Statistics · Estimation
Confidence Intervals Explained
Point estimates give you one number — confidence intervals tell you how much to trust it. Here’s exactly what they mean and how to calculate them.
You poll 500 voters and find 52% support a particular policy. But you didn’t ask everyone — so how confident can you be that the true population support is 52%? A confidence interval answers this question by providing a range of plausible values for the true population parameter, with an associated level of confidence.
📘 Key Term
A confidence interval (CI) is a range of values, calculated from sample data, that is likely to contain the true population parameter with a stated probability (the confidence level). A 95% CI means that if we repeated the sampling process 100 times, approximately 95 of those intervals would contain the true parameter.
The General Formula
CI = x̄ ± Z* × (σ/√n)
Where: x̄ = sample mean, Z* = critical value for confidence level (1.96 for 95%), σ = population standard deviation, n = sample size. When σ is unknown and n is small, use the t-distribution instead.
Common Critical Values
| Confidence Level | Z* Critical Value | Meaning |
|---|---|---|
| 90% | 1.645 | 90 of 100 intervals capture the true value |
| 95% | 1.960 | 95 of 100 intervals capture the true value |
| 99% | 2.576 | 99 of 100 intervals capture the true value |
Worked Example: Average Wage
Sample mean wage: x̄ = £32,500 | σ = £6,000 | n = 100
95% CI = £32,500 ± 1.96 × (£6,000/√100)
= £32,500 ± 1.96 × £600 = £32,500 ± £1,176
95% CI = (£31,324, £33,676)
💡 Key Insight
Wider confidence intervals reflect more uncertainty — you can reduce width by increasing sample size (n) or accepting a lower confidence level. Doubling n narrows the interval by a factor of √2 ≈ 1.41. This trade-off between precision and confidence is fundamental to survey design and experimental planning.
⚠️ Common Error
Students often say: ‘there is a 95% probability the true mean lies within this interval.’ This is wrong once the interval is calculated — the true mean either does or doesn’t lie in the interval. The 95% refers to the long-run frequency of the procedure: 95% of all intervals constructed this way will contain the true parameter. The specific interval already computed is fixed.
Q1. A sample of 64 households has a mean monthly expenditure of £1,850 with a known population standard deviation of £240. Construct a 95% confidence interval for the true population mean. [5 marks]
Answer: CI = 1850 ± 1.96 × (240/√64) = 1850 ± 1.96 × 30 = 1850 ± 58.8. 95% CI = (£1,791.20, £1,908.80). We are 95% confident the true mean monthly household expenditure lies between £1,791 and £1,909. Note: this means 95% of intervals constructed using this method would capture the true population mean.
References
1. Moore, D.S., McCabe, G.P. and Craig, B.A. (2021) Introduction to the Practice of Statistics. W.H. Freeman.
2. Triola, M.F. (2022) Elementary Statistics. Pearson.
3. Freedman, D., Pisani, R. and Purves, R. (2007) Statistics. W.W. Norton.
4. Field, A. (2018) Discovering Statistics Using IBM SPSS Statistics. SAGE.
5. Gelman, A. and Hill, J. (2006) Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press.