What Is a Confidence Interval?
A confidence interval (CI)is a range of values, derived from sample data, that is likely to contain the true value of an unknown population parameter. It provides a measure of uncertainty around a sample estimate. Instead of saying “the average is 50,” a confidence interval says “we are 95% confident the true average lies between 47.2 and 52.8.” This concept was formalized by Jerzy Neyman in 1937 and is foundational to inferential statistics.
How Are Confidence Intervals Calculated?
For a Population Mean
CI = x̄ ± z × (σ / √n)
- x̄ = sample mean
- z = critical value (1.96 for 95% confidence)
- σ = standard deviation (population or sample)
- n = sample size
For a Population Proportion
CI = p̂ ± z × √(p̂(1 − p̂) / n)
- p̂ = sample proportion (successes / total)
- z = critical value
- n = sample size
Worked Example
A professor tests n = 36 students and finds a mean score of x̄ = 78 with a population standard deviation of σ = 12. At 95% confidence (z = 1.96):
- Standard Error: SE = 12 / √36 = 12 / 6 = 2.0
- Margin of Error: ME = 1.96 × 2.0 = 3.92
- 95% CI: 78 ± 3.92 = [74.08, 81.92]
Interpretation: We are 95% confident the true population mean score lies between 74.08 and 81.92.
Understanding Your Results
The margin of error tells you the precision of your estimate — a smaller margin means more precision. It depends on three factors:
- Confidence level: Higher confidence → wider interval → larger margin of error.
- Sample size (n): Larger sample → narrower interval → smaller margin of error. The relationship follows √n, so quadrupling n halves the margin.
- Variability (σ): More variability in the data → wider interval.
z-Distribution vs. t-Distribution
When the population standard deviation (σ) is known or the sample size is large (n ≥ 30), the z-distribution (standard normal) is used. When σ is unknown and n is small (< 30), the t-distribution is used instead. The t-distribution has heavier tails, producing wider intervals that account for the additional uncertainty of estimating σ from the sample. As degrees of freedom (df = n − 1) increase, the t-distribution approaches the z-distribution.
Critical Values Reference Table
| Confidence Level | z-Score | Significance (α) | Tail Area |
|---|---|---|---|
| 90% | 1.645 | 0.10 | 0.05 each tail |
| 95% | 1.960 | 0.05 | 0.025 each tail |
| 99% | 2.576 | 0.01 | 0.005 each tail |
Common Misconceptions
- Wrong: “There is a 95% probability that the true mean is in this interval.” The true mean is either in the interval or it isn't — it's not random.
- Correct: “If we repeated this procedure many times, 95% of the resulting intervals would contain the true mean.”
- Wrong: “A wider interval is better because it's more likely to contain the true value.” Width is a trade-off with precision — a CI of [0, infinity] is always “correct” but useless.
Real-World Applications
- Political polling: “Candidate A leads with 52% ± 3% at 95% confidence.” The true support could be 49–55%.
- Medical trials: “The drug reduced symptoms by 15% ± 4%.” If the CI includes 0%, the result is not statistically significant.
- Quality control: “Average weight of cereal boxes: 500g ± 2g.” Ensures manufacturing stays within spec.
- A/B testing: “Version B increased conversion by 3.2% [95% CI: 1.1%, 5.3%].” Since the CI doesn't include 0, the improvement is significant.
Sources and References
- Neyman, J. (1937). “Outline of a theory of statistical estimation based on the classical theory of probability.” Philosophical Transactions of the Royal Society A, 236(767), 333–380.
- Moore, D.S., McCabe, G.P., & Craig, B.A. (2017). Introduction to the Practice of Statistics (9th ed.). W.H. Freeman.
- Agresti, A. & Coull, B.A. (1998). “Approximate is better than ‘exact’ for interval estimation of binomial proportions.” The American Statistician, 52(2), 119–126.
- Student [Gosset, W.S.] (1908). “The probable error of a mean.” Biometrika, 6(1), 1–25.