Why does halving the margin of error quadruple the sample size?

Because the margin of error appears squared in the sample size formula, n = z² p(1-p)/e². Halving e makes e² four times smaller, so n grows four times: from about 385 at ±5 points to about 1,537 at ±2.5 points.

Does margin of error cover all survey errors?

No. It covers only random sampling error, the chance wobble from surveying a sample instead of everyone. It says nothing about bias: a convenience sample, a leading question, or high non-response can be badly wrong while showing a tiny margin of error.

Concept explainer

What is margin of error?

Q: What is a good margin of error for a survey?

Plus or minus 5 percentage points at 95 percent confidence is the common convention for program surveys, needing about 385 respondents. Use ±3 points (about 1,067 respondents) when small differences matter, and ±10 points (about 97) only for rough monitoring reads where direction matters more than the exact rate.

Margin of error is the plus-or-minus band around a survey estimate: how far the number from your sample could reasonably sit from the true number in the whole population, just from the luck of who got sampled. "62 percent, ±5 points" means the truth is plausibly anywhere between 57 and 67 percent. It shrinks as the sample grows, but slowly: with the square root of n.

Margin of error is the price tag on precision. Before fieldwork, you choose the margin you can live with, and that choice, more than anything else, sets your sample size and budget. After fieldwork, it is the honesty label on every number you report.

The formula

e = z × √( p(1 - p) / n ) z is 1.96 at 95 percent confidence, p is the proportion (0.5 gives the widest, safest margin), n is the sample size. Rearranged for planning: n = z² × p(1 - p) / e². For clustered samples multiply n by the design effect.

Sample sizes by margin of error

At 95 percent confidence, for a proportion near 50 percent, large population:

Margin of error	Sample needed	Rough use case
±10 points	about 97	Quick monitoring read
±7 points	about 196	Small program check
±5 points	about 385	Standard program survey
±3 points	about 1,067	When small differences matter
±2.5 points	about 1,537	High-stakes estimates

The square law: why precision gets expensive

Because e is squared in the planning formula, every halving of the margin quadruples the sample. From ±5 to ±2.5 points takes you from 385 to about 1,537 respondents. The first thousand rupees of precision are cheap; the last are brutal. This is why a good sampling plan starts from the decision you need to make ("can we tell 40 percent from 50 percent?") and works backwards, rather than demanding maximum precision by reflex.

Choose the margin your decision actually needs; every extra point of precision is paid for in fieldwork.

What margin of error does not cover

It measures only random sampling error, the wobble of chance. It is silent about bias. A convenience sample of the easiest-to-reach schools can report a beautifully tight margin and still be systematically wrong, because everyone in it was selected the same skewed way. No sample size fixes bias; only better selection does. Treat a tiny margin on a non-random sample as decoration, not evidence.

Frequently asked questions

What is a good margin of error for a survey?

±5 percentage points at 95 percent confidence is the workhorse for program surveys (about 385 respondents). Go tighter (±3, about 1,067) when decisions hinge on small differences; accept ±10 (about 97) for quick directional checks.

Why does halving the margin quadruple the sample?

The margin sits squared in the denominator of n = z²p(1-p)/e². Halving e shrinks e² by four, so n grows by four. Precision compounds in cost, which is why it should be chosen deliberately, not maximally.

Does a small margin of error mean the survey is right?

Only if the sample was drawn fairly. Margin of error assumes random selection; it cannot see bias from convenience sampling, non-response, or leading questions. A biased survey with a tiny margin is precisely wrong.

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