Standard Error
What is standard error?
Standard error (SE) quantifies how much a sample statistic (commonly the sample mean) is expected to vary from the true population parameter. It is the standard deviation of the sampling distribution of a statistic and indicates how accurately a sample represents the population. Larger samples generally produce smaller SEs.
Formula and calculation
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When the population standard deviation (σ) is known:
SE = σ / sqrt(n)
where n is the sample size. -
When σ is unknown (most practical situations), substitute the sample standard deviation (s):
SE ≈ s / sqrt(n) -
Relative standard error (RSE) expresses SE as a percentage of the estimate:
RSE = (SE / estimate) × 100%
Standard error vs. standard deviation
- Standard deviation measures the spread of individual data points around their mean.
- Standard error measures the spread of sample estimates (e.g., sample means) around the true population value.
- SE normalizes variability by sample size: as n increases, SE decreases even if the underlying SD remains the same.
Role in confidence intervals and hypothesis testing
- Confidence intervals: SE is used to build intervals around an estimate (e.g., mean ± zSE or mean ± tSE). Smaller SEs produce narrower, more precise intervals.
- Hypothesis testing: test statistics (z or t) divide the difference between an observed statistic and a hypothesized value by the SE. A smaller SE increases the test statistic magnitude, making it easier to detect statistically significant differences.
Limitations and assumptions
- SE assumes samples are randomly and representatively drawn. Biased or nonrandom samples can produce misleading SEs.
- SE estimates rely on correct variability estimates; with very small samples, s may poorly estimate σ, reducing SE reliability. In such cases, use t-distribution adjustments.
- SE calculations often assume approximate normality of the sampling distribution (central limit theorem). For highly skewed data or extreme outliers, the SE may not reflect true uncertainty.
Example
- Sample of 50 observations, sample standard deviation s = 1.0, estimate = −0.20:
- SE = 1.0 / sqrt(50) = 1.0 / 7.07 ≈ 0.141
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Approximate 95% CI (using ~1.96×SE): −0.20 ± 0.28 → roughly (−0.48, 0.08). If using ±1 SE for a quick sense: −0.20 ± 0.14 → (−0.34, −0.06).
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Sample of 100 observations, s = 0.90, estimate = −0.25:
- SE = 0.90 / sqrt(100) = 0.90 / 10 = 0.09
- Tighter CI than with n = 50, reflecting greater precision from a larger sample.
Practical uses
- Assess precision of estimates (means, proportions, regression coefficients).
- Compare study results: differing SEs explain why similar point estimates can lead to different confidence intervals.
- Inform sample-size planning: desired SE (or CI width) helps determine required n.
- In finance and research, use SE to judge reliability of historical averages, effect sizes, or model coefficients.
Quick FAQs
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What does a small SE mean?
A small SE indicates the sample estimate is likely close to the true population value. -
How is SE computed?
Divide the standard deviation (population σ or sample s) by the square root of the sample size n. -
Is SE the same as standard deviation?
No. SD describes variability of individual observations; SE describes variability of an estimator across repeated samples.
Key takeaways
- SE measures the expected sampling variability of an estimate and shrinks as sample size increases.
- Use SE to form confidence intervals and compute test statistics for hypothesis testing.
- Interpret SE carefully: its usefulness depends on sample representativeness, sample size, and distributional assumptions.