Learn
Interactive·

Confidence Intervals (CI)

A single estimate is useful, but a range shows what values remain plausible.

A logistics company samples 100 orders and estimates a mean delivery time of 34.0 hours.

The true population mean is not necessarily exactly 34.0 hours. A different sample would produce a different estimate.

Estimated mean delivery time

31.2 hr34.0 hr36.8 hr

Key question

What range of population means is compatible with the data?

1

Concept

Turn one estimate into a range

A confidence interval surrounds an estimate with uncertainty

A 95% confidence interval of [31.2, 36.8] hours means that values in this range are reasonably compatible with the observed data and model.

The center is the point estimate. The endpoints show the uncertainty around that estimate.

2

Why It Matters

What does 95% mean?

95% is not the probability of this one fixed interval

If the same procedure were repeated over many random samples, about 95% of the resulting intervals would contain the true population mean.

A common mistake

❌ There is a 95% probability that the true mean lies in this interval.

✓ The procedure captures the true mean in about 95% of repeated samples.

3

How It Works

What controls interval width?

Larger samples and less variable data produce narrower intervals

ConditionInterval
Small sample · high variabilityWide
Large sample · same variabilityNarrower
Same sample · lower variabilityNarrower

Increasing the confidence level from 95% to 99% makes the interval wider because the method must capture the true value more often.

4

Example

How should it be interpreted?

Read both the location and the width

IntervalInterpretation
[31.2, 36.8]Relatively precise
[20.0, 48.0]Substantial uncertainty
[-0.4, 3.2]A difference of zero remains plausible

Narrow intervals indicate greater precision. For differences, whether the interval includes zero is often important.

5

Interactive

Change the interval

Change the interval

The estimate stays at 34.0 hours. Only the interval around it responds to the three inputs.

31.236.8242832364044

Standard error

1.43 hr

Margin of error

±2.80

Interval width

5.61 hr

95% CI [31.2, 36.8] · multiplier 1.960

A standard error of 1.43 hours times 1.96 gives a margin of ±2.80, so the 95% interval runs from 31.2 to 36.8 hours. Four times the orders would halve that width.

Raising the level widens the interval. It buys coverage by saying less.

  • •
    What happens when n increases?
  • •
    What happens when 95% becomes 99%?
  • •
    What happens when variability increases?

Key takeaways

  • •
    A confidence interval shows uncertainty around a point estimate.
  • •
    Interval width depends on standard error and confidence level.
  • •
    Narrower intervals indicate greater precision.
  • •
    For a difference, check whether the interval includes zero.

Confidence intervals show both the estimated size of an effect and how precisely it is known.

Go Deeper

How are confidence intervals connected to hypothesis tests?

Logic of Hypothesis Testing
Logic of Hypothesis Testing