Back to the coffee chain. Each sample gives one average — but a smart analyst reports a
range around it: a confidence interval. The famous claim is "95% confident the true
average is in here." But 95% of what, exactly? Let's draw a hundred intervals and watch.
Step 1
Draw one interval
One sample → one range around its mean
Step 2
Draw a hundred
Watch how many catch the truth
Step 3
Change the confidence
Wider nets, fewer misses
A hundred analysts, a hundred samples
The gold line is the true average (which a real analyst never sees). Each horizontal bar is one
sample's interval. Green = it caught the truth. Red = it missed.
interval that caught the true mean interval that missed true mean
Intervals drawn0
Caught the truth0
Missed0
Catch rate—
Notice: your single interval either caught the truth or didn't — you can't tell which from one sample alone. The "95%" was never about this one interval.
The pattern: across many intervals, close to 95% of them catch the true mean and about 5% miss (the red ones). The 95% is the long-run success rate of the method — a property of the procedure, not of any single result.
So what does "95% confidence" really mean?
It means: if you repeated this whole sampling-and-interval process many times, about 95% of the intervals would contain the true value.
Careful: for the one interval you actually computed, it's not "a 95% chance the truth is inside it."
The truth is fixed; your interval either caught it or didn't. The 95% is how often the method succeeds in the long run.
You compute a single 95% confidence interval: [11.2, 13.0]. Which statement is correct?
AThere's a 95% probability the true mean is between 11.2 and 13.0.
B95% of customers spend between 11.2 and 13.0.
CThe method that produced this interval catches the true mean about 95% of the time.
Catch the Mean · built for teaching · Jan Erik Meidell