# Confidence Intervals - Statistics Teaching Tools

## What is a confidence interval?

A confidence interval is a range around a measurement that conveys how precise the measurement is. For most chronic disease and injury programs, the measurement in question is a proportion or a rate (the percent of New Yorkers who exercise regularly or the lung cancer incidence rate). Confidence intervals are often seen on the news when the results of polls are released. This is an example from the Associate Press in October 1996:

The latest ABC News-Washington Post poll showed 56 percent favored Clinton while 39 percent would vote for Dole. The ABC News-Washington Post telephone poll of 1,014 adults was conducted March 8-10 and had a

margin of error of plus or minus 3.5 percentage points.(Emphasis added).

Although it is not stated, the margin of error presented here was probably the 95 percent confidence interval. In the simplest terms, this means that there is a 95 percent chance that between 35.5 percent and 42.5 percent of voters would vote for Bob Dole (39 percent plus or minus 3.5 percent). Conversely, there is a 5 percent chance that fewer than 35.5 percent of voters or more than 42.5 percent of voters would vote for Bob Dole.

The precise statistical definition of the 95 percent confidence interval is that if the telephone poll were conducted 100 times, 95 times the percent of respondents favoring Bob Dole would be within the calculated confidence intervals and five times the percent favoring Dole would be either higher or lower than the range of the confidence intervals.

Instead of 95 percent confidence intervals, you can also have confidence intervals based on different levels of significance, such as 90 percent or 99 percent. Level of significance is a statistical term for how willing you are to be wrong. With a 95 percent confidence interval, you have a 5 percent chance of being wrong. With a 90 percent confidence interval, you have a 10 percent chance of being wrong. A 99 percent confidence interval would be wider than a 95 percent confidence interval (for example, plus or minus 4.5 percent instead of 3.5 percent). A 90 percent confidence interval would be narrower (plus or minus 2.5 percent, for example).

## What does a confidence interval tell you?

he confidence interval tells you more than just the possible range around the estimate. It also tells you about how stable the estimate is. A stable estimate is one that would be close to the same value if the survey were repeated. An unstable estimate is one that would vary from one sample to another. Wider confidence intervals in relation to the estimate itself indicate instability. For example, if 5 percent of voters are undecided, but the margin of error of your survey is plus or minus 3.5 percent, then the estimate is relatively unstable. In one sample of voters, you might have 2 percent say they are undecided, and in the next sample, 8 percent are undecided. This is four times more undecided voters, but both values are still within the margin of error of the initial survey sample.

On the other hand, narrow confidence intervals in relation to the point estimate tell you that the estimated value is relatively stable; that repeated polls would give approximately the same results.

## How are confidence intervals calculated?

Confidence intervals are calculated based on the standard error of a measurement. For sample surveys, such as the presidential telephone poll, the standard error is a calculation which shows how well the poll (sample point estimate) can be used to approximate the true value (population parameter), i.e. how many of the people surveyed said they would vote for Dole versus how many people actually would vote for Dole in the election. Generally, the larger the number of measurements made (people surveyed), the smaller the standard error and narrower the resulting confidence intervals.

Once the standard error is calculated, the confidence interval is determined by multiplying the standard error by a constant that reflects the level of significance desired, based on the normal distribution. The constant for 95 percent confidence intervals is 1.96.