STANDARD ERROR OF A PROPORTION:


The proportion is the mean of a random variable, \(X,\) that is \(x_i =1\) when the individual has a trait, and \(x_i = 0\) otherwise. The standard deviation involves the expression \(\displaystyle \sum_{i=1}^n (x_i-\bar x)^2\).

Let’s suppose there are \(m\) \(x_i = 1\)’s (and \(n-m\) \(x_i = 0\)’s) among the \(n\) subjects. Then, the mean corresponds to the proportion \(\bar x = \hat p = \frac{m}{n}\) and \(x_i-\bar x\)is equal to \(1-\frac{m}{n}\) for \(m\) observations and \(0-\frac{m}{n}\) for \(n-m\) observations. Hence,

\[\begin{align}\sum_{i=1}^n (x_i-\bar x)^2 &= m\left(1-\frac{m}{n}\right)^2 + (n-m)\left(0-\frac{m}{n}\right)^2\\[2ex] &=m\left(1-2\frac{m}{n}+\frac{m^2}{n^2}\right) +(n-m)\left(\frac{m^2}{n^2}\right)\\[2ex] &=m-2\left(\frac{m^2}{n}\right)+\frac{m^3}{n^2}+\frac{m^2}{n}-\frac{m^3}{n^2}\\[2ex] &=m-\frac{m^2}{n}\\[2ex] &=m\left(1-\frac{m}{n}\right)\\[2ex] &=n\hat p(1-\hat p) \end{align}\].

The sample variance is,

\[\mathrm{var}=\frac{\sum_{i=1}^n (x_i-\bar x)^2}{n}=\hat p(1-\hat p)\]

The sample proportion is the mean of \(n\) observations, so the standard error of the proportion is calculated like the standard error of the mean: the SD divided by the square root of the sample size or

\[\mathrm{SE}=\sqrt{\frac{\hat p(1-\hat p)}{n}}\]


References:

The Standard Error of a Proportion


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