Confidence interval and standard error of the mean
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Confidence Intervals (CI)
The confidence interval is a key concept in medical statistics and is often misunderstood.
Formal definition:
A confidence interval is a range of values for a population parameter (such as the mean or proportion) constructed so that this range has a specified probability of including the true value of that parameter. This specified probability is called the confidence level, and the ends of the interval are called the confidence limits.Plain English:
A confidence interval gives a range within which the true value (e.g. the true mean or effect size) is likely to lie, based on your sample data. A 95% confidence level means that, if you repeated your study many times, 95% of the calculated intervals would contain the true population value.Why are confidence intervals useful?
- They help quantify the precision of an estimate (e.g. mean difference, risk ratio).
- A narrow interval indicates greater precision, whilst a wider interval indicates more uncertainty.
- They help assess statistical significance:
- If the confidence interval for a mean difference includes 0, the result is not statistically significant. For example, a confidence interval of –2 to 10 mmHg includes 0 and is therefore not statistically significant.
- If the confidence interval for a ratio (e.g. relative risk, odds ratio) includes 1 the result is not statistically significant
- They aid clinical interpretation: the range of values provides context on how meaningful the result might be.
How is a confidence interval calculated?
Confidence intervals are built around the sample mean using the standard error of the mean (SEM).
Calculating SEM:
To calculate SEM we can use the equation below:SEM = s / √n
Where s = sample standard deviation, n = sample size
SEM reflects how much sample means vary from the true population mean on repeated sampling. As sample size increases, SEM decreases & confidence intervals become narrower and more precise
To calculate a 95% confidence interval for the mean:
- Lower limit = mean − (1.96 × SEM)
- Upper limit = mean + (1.96 × SEM)
Note:
For small samples (typically n < 30), use a value from the Student’s t-distribution instead of 1.96.
For other confidence levels (e.g. 90% or 99%), use a different multiplier (e.g. 1.645 for 90%, 2.58 for 99%).
Example:
A study finds the average systolic blood pressure in a sample is 130 mmHg, with a SEM of 2 mmHg.→ 95% CI = 130 ± (1.96 × 2) = 126.1 to 133.9 mmHg
This means we are 95% confident that the true population mean lies between 126.1 and 133.9 mmHg.
Important distinction:
- Standard deviation (SD) measures the variation between individual data points in a sample.
- Standard error (SE or SEM) measures the precision of a sample estimate (like a mean).
Therefore SD is unaffected by sample size whilst SEM decreases as sample size increases.