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Confidence Intervals Explained

A 95% confidence interval does not mean there is a 95% probability the true value lies within it. This is one of the most persistently misunderstood concepts in statistics. Let's build the correct intuition from the ground up.

In this lesson

  1. What a confidence interval is
  2. How they are constructed
  3. The correct interpretation
  4. Common misinterpretations
  5. What affects interval width

1 What a Confidence Interval Is

A confidence interval is a range of values constructed from sample data that is designed to capture the true population parameter a specified percentage of the time. The 95% refers to the procedure, not to any single interval.

Here's the concrete setup: you want to know the average height of all adults in a country. You can't measure everyone, so you take a sample of 1,000 people and measure their heights. You calculate the sample mean and build a confidence interval around it.

The Key Idea

A 95% confidence interval means: if you repeated this sampling procedure many times and built a confidence interval each time, 95% of those intervals would contain the true population mean. The 95% describes the long-run reliability of the procedure.

2 How Confidence Intervals Are Constructed

The basic formula for a confidence interval for a mean: CI = x̄ ± z* × (σ/√n), where x̄ is the sample mean, z* is the critical value (1.96 for 95% confidence), σ is the standard deviation, and n is the sample size.

Building a 95% CI
Sample of 100 students, mean test score = 75, standard deviation = 10
1Standard error = σ/√n = 10/√100 = 10/10 = 1
2For 95% confidence, z* = 1.96
3Margin of error = 1.96 × 1 = 1.96
4CI = 75 ± 1.96 = (73.04, 76.96)
Answer: We are 95% confident the true mean score is between 73.04 and 76.96

The margin of error (±1.96 in this case) is half the width of the interval. It's what news reports mean when they say "accurate to within ±3 percentage points." Smaller margins of error require larger samples.

3 The Correct Interpretation

After you've built a specific 95% CI of (73.04, 76.96), the true mean either is or isn't in that interval. There's no probability involved for that specific interval — it's a fixed range and the true value is a fixed number. The 95% describes the process that generated the interval.

The correct statement: "I used a procedure that produces intervals containing the true value 95% of the time. This is one of those intervals." Not: "There is a 95% probability the true value is in this specific range."

The practical takeaway: treat a 95% CI as a plausible range for the true parameter. Values inside the interval are consistent with your data. Values outside would be surprising given what you observed.

4 Common Misinterpretations

'95% chance the true value is in this interval'

Once the interval is calculated, there's no randomness left. The true value is fixed. The correct framing is about the procedure's reliability over many repetitions, not the probability for this specific interval.

'95% of the data falls in this interval'

The confidence interval is about where the population mean might be, not where individual data points fall. For individual data points, you'd use a prediction interval, which is much wider.

Wider is not always worse

A 99% confidence interval is wider than a 95% one — that's how it achieves higher confidence. The tradeoff is precision vs reliability. For medical decisions you might want 99% confidence; for a quick business estimate, 90% might be fine.

5 What Affects Interval Width

Three things determine how wide a confidence interval is. Sample size is the most controllable: doubling the sample size reduces the margin of error by a factor of √2 (about 29%). To halve the margin of error you need to quadruple the sample size.

Confidence level: higher confidence (99% vs 95%) requires a wider interval. You're casting a bigger net to be more sure of catching the true value.

Population variability: more variable populations produce wider intervals. You can't control this directly — it's a property of what you're measuring.

Practice Problems

A poll reports that 52% support a candidate with a 95% CI of (49%, 55%). Is the candidate definitely winning?
No — the interval includes values below 50% (specifically 49%), meaning a majority opposition is consistent with the data. The result is not statistically distinguishable from a 50-50 split.
If you increase your sample size from 100 to 400, what happens to the margin of error?
The margin of error is halved. Margin of error is proportional to 1/√n. √400 = 20, √100 = 10. So the standard error halves and so does the margin of error.
A 95% CI and 99% CI are calculated from the same data. Which is wider?
The 99% CI is wider. To be 99% confident instead of 95% confident, you need a larger z* value (2.576 vs 1.96), which produces a wider interval.