Sample size determination

Sample size determination is the formula-based calculation of how many observations a survey needs to reach a target margin of error at a chosen confidence level, given the population's estimated variability.

ByHoang TruongUpdated July 5, 2026

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A hotel chain wants its estimate of average hold time accurate to within 2 minutes at 95 per cent confidence, with a population standard deviation of 15 minutes. Plugging in, n = (1.96 × 15 ÷ 2)² = 216.09, which must always round up rather than down, so 217 calls need to be sampled to guarantee the target precision.

Where it fits

SubjectData Analysis & StatisticsCoreTopicEstimation & Sampling DistributionsCore

The formula

LaTeX

n = \left(\dfrac{z \sigma}{E}\right)^2

Variables

$n$: Required sample size (observations)
$z$: z-score for the chosen confidence level
$σ$: Population standard deviation (€)
$E$: Target margin of error (€)

Gives the minimum sample size needed to estimate a population mean within a target margin of error at a chosen confidence level.

LaTeX

n = \dfrac{z^2\, p(1-p)}{E^2}

Variables

$n$: Required sample size (observations)
$z$: z-score for the chosen confidence level
$p$: Estimated population proportion
$E$: Target margin of error, expressed as a proportion

Gives the minimum sample size needed to estimate a population proportion within a target margin of error; p is often set to 0.5 for a conservative estimate.

Check yourself

PracticeCORE

A hotel chain wants to estimate the average number of minutes guests spend on hold with its call centre. It targets a margin of error of 2 minutes at 95% confidence (z = 1.96), and past call logs suggest a population standard deviation of 15 minutes. How many calls must be sampled?

Select an answer to check your understanding.