Confidence Interval

TL;DR

• A confidence interval gives a sample-based range that estimates a population parameter.
• The interval’s width reflects the chosen confidence level and the sample size.
• It quantifies uncertainty but does not guarantee the true value lies inside the interval.

Definition

A confidence interval is a range of values that is calculated from a sample dataset and is used to estimate the population parameter. It provides a measure of how certain we can be that the true population value lies within the calculated range.

Explanation

A confidence interval is computed from sample data (for example, a sample mean or sample proportion) and a chosen confidence level (commonly 95%). The interval indicates a range of plausible values for the corresponding population parameter. The width of the interval depends on the variability in the sample data, the sample size, and the chosen confidence level: higher confidence levels produce wider intervals, and larger sample sizes produce narrower intervals. A confidence interval expresses uncertainty about an estimate but does not guarantee that the true population parameter falls inside the calculated range.

Examples

Estimating an average (height)

We take a random sample of 100 students. The sample average height is 175 cm and the sample standard deviation is 5 cm. A 95% confidence interval for the average height is calculated as:

$$175\ \text{cm} \pm 1.96 \times \left(\frac{5\ \text{cm}}{\sqrt{100}}\right)$$

This gives a range of 166.4 cm to 183.6 cm, meaning we can be 95% confident that the true average height of all the students lies within this range.

Estimating a proportion (obesity)

We take a random sample of 1000 adults and observe 200 are obese. A 95% confidence interval for the proportion obese is:

$$\frac{200}{1000} \pm 1.96 \times \sqrt{\frac{200}{1000}\left(1 - \frac{200}{1000}\right)/1000}$$

This yields a range of 0.16 to 0.24, meaning we can be 95% confident that the true proportion of adults in the city who are obese lies within this range.

Use cases

• Estimating population parameters (such as means or proportions) from sample data.
• Making predictions about population values based on sample-based uncertainty quantification.

Notes or pitfalls

• A confidence interval is not a fixed value; it is a range computed from the sample and the chosen confidence level.
• Increasing the confidence level (e.g., from 95% to 99%) produces a wider interval; decreasing it (e.g., to 90%) produces a narrower interval.
• Increasing the sample size typically narrows the confidence interval.
• A confidence interval does not guarantee the true population parameter lies within it; there is always a chance the true value falls outside the interval.

Related terms

• population parameter
• sample
• standard deviation
• proportion
• confidence level