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Standard error: importance and formulas

Short answer

Standard error is the standard deviation of a statistic's sampling distribution, measuring sampling variability. It is used for reliability, confidence intervals, hypothesis tests, sample size determination, comparing samples, and control limits. For sample mean: sigma/√n; for sample proportion: sqrt(PQ/n).

Define Standard error and explain its importance in inferential statistics and write down the formula of standard error of sample mean and sample proportion.

Standard error (SE): the standard deviation of the sampling distribution of a statistic. It measures how much the statistic (for example x̄ or p̂) varies from sample to sample because of chance.

Importance in inferential statistics

  1. Reliability of an estimate: a small standard error means the sample statistic is close to the population parameter; a large one means it is unreliable.
  2. Confidence intervals: the interval is statistic ± z × SE (or t × SE).
  3. Testing of hypothesis: the test statistic is (statistic − hypothesised value)/SE, such as z = (x̄ − μ₀)/(σ/√n).
  4. Sample size: SE decreases as n increases (proportional to 1/√n), so it is used to find the sample size needed for a required precision.
  5. Comparing samples: the difference between two statistics is judged against its standard error to decide if it is significant or only due to sampling fluctuation.
  6. Control limits: limits such as μ ± 3 SE are used in quality control charts.

Formulas

StatisticStandard error
Sample mean x̄SE(x̄) = σ/√n (σ unknown, large n: s/√n)
Sample mean, finite populationSE(x̄) = (σ/√n) √((N − n)/(N − 1))
Sample proportion p̂SE(p̂) = √(PQ/n) (if P unknown: √(p̂q̂/n))

where σ = population standard deviation, s = sample standard deviation, P = population proportion, Q = 1 − P, n = sample size, N = population size.

✓ Answer checked by Claude Sonnet 5.5

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