spread of individual data points around the mean.
  • Standard error (SE): Spread of the sample mean (or statistic) around the true population value.

SD = variability in data.
SE = variability in the estimate.

In short:
The Standard Error (SE) measures how much a sample statistic (like a mean or proportion) is expected to vary across repeated samples. It’s smaller with larger samples, and it’s central to confidence intervals and hypothesis testing.