Probability, Standard Error & Confidence Intervals
Separate variation among people from uncertainty about an estimate, and interpret intervals without overclaiming.
Quick revision ↓ Medical Dictionary
01Chance and conditional probability
Probability describes uncertainty within a specified model or reference situation. Conditional probability asks about an event given particular information. The probability of a positive test in people with disease is different from the probability of disease in people with a positive test. Keep the conditioning group explicit. Confusing these directions can badly distort diagnostic reasoning, especially when the target condition is uncommon.
02Sampling variation
Different random samples from the same population usually produce different estimates. Standard error quantifies the sampling variability of an estimator under model assumptions. For an independent sample mean, its estimated standard error is SD divided by the square root of the sample size. Larger samples generally improve precision, but they do not automatically correct systematic bias, poor measurements or an inappropriate sampling frame.
03Interpreting a confidence interval
A frequentist 95% confidence interval comes from a procedure that would cover the fixed true parameter in 95% of repeated samples under its assumptions. It is commonly used to describe values compatible with the data and model. It is not a 95% probability assigned to this fixed parameter by that procedure, nor a range containing 95% of individual patients' measurements.
04Width and assumptions
Intervals narrow when more independent information is available and widen when variability is greater. A 99% confidence interval is generally wider than a 95% interval computed from the same data by the same method. Formula-based intervals depend on assumptions about sampling and the model. A narrow interval around a biased estimate remains misleadingly precise rather than trustworthy.
05Worked precision example
Suppose an independent sample of 100 patients has mean score 50 and SD 10. The estimated SE is 1. A large-sample normal approximation gives a 95% interval of roughly 48.04 to 51.96; a t-based interval is slightly wider. This is uncertainty about the population mean under the assumptions, not a prediction interval for the score of the next individual patient.
High-Yield Revision
- SE = SD / square root of n for an independent sample mean.
- More confidence usually means a wider interval.
- Precision cannot compensate for bias.
Common Pitfalls
- A confidence interval for a mean is not an interval for individual patients.
- Do not interpret a non-significant interval as proof of equivalence.
References & Further Reading
Use the full methods and relevant chapters when appraising a paper. These sources support further study; the examples and questions above are original teaching exercises.
- Oxford CEBM — Critical appraisal tools
- Cochrane Handbook — Methods for evidence synthesis
- CDC — Analysing and interpreting epidemiological data
- ASA — Statement on p-values and statistical significance
- Oxford CEBM — Likelihood ratios
- Cochrane — GRADE and summary of findings
- WMA — Declaration of Helsinki, 2024 revision
- EQUATOR — Reporting guideline library
- NIH OBSSR — Mixed-methods research resources
- Institute for Healthcare Improvement — Model for Improvement