Module 07 · Determining sample size

Sample size for a population mean (unknown $σ$ )

Use the sample standard deviation $s_{p}$ from a pilot study to plan your sample. The $t$ critical value adjusts with the planned number of observations.

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Pilot estimate ·

s_{p}

Planned sample size99

Estimated margin of error1.99

Intermediate calculations:

\frac{α}{2}

0.025

Degrees of freedom

n - 1

Critical value

t_{1 - α / 2, n - 1}

1.984

Estimated standard error

\frac{s_{p}}{\sqrt{n}}

1.0050

Plan for 99 observations. Using the pilot standard deviation, the estimated margin of error is approximately 1.99; the final sample may have a different spread and margin of error.

With pilot standard deviation 10, target margin of error 2, and 95 percent confidence, the planned sample size is 99. The planned degrees of freedom are 98, the t critical value is 1.984, and the estimated margin of error is approximately 1.99. This is a planning estimate based on the pilot spread.

The idea

When $σ$ is unknown, a representative pilot study provides an estimate $s_{p}$ . Enter its sample standard deviation, calculated using the pilot’s sample size minus one in the variance denominator. Use the same measurement units as your target margin of error $E$ .

The planner finds the smallest whole number $n \geq 2$ satisfying $t_{1 - α / 2, n - 1} \frac{s_{p}}{\sqrt{n}} \leq E$ . Because the critical value itself depends on $n$ , the calculation checks candidate sizes rather than rounding a single fixed-critical-value formula.

The $t$ degrees of freedom are $n - 1$ for the planned study. The pilot supplies the spread estimate. With $s_{p} = 10$ , $E = 2$ , and 95% confidence, the resulting plan is 99 observations.

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Reduce the target margin from 2 to 1 while holding the pilot standard deviation and confidence level fixed. The required count rises to roughly four times as many observations. The relation is approximate here because the $t$ critical value also changes with the planned sample size.

$E$ is half the interval’s total width. This plan treats $s_{p}$ as the future spread estimate: a more variable final sample can produce a wider interval than the target. Consider whether the pilot represents the population and conditions of the planned study.

The planned interval assumes independent observations and a normal population or an adequate approximation for the mean; a small or unrepresentative pilot can give an unreliable spread estimate. Reference: Penn State STAT 800, sample-size computation ↗

Next, use sampling uncertainty to evaluate a claim.Explore hypothesis testing

Sample size for a population mean (unknown σ)

See how precision improves

Plan your sample

Intermediate calculations:

The idea

TRY THIS More precision, more data

Sample size for a population mean (unknown $σ$ )