*56*

A **confidence interval** is a range of values that is likely to contain a population parameter with a certain level of confidence.

One question students often have is:

*What is considered a good confidence interval?*

The answer: In general, narrow confidence intervals are more desirable since this provides us with a narrow range of values that we’re confident contains some population parameter.

For example, suppose we want to estimate the mean height of a certain species of plant and we create the following 95% confidence interval:

95% Confidence Interval = [12.5 inches, 60.5 inches]

Compare this to the following 95% confidence interval:

95% Confidence Interval = [34 inches, 39 inches]

The second confidence interval is much narrower and gives us a more precise idea of what the true population mean height may be.

However, in order to obtain a narrow confidence interval we must increase our sample size which is not always practical in real-world research.

To illustrate this, consider the following example.

**Example: Calculating a Confidence Interval**

To calculate a confidence interval for a population mean, we can use the following formula:

**Confidence Interval = ****x ± z*(s/√n)**

where:

**x:**sample mean**z:**the chosen z-value**s:**sample standard deviation**n:**sample size

The z-value that you will use is dependent on the confidence level that you choose. The following table shows the z-value that corresponds to popular confidence level choices:

Confidence Level | z-value |
---|---|

0.90 | 1.645 |

0.95 | 1.96 |

0.99 | 2.58 |

For example, suppose we collect a random sample of 25 plants with the following information:

- Sample size
**n = 25** - Sample mean height
**x = 36.5 inches** - Sample standard deviation
**s = 18.5 inches**

Here is how to find calculate the 95% confidence interval for the true population mean height:

**95% Confidence Interval: **36.5 ± 1.96*(18.5/√25) = **[29.248, 43.752]**

We interpret this interval to mean that we’re 95% confident that the true population mean height for this species of plant is between 29.248 inches and 43.752 inches.

Now suppose instead we collect the following random sample of 100 plants with the following information:

- Sample size
**n = 100** - Sample mean height
**x = 36.5 inches** - Sample standard deviation
**s = 18.5 inches**

Here is how to find calculate the 95% confidence interval for the true population mean height:

**95% Confidence Interval: **36.5 ± 1.96*(18.5/√100) = **[32.874, 40.126]**

We interpret this interval to mean that we’re 95% confident that the true population mean height for this species of plant is between 32.874 inches and 40.126 inches.

Notice that by simply increasing the sample size we were able to produce a more narrow confidence interval for the population mean.

In a real-life situation, a researcher would prefer this second interval because it gives them a more precise idea of the range of values that could contain the true population mean.

However, it’s often more time-consuming and resource-intensive to gather larger sample sizes, so in reality it’s not always practical to do so.

Also keep in mind that some datasets simply have more variability in the data, which causes high values for the sample standard deviation. This naturally results in wide confidence intervals.

Thus, in order to create a “narrow” confidence interval the only variable that researchers can actually control is the sample size.

**Conclusion**

Here’s a quick summary of the main points made in this article:

**1.** Researchers often consider a “good” confidence interval to be one that is narrow.

**2.** By increasing the sample size used, researchers can produce narrower confidence intervals.

**3.** What is considered a “narrow” confidence interval varies from one field to the next because some types of data naturally have higher variability than others.

**Related:** The Relationship Between Sample Size and Margin of Error

**Additional Resources**

The following tutorials provide additional information about confidence intervals:

An Introduction to Confidence Intervals

How to Report Confidence Intervals

4 Examples of Confidence Intervals in Real Life