Distribution of proportion

The proportion of successes from a random sample with probability π of success, p , has a distribution with mean and standard deviation

μp  =  π

σp  = 

Distribution of estimation error

The estimation error is p - π and its distribution has the same shape as that of p, but is shifted to have mean zero. The bias and standard error 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  = 

Example

In a random sample of n = 36 values, there were x = 17 successes. Our best estimate of π is the sample proportion, p  = 17/36. Using this estimate, the distribution of the number of successes in similar samples would be

X ~ binomial (n = 36, π = 17/36)

The proportion of successes in similar samples would have a scaled form of this distribution

and the error distribution would shift this to have mean zero:

From this error distribution, it is unlikely that our estimate of the proportion of successes (17/36) would be in error by more than 0.2.