Worst-case standard error

When estimating a population proportion (or probability) π, the corresponding sample proportion p is used as a point estimate. It has standard error

This standard error takes its maximum value when π = 0.5 and this provides a worst-case value for the standard error. Whatever the value of π, the standard error of the estimate will be less than

The diagram below shows the estimated standard error from a sample of size n = 100.

Drag the slider to investigate how the standard error depends on p and observe that it is maximum when p = 0.5. Note also that the resulting confidence interval for π is also widest for this value.

Reporting the accuracy from a survey

Public opinion polls and other surveys often ask questions about a range of topics. Each question can be modelled as a random sample from some categorical population. Several proportions are usually estimated from the data that are collected and each of these point estimates can be associated with a standard error and confidence interval.

Rather than giving separate confidence intervals for the individual proportions, the poll results are usually accompanied by a single value called the margin of error of the poll. This is the worst-case '±' value for a 95% confidence interval that arises when p = 0.5.

The '±' values of the 95% confidence intervals for all proportions reported in the poll will be less than this value — sometimes considerably less.

The diagram below shows the 95% confidence intervals that would arise from samples of size n = 100 for all possible values of p. Drag the slider to see the widths of the confidence intervals.

Observe that the confidence interval is widest when p = 0.5 and narrowest when p is close to 0 or 1. (Note however that our guidelines on sample size imply that we should not be using this type of confidence interval for n = 100 when p is under 0.05 or over 0.95.)

Select Margin of error from the pop-up menu. Observe that


Example

A numerical example illustrates the use of margin of error in a survey.

Customer satisfaction survey

The table below shows the results from a survey of customers of the company Thermatrix. The company prides itself on good customer relations and intends to conduct a customer satisfaction survey at least one a year. The survey involved 115 customers who were randomly selected.

Question    Yes    No
In general, were deliveries on time? 63 52
Were the contact people at Thermatrix helpful and courteous? 86 29
Was the pricing structure fair to your company? 101 14
Would you recommend Thermatrix to other companies? 105 10

Since the sample size is n = 115, the margin of error for the poll is 0.093 — i.e. 9.3%.

Since approximately 1/2 of the respondents replied that deliveries were on time, a 95% confidence interval for the probability that a customer believes that deliveries are on time is approximately (63/115 ± 0.093). However properly calculated confidence intervals for the other questions would be narrower than this.