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Sample Size, Margin of Error & the Cost of Precision

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Solving the margin of error backward

The margin of error for a proportion is ME = z √(p(1 − p)/n). Designing a study means fixing the ME you want and solving for n: n = (z/ME)² · p(1 − p). Because n appears under a square root in the original expression, it appears squared when solved for — which is the whole reason precision is expensive. And because n must be a whole number of subjects, you always round up; rounding down would leave the margin of error slightly larger than specified.

Required sample size for a proportion
n = (z*/ME)² · p*(1 − p*) with p* = 0.5 when no prior estimate exists always round UP
For a mean the analogous expression is n = (z*σ/ME)², which requires an estimate of σ from a pilot study or prior data.

Why p* = 0.5 is the safe choice

The quantity p(1 − p) is largest at p = 0.5, where it equals 0.25, and falls off toward zero as p approaches 0 or 1. So using 0.5 when you have no prior information produces the largest required sample size, which guarantees the margin of error will be no worse than specified whatever the true proportion turns out to be. It is a deliberately conservative choice: if the true p is far from 0.5 you will have collected more data than strictly necessary, which is the right way to be wrong.

The fourfold rule

Since ME is proportional to 1/√n, cutting the margin of error in half requires four times the sample. Cutting it to a third requires nine times. This is the dominant practical constraint in survey design: a national poll with a 3% margin of error needs roughly 1,100 respondents, and pushing that to 1.5% needs about 4,300 — nearly four times the cost for a gain most readers would not notice. It is also why polls cluster around the same sample size regardless of who commissions them.

Worked example

A pollster wants a 95% confidence interval for a population proportion with a margin of error no greater than 0.03, with no prior estimate of p. Find the required sample size, then find it for a margin of error of 0.015.

  1. 1.Use z = 1.96 for 95% confidence and the conservative p = 0.5, so p(1 − p) = 0.25.
  2. 2.n = (1.96/0.03)²(0.25) = (65.33)²(0.25) = 4268.4(0.25) = 1067.1, so round up to 1068.
  3. 3.For ME = 0.015: n = (1.96/0.015)²(0.25) = (130.67)²(0.25) = 17071(0.25) = 4267.8, so round up to 4268.
  4. 4.The ratio is about 4268/1068 ≈ 4.0, confirming the fourfold rule.
Answer: About 1,068 respondents for a 3% margin of error and about 4,268 for 1.5% — four times the sample for twice the precision, which is the square-root relationship in practice.
Checkpoint

When planning a study with no prior estimate of p, using p* = 0.5 is recommended because it —

Checkpoint

A sample size calculation produces n = 384.2. The correct sample size to use is —

Answer the 2 checkpoints as you read.

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