Math · Introductory statistics · Worked example

Choose a sample size for a margin of error

A survey should estimate a mean within ±3 points with 95% confidence. A pilot study suggests σ ≈ 15. How many people are needed?

Solve the margin for n

For planning, use z* = 1.96 and the pilot estimate of σ: the margin z*σ/√n must be at most 3.

n = {(\frac{z^{*} σ}{E})}^{2}

{(\frac{1.96 \cdot 15}{3})}^{2} = 96.04

A fractional person is not possible, and rounding down would make the margin slightly too wide, so round up to 97.

At least 97 people.

How many people are needed to halve the margin to ±1.5?

Show the answer and explanation

Halving E quadruples n: (1.96 × 15 ÷ 1.5)² = 384.16, which rounds up to 385.

{(\frac{1.96 \cdot 15}{1.5})}^{2} = 384.16

In Math, change the margin from 3 to 2: the sample size becomes (1.96 × 15 ÷ 2)² = 216.09, so 217 people.

Sources and scope

Authored study material. Tool results depend on the stated inputs and model assumptions.

Make it concrete

Open the example inputs, change a value and keep a useful result on your board.

Your existing work stays on this device. Examples open as editable copies.