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  = 

Management succession plans

Many small companies do not worry about the consequences of executives resigning, despite the disruption that this can cause to the company.

Coopers & Lybrand surveyed 210 chief executives of fast-growing small companies and the table below shows the number whose companies had a management succession plan to deal with such departures.

Management succession plan?     Frequency
Yes 107
No 103
Total     210

What is the probability that such a company will have a management succession plan?

There is some underlying probability, π, that a small fast-growing company will have a management succession plan, and our best estimate is the sample proportion, 107/210 = 0.510.

How accurate is this estimate?

The number with a succession plan should have a binomial distribution,

X  ~  binomial (n = 210,  π)

The diagram below initially shows this distribution with π replaced by our best estimate, p = 0.510.

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.

From the error distribution (or from the standard error), it is unlikely that the estimate of the probability that this type of company has a succesion plan, p = 0.510, will be more than 0.1 in error.