Confidence Intervals in Health Research
A confidence interval (CI) is a range of plausible values for an unknown population parameter, calculated from sample data. It communicates both the estimated effect and its precision.
Learning objectives
- Define confidence level, point estimate, standard error and margin of error.
- Calculate and interpret simple confidence intervals.
- Relate CI width to sample size and variability.
- Use CIs to assess statistical and clinical significance.
Key concepts
- Point estimate: a single sample value, such as a mean, proportion or risk ratio.
- Standard error: the estimated sampling variability of the point estimate.
- Confidence level: the long-run percentage of intervals that would contain the true parameter under repeated sampling.
- Margin of error: critical value × standard error.
General form: estimate ± critical value × standard error. For a large-sample 95% CI, the critical z-value is approximately 1.96.
Mean
When appropriate: x̄ ± 1.96(SE), where SE = s/√n for a large sample.
Proportion
Approximate 95% CI: p ± 1.96√[p(1−p)/n], provided sample conditions are suitable.
Worked example
A sample has mean systolic blood pressure 128 mmHg with SE 3 mmHg. The 95% CI is 128 ± 1.96×3 = 128 ± 5.88, or approximately 122.1–133.9 mmHg.
Interpretation: the method gives a plausible range for the population mean. Do not say there is a 95% probability that this fixed interval contains the fixed true mean.
What affects interval width?
| Factor | Effect on CI |
|---|---|
| Larger sample size | Narrower interval, greater precision |
| Greater variability | Wider interval, lower precision |
| Higher confidence level | Wider interval |
| Smaller standard error | Narrower interval |
Confidence intervals for effect measures
For a mean difference, a 95% CI excluding 0 suggests statistical significance at the 5% two-sided level. For risk ratios, odds ratios and hazard ratios, a CI excluding 1 suggests significance. The location and width of the interval show the direction, magnitude and precision of the effect.
Clinical versus statistical importance
A narrow CI around a very small effect may be statistically significant but clinically unimportant. A wide CI crossing the null may include both important benefit and harm, indicating imprecision rather than proof of no effect.
Common mistakes
- Interpreting 95% confidence as certainty about one realised interval.
- Using normal approximations with very small samples or rare outcomes without checking conditions.
- Reporting only p-values and omitting effect size and precision.
- Assuming a wide interval means no association.
Review questions
- What happens to a CI when sample size increases?
- What is the null value for a risk ratio?
- Interpret a risk ratio of 0.70 with 95% CI 0.55–0.90.
Return to the Clinical Medicine Year 2 curriculum
References
- World Health Organization. Basic Epidemiology.
- BMJ Statistics Notes: Confidence intervals.
Educational content only; consult a statistician for complex analyses.
