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Confidence Intervals And Confidence Levels

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A confidence interval is a measure of estimation. It is an estimated range of values that is likely to include the population parameter being calculated. For instance, instead of estimating the mean age of a certain population to be a single value like 25.5 years, we could say that the mean age is somewhere between 23 and 28. This confidence interval contains the single value we are estimating; yet it gives us a wider net to be right.

When we use confidence intervals to estimate a number, or population parameter, we can also estimate just how accurate our estimate is. The likelihood that our confidence interval will contain the population parameter is called the confidence level. For example, how confident are we that our confidence interval of 23 – 27 years of age contains the mean age of our population? If this range of ages was calculated with a 95% confidence level, we could say that we are 95% confident that the mean age of our population is between 23 and 28 years. Or, the chances are 95 out of 100 that the mean age of the population falls between 23 and 28 years.

Confidence levels can be constructed for any level of confidence, however the most commonly used are 90 percent, 95 percent, and 99 percent. The larger the confidence level is, the narrower the confidence interval. For instance, when we used a 95 percent confidence level, our confidence interval was 23 – 28 years of age. If we use a 90 percent confidence level to calculate the confidence level for the mean age of our population, our confidence interval might be 25 – 26 years of age. Conversely, if we use a 99 percent confidence level, our confidence interval might be 21 – 30 years of age.

Calculating The Confidence Interval

There are four steps to calculating the confidence level for means:

1. Calculate the standard error of the mean.

2. Decide on the level of confidence (i.e. 90 percent, 95 percent, 99 percent, etc.). Then, find the corresponding Z value. This can usually be done with a table in the appendix in a statistics book. For reference, the Z value for a 95 percent confidence level is 1.96 while the Z value for a 90 percent confidence level is 1.65 and the Z value for a 99 percent confidence level is 2.58.

3. Calculate the confidence interval.

4. Interpret the results.

The formula for calculating the confidence interval is: CI = sample mean +/- Z score (standard error of the mean).

If we estimate the mean age for our population to be 25.5, we calculate the standard error of the mean to be 1.2, and we choose a 95 percent confidence level (remember, the Z score for this is 1.96), our calculation would look like this:

CI = 25.5 – 1.96(1.2) = 23.1 and
CI = 25.5 + 1.96(1.2) = 27.9.

Thus, our confidence interval is 23.1 to 27.9 years of age. This means that we can be 95 percent confident that the actual mean age of the population is not less than 23.1 and is not greater than 27.9. In other words, if we collect a large amount of samples (say, 500) from the population of interest, 95 times out of 100, the true population mean would be included within our computed interval. With a 95 percent confidence level, there is a 5 percent chance that we are wrong. Five times out of 100, the true population mean will not be included in our specified interval.

References

Frankfort-Nachmias, C. & Leon-Guerrero, A. (2006). Social Statistics for a Diverse Society. Thousand Oaks, CA: Pine Forge Press.

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