Calculate standard error, margin of error, and confidence intervals for statistical analysis
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Standard error (SE) measures the precision of a sample statistic as an estimate of the population parameter. It quantifies how much the statistic varies from sample to sample.
Where: σ = population standard deviation, n = sample size
When σ is unknown, use sample standard deviation (s)
Where: p = sample proportion, n = sample size
For two independent samples with standard deviations s₁ and s₂
For two independent sample proportions
The margin of error (ME) is the maximum expected difference between the true population parameter and sample estimate:
Confidence intervals provide a range likely to contain the true population parameter:
| Statistic | Confidence Interval Formula |
|---|---|
| Mean | x̄ ± (Z × SE) |
| Proportion | p ± (Z × SE) |
| Difference (Means) | (x̄₁ - x̄₂) ± (Z × SE) |
| Difference (Props) | (p₁ - p₂) ± (Z × SE) |
| Measure | What It Measures |
|---|---|
| Standard Deviation (σ, s) | Spread of individual data points around the mean |
| Standard Error (SE) | Variability of sample statistic across different samples |
Scenario: Sample mean = 105, SE = 2.5, 95% CI
Margin of Error: 1.96 × 2.5 = 4.9
95% CI: [100.1, 109.9]
Interpretation: We are 95% confident the true population mean lies between 100.1 and 109.9
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SD measures variability of data; SE measures variability of a statistic (e.g., mean) across repeated samples and shrinks with larger n.
Confidence intervals are built from estimates ± critical value × SE (e.g., mean ± t*·SE).
For differences in means under equal variance assumptions; otherwise use Welch’s (unpooled) approach.
SE typically scales like SD/√n, so it decreases as sample size grows, reflecting more precise estimates.