Module 07 · Determining sample size

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

Choose the precision you need, then find how many observations to collect. Start with the known population standard deviation $σ$ , a target margin of error, and a confidence level.

See how precision improves

Known spread ·

σ

Required sample size97

Margin of error at this

n

1.99

Intermediate calculations:

\frac{α}{2}

0.025

Critical value

z_{1 - α / 2}

1.960

Before rounding

{(\frac{z_{1 - α / 2} σ}{E})}^{2}

96.04

Standard error

\frac{σ}{\sqrt{n}}

1.0153

Collect at least 97 independent observations. At 95% confidence, the margin of error is approximately 1.99, meeting the target of 2.

With known population standard deviation 10, target margin of error 2, and 95 percent confidence, the required sample size is 97. The unrounded requirement is 96.04, the critical value is 1.960, and the achieved margin of error is approximately 1.99.

The idea

The margin of error $E$ is the distance from the sample mean to either confidence limit: half the interval’s total width. Decide what precision is useful before collecting the data. For example, a target of 2 minutes means an interval no wider than 4 minutes.

Starting from $E = z_{1 - α / 2} σ / \sqrt{n}$ , solve for the required count: $n = ⌈ {(\frac{z_{1 - α / 2} σ}{E})}^{2} ⌉$ . The ceiling brackets mean round upward to a whole number.

With $σ = 10$ , $E = 2$ , and 95% confidence, the calculation gives approximately 96.04. Choose 97 observations; rounding down to 96 would exceed the target margin.

TRY THIS More precision, more data

Keep $σ$ and the confidence level fixed. Reducing $E$ from 2 to 1 multiplies the unrounded sample-size requirement by four. The whole-number counts are approximately four times as large because each must be rounded up.

A higher confidence level or a more variable population also requires more observations. The sample mean does not enter this planning calculation.

This calculation assumes independent observations and a known population standard deviation, with a normal population or an adequate normal approximation for the sample mean. Reference: NIST / SEMATECH, sample sizes required ↗

What if the population spread is unknown?Plan with a pilot sample

Sample size for a population mean (known σ)

See how precision improves

Plan your sample

Intermediate calculations:

The idea

TRY THIS More precision, more data

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