A target analogy of the confidence intervals

The logic behind a confidence interval is like guessing where the bullseye is by looking at where a dart lands.

Standard Error (SE): σ / √n
Margin of Error (MOE): Z × SE
Confidence Interval: x̄ ± MOE

Shooter Skill (Standard Error)

Range (Multiplier)

1. Multiple shots (samples)

We see the True Mean (Red Center) casting a range that traps predictable sample means (Blue Dots).

Note: This central circle shows the Margin of Error based on the Standard Error (SE). It visualizes the sampling distribution, not a Confidence Interval.

2. One shot (sample)

You only get one sample. We reverse it: Draw the range around the single dart. Does it capture the invisible center?

Click on any dot or line in either graph to select it and test if its confidence interval captures the true mean!

3. The Long Run

Drawing ranges around all 100 samples. Green ranges succeed. Red ranges fail.

Target

Range: 68% (1 SE)
Hits: -- / 100

Confidence interval plot