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 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 poll.

The table below shows the results from a public opinion poll conducted by the Sun newspaper in the UK at the start of May 2014.

Party    Number intending to vote    Percentage
Conservative
Labour
Liberal-Democrat
UK Independence Party
Other
465
443
97
217
65
34.6
33.0
7.2
16.1
4.9

Although the table excludes respondents who were undecided, refused to answer the question or indicated that they would not vote, the data will be, at least approximately, a random sample from the population of prospective voters.

Since the sample size is n = 1,344, the margin of error for the poll is 0.027 — i.e. 2.7%.

The proportions intending to vote for the Conservative and Labour parties were close to 1/2, so a 95% confidence interval for the percentages voting for these parties would be approximately (34.6 ± 2.7) and (33.0 ± 2.7). However properly calculated confidence intervals for the other candidates would be narrower than this.


We finally note that this telephone survey was not really a random sample of potential voters. Not only did it exclude those who were undecided or refused to answer, but the distribution of ages, genders and other characteristics did not match the distributions that would be likely from all voters. We cannot explain the methodology used here, but the survey company adjusted the figures to be more representative and the published data were as follows.

Party    Number intending to vote    Percentage
Conservative
Labour
Liberal-Democrat
UK Independence Party
Other
464
482
116
184
98
34.5
35.9
8.6
13.7
7.3