How to Interpret Confidence Intervals: A Complete Guide

If you work with data, whether you are a scientist, statistician, marketer, or policymaker, it is essential to know how to interpret confidence intervals. But, what exactly is a confidence interval, and why is it so important? If you are not entirely sure or need a refresher, here is a quick explanation.

In short, a confidence interval is a statistical tool used to estimate the reliability of a study's results. It represents a range of values that, with a certain degree of confidence, is assumed to contain the true value of a population parameter (like a mean or proportion).

But, for now, let's focus on what matters most in practice: how to interpret a confidence interval, apply it effectively using concrete examples, and avoid common misunderstandings.

Confidence level and its meaning

Confidence intervals come with confidence levels (also known as confidence coefficients) crucial to their interpretation. The most common confidence coefficients are 0.90, 0.95, and 0.99 (or 90%, 95%, and 99% respectively). A confidence level of 95% suggests that if we were to take 100 different samples and construct a confidence interval for each, we expect approximately 95 of those intervals to contain the true population parameter.

Confidence interval width and uncertainty

Another essential thing to remember when interpreting confidence intervals is to pay attention to their width, which reflects the level of uncertainty. The greater the uncertainty, the wider the interval. If two estimates are equal, but one has a 95% confidence interval wider than the other, the estimate with the wider interval has greater uncertainty. Factors such as variability and, most importantly, sample size influence the width of a confidence interval. You can explore how sample size affects the interval using this sample size calculator 🇺🇸.

Importance of upper and lower limits

Every confidence interval consists of two bounds:

• Lower bound: This marks the start of the interval and indicates the lowest value within which the population parameter is likely to be found with the specified confidence level; and
• Upper bound: Conversely, this marks the end of the interval, indicating the highest value expected to encompass the population parameter.

These limits define the range of plausible values for the parameter based on the sample data. It is essential to understand that the true parameter of the population is fixed and that the interval captures this parameter with a certain level of confidence, rather than the parameter moving within the interval.

Example 1: Estimating pitch speed

A baseball coach wants to know his league's average pitch speed. He recorded the speed in mph of each ball thrown in a random sample of 100 pitches and constructed a confidence interval at a 95% confidence level to estimate the average speed. The confidence interval obtained is [68 mph, 75 mph].

How do you interpret this confidence interval?

If the coach takes another sample of 100 throws, the probability that the sample mean will be between 68 and 75 mph equals 95%.

Example 2: Effectiveness of a new drug

A pharmaceutical company conducted a clinical trial to compare the efficacy of a new drug with that of a standard treatment. After 3 months, the mean change in systolic blood pressure for the new drug was -15 mmHg, with a standard deviation of 10 mmHg, in a sample of 100 patients. The 95% confidence interval is [-17 mmHg, -13 mmHg].

How do you interpret this confidence interval?

We are 95% confident that the true mean change in systolic blood pressure for the population treated with the new drug is between -17 and -13 mmHg.

Example 3: Impact of training on productivity

A company evaluated the impact of a new training program on employee productivity, measured by the number of units produced per hour. After training, the average increase in productivity was 3 units/hour, with a standard deviation of 1.5 units/hour, based on a sample of 50 employees. The 99% confidence interval is [2.45, 3.55] units/hour.

How do you interpret this confidence interval?

We are 99% confident that all employees' mean productivity gain lies between 2.45 and 3.55 units/hour.

Misunderstandings about confidence intervals can lead to erroneous conclusions and misuse of statistical data.

• Misinterpretation #1: Treating the confidence level as the probability that the interval contains the parameter

  A common misconception is, for instance, that a 95% confidence level implies a 95% probability that the interval contains the true population parameter. This interpretation is incorrect because the interval either contains the parameter or does not. The 95% confidence level refers to the long-term proportion of these intervals containing the parameter if we repeated the study many times.

• Misinterpretation #2: Assuming the parameter is most likely near the center of the interval

  Another common mistake is assuming that the population parameter will likely be located anywhere within the interval. The confidence interval provides a range of plausible values. However, this does not suggest that the parameter is more likely to be found near the center of the interval than at the ends.

• Misinterpretation #3: Thinking overlapping intervals mean no statistical difference

  People sometimes misinterpret overlapping confidence intervals between two groups as proof that no significant difference exists between them. While overlap may suggest that further investigation is needed, it does not in itself determine statistical significance.

• Misinterpretation #4: Using confidence intervals to predict individual outcomes

  Using confidence intervals to predict future observations is a misuse of the concept. Confidence intervals estimate a population parameter, not the range of future individual observations. Want to learn more about it? Check out our article: Prediction interval vs. Confidence interval: What’s the difference?

Understanding these aspects is essential for accurately interpreting confidence intervals:

• Confidence level: This represents the frequency with which intervals constructed in the same way from different samples would encompass the true population parameter.
• Interval limits: This is not the probability that the parameter lies at the extremes, but rather the range covering the parameter.
• Fixed parameter: The population parameter is a fixed quantity, and the confidence interval provides a sample-based range for where that fixed value lies.

This article was written by Claudia Herambourg and reviewed by Steven Wooding.