How do you interpret a 95% confidence interval

Confidence intervals translate sample uncertainty into a range that likely contains the true population parameter. For a 95% interval, if we could repeat the experiment many times, about 95% of those intervals would capture the unknown value. The interval is centered on the point estimate and expands outward by a margin that depends on variability and sample size. The width shrinks when the sample is larger or the data are less variable, and it widens when the opposite holds. Thus the interval conveys both the estimate and its precision.

How a CI is constructed

To build a CI we start with a point estimate, such as the sample mean $\bar{x}$. Assuming the sampling distribution is approximately normal, we add and subtract a critical value times the standard error: $\bar{x} \pm z_{\alpha/2}\,SE$. For a 95% confidence level the critical value $z_{0.025}$ is 1.96. The standard error reflects the spread of the estimator and is calculated as $s/\sqrt{n}$ for a mean, where $s$ is the sample standard deviation and $n$ the sample size.

Key points to remember when reading a CI

• The interval is random, the parameter is fixed.
• A 95% CI does not guarantee the parameter is inside this particular interval.
• Wider intervals indicate more uncertainty.
• The confidence level applies to the long‑run performance of the method.

Suppose a researcher measures the systolic blood pressure of 40 patients and obtains a sample mean of 128 mmHg with a standard deviation of 12 mmHg. The standard error is $SE = 12/\sqrt{40} \approx 1.90$. The margin of error is $1.96 \times 1.90 \approx 3.72$. Therefore the 95% confidence interval for the true mean pressure is $128 \pm 3.72$, or (124.28, 131.72) mmHg. This means we are 95% confident that the average blood pressure in the whole population lies between 124.28 and 131.72 mmHg.

Numbers used in the example above

Procedure to construct a 95% CI for a mean

1. 1Compute the sample mean $\bar{x}$.
2. 2Calculate the sample standard deviation $s$.
3. 3Find the standard error $SE = s/\sqrt{n}$.
4. 4Multiply $SE$ by the critical value 1.96 to get the margin of error.
5. 5Add and subtract the margin from $\bar{x}$ to obtain the lower and upper bounds.