Standard error of a fifty-fifty survey result

Question

In a survey of n = 100 respondents, a fraction \hat{p} = 0.50 answered "yes." What is the standard error of the sample proportion?

Accepted Answer

A proportion is an average of 0s and 1s, so it has a standard error. For a proportion it is 	ext{SE} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}. At \hat{p} = 0.50 the outcome spread \hat{p}(1-\hat{p}) = 0.25 is as large as it can get, so a fifty-fifty split is the noisiest case. With n = 100: 	ext{SE} = \sqrt{\frac{0.50 	imes 0.50}{100}} = \sqrt{\frac{0.25}{100}} = \sqrt{0.0025} = 0.05. So the 50\% result has a standard error of exactly 5 percentage points. Takeaway. The standard error of a proportion is \sqrt{\hat{p}(1-\hat{p})/n}, and it peaks at \hat{p} = 0.5: a fifty-fifty split is the hard…

Standard error of a fifty-fifty survey result

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