Confidence intervals are everywhere in statistics. They are meant to show how sure we are about a number, like an average or a proportion. But here is the catch: they do not actually tell you how confident you should be about the specific interval you have right now. That misunderstanding creates what I call The Illusion of Confidence Intervals.
What Exactly Is a Confidence Interval?
A confidence interval is a range of values based on your sample data, used to estimate a number for the whole population. When someone says a 95 percent confidence interval, it means that if you repeated the same experiment many times and calculated a new interval each time, about 95 percent of those intervals would include the true value.
Still sounds a bit complex? Think of it this way. Imagine running the same experiment one hundred thousand times. Each time, you calculate a new confidence interval. About ninety-five thousand of those intervals would include the true value you are trying to estimate.
The Misinterpretation of a Single Confidence Interval
Many people believe that a 95 percent confidence interval means there is a 95 percent chance the true value is inside their specific interval. That is not true. The true value does not move. It is fixed. Your interval either includes it or it does not. The 95 percent only applies to the method. It tells you how often the method works when used many times, not how likely your one result is correct.
This misunderstanding often leads to overconfidence. Saying we are 95 percent confident makes it sound like we are assigning probability to the truth being in the interval, but that is not what is happening.
Why Should You Care?
Why does this matter? Because what you think a confidence interval means is wrong. It doesn’t tell you how likely your number is actually right. It just shows how often your method works if you do it over and over. Knowing this helps you explain things better and stop people from getting the wrong idea about how certain your results really are.
The Right Way to Think About Confidence Intervals
Think of confidence intervals as a way to test how reliable your method is, not a statement about your one result. The confidence comes from repetition, from knowing that your process works most of the time across many experiments.
Now, if you really want to say there is a 95 percent chance the true number is in this range, you are thinking about something else. That would be a Bayesian credible interval.
Bayesian Credible Interval
Bayesian thinking starts with a prior belief. This is basically your best guess about the true value based on what you already know or think. It’s like your starting point.
Then, you gather new data. With that data, you update your original guess. This is done using something called Bayes’ theorem, which combines your prior belief with the new information to give you an updated belief or posterior.
From that updated belief, you get a credible interval. This interval tells you the probability that the true value is inside that range. For example, there’s a 95 percent chance the true value is within the interval.
Most people think confidence intervals work the same way, but they don’t. Confidence intervals don’t give you an actual probability for the true value being in your interval. That’s what a Bayesian credible interval does.
Conclusion
Confidence intervals tell you how often your method will include the true value across many experiments. They do not tell you the chance that the true value is inside the one interval you calculated. Once you have your result, the value either is in there or it is not. There is no probability anymore.
