Formula & Calculator
Confidence Interval for the Mean
Calculates a range of values likely to contain the true population mean, based on a sample mean and desired confidence level.
Interpretation
CI = x̄ ± z·(σ/√n). Range of plausible values for population mean, with a given confidence level. Used in inferential statistics and research.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| CI | Confidence interval | |
| x̄ | Sample mean | |
| z | Z-score for confidence level | |
| σ | Population standard deviation | |
| n | Sample size |
What it means
A confidence interval for the population mean provides a range of values that is likely to contain the true mean, based on a sample statistic. The formula uses the sample mean x̄, a critical value z (from normal distribution) for the desired confidence level, and the standard error σ/√n. It assumes known population standard deviation. For unknown σ, the t‑distribution is used. The interval width reflects the uncertainty due to sampling. It is widely used in reporting study results, in quality control, and in survey analysis. Understanding confidence intervals is crucial for interpreting the precision of estimates and for making inferential conclusions.
Worked example
Confidence Interval for Mean – Two Examples
Real‑World| Parameter | Value |
|---|---|
| x̄ | 50 |
| σ | 10 |
| n | 100 |
| z | 1.96 |
| Parameter | Value |
|---|---|
| x̄ | 100 |
| σ | 20 |
| n | 25 |
| z | 1.96 |
Common mistakes
- Confidence interval for the mean: Estimate of the population mean with a given confidence level.
- z: The z‑score for the desired confidence level (e.g., 1.96 for 95%).
- σ: Population standard deviation – if unknown, use t‑distribution and sample standard deviation.
- n: Sample size – must be large enough for z (or use t).
- Interpretation: The interval has a (1−α)·100% chance of containing the true mean.
Applications
A confidence interval for the mean provides a range of plausible values for the population mean, based on sample data, with a specified level of confidence (e.g., 95%). It is a cornerstone of inferential statistics, used to quantify the uncertainty of estimates. Engineers use confidence intervals to assess process performance, to compare designs, and to validate simulation models. In research, they report the precision of estimates. By constructing confidence intervals, professionals can make informed decisions, recognise the limitations of sample data, and communicate the reliability of findings. The interval width is determined by the sample size, standard deviation, and confidence level. Understanding confidence intervals is essential for evidence‑based decision‑making.
- Quality control – estimating true mean of a process
- Research – reporting means with uncertainty
- Engineering design – comparing alternative designs
- Market research – estimating population parameters
- Risk analysis – assessing uncertainty in predictions
Frequently Asked Questions
CI = x̄ ± z·(σ/√n) (if σ known). It gives a range of values that likely contains the population mean μ, with a certain confidence level (e.g., 95%).
If we repeated the sampling many times, 95% of such intervals would contain the true population mean. It is not a 95% probability that the particular interval contains μ.
The width depends on the standard deviation (σ), sample size (n), and the confidence level (z). Larger σ and higher confidence widen it; larger n narrows it.
A confidence interval is for the population mean. A prediction interval is for an individual future observation, which is wider because it includes individual variability.
Use the t‑distribution: CI = x̄ ± t·(s/√n), with df = n−1. This accounts for the extra uncertainty from estimating σ.
The margin of error is E = z·(σ/√n) (or t·(s/√n)). It is half the width of the interval.
Larger n decreases the standard error (σ/√n), leading to a narrower interval, all else being equal.
Higher confidence level (e.g., 99% vs 95%) requires a larger z‑value, resulting in a wider interval.
The data should be approximately normally distributed, or the sample size should be large enough (n ≥ 30) for the CLT to apply. The population standard deviation must be known.
If the interval contains zero, it suggests that the true mean difference could be zero, meaning no statistically significant effect at the given confidence level.