Distribution of proportion

In a random sample from a categorical population with probability π of success, the number of successes, x , has a binomial distribution,

X  ~  binomial (n,  π)

The sample proportion,  p  =  x / n, has a distribution with the same shape but scaled by a factor 1/n. From the properties of the binomial distribution, its distribution has mean and standard deviation

μp  =  π

σp  = 

Distribution of estimation error

When the proportion p is used to estimate π, the estimation error is p - π. The error distribution therefore has the same shape as that of p, but is shifted to have mean zero. The bias and standard error of the sample proportion are therefore

bias  =  μerror  =  0

standard error  =  σerror  = 

Standard error from data

Unfortunately, the formula for the standard error of p involves π, and this is unknown in practical problems. To get a numerical value for the standard error, we therefore replace π with our best estimate of its value, p .

bias  =  μerror  =  0

standard error  =  σerror  = 

Rice survey

In the rice survey, a proportion p = 17/36 = 0.472 of the n = 36 farmers used 'Old' varieties. The number using 'Old' varieties should have a binomial distribution,

X  ~  binomial (n = 36,  π)

The diagram below initially shows this distribution with π replaced by our best estimate, p = 0.472.

Use the pop-up menu to display the (approximate) distributions of the sample proportion, p, and the estimation error. Observe that all three distributions have the same basic shape — only the scale on the axis changes.

We estimated that a proportion 0.472 of farmers in the region use 'Old' varieties. From the error distribution, it is unlikely that this estimate will be in error by more than 0.2.