# Why do we need a smaller sample size to detect a smaller proportion?

The plot below shows the sample size needed to detect a proportion with a precision 0.01 for various true proportions: This assumes an infinite population size, and the confidence intervals are fixed at 0.95.

If the true proportion is very small (or equivalently, very large) then we need a sample size of under 1000 to detect the proportion. However, if the true proportion is 0.5 then we need a sample size of almost 10,000.

I understand mathematically why this is from using the formula to calculate population size. But I would like an intuitive understanding of why this is the case.

It seems more intuitive to me that small proportions would need larger samples to detect them. Is this because I have kept the precision fixed (1% ± 1% is a much larger percentage error than 50% ± 1%)?

• Somebody informs you of the truth, "red and green balls in this sack are very disproportional" and asks, "tell me which colour dominates". How many draws of a ball from the sack do yo think you will need before you can answer which colour prevails? Many draws or few draws? And what if they inform you of the truth "red and green balls are about balanced, yet one colour is a bit ahead"."Which is it?" Will you draw few or many times before you can answer it with confidence? Aug 24, 2022 at 11:52
• (in this example, the "information about the truth" corresponds to the assumption of the population proportion; and the question "who dominates" is like asking "is it far away from 50%", i.e. in a way akin to estimating a proportion) Aug 24, 2022 at 12:06
• Suppose the actual $p$ was very close to $0$: you could reach the conclusion it is likely to be less than $0.01$ with about $300$ all failures since $0.99^{300}\approx 0.049$. That is related to absolute precision. But finding the proportion with a $1\%$ relative precision might be much harder in that case and need a much larger sample size than if the proportion is close to $0.50$ Aug 24, 2022 at 12:59
• Thanks for the comments - Henry, yours in particular really cleared things up for me. Makes sense! Aug 25, 2022 at 8:47