Confidence Interval (Mean)
95% CI
[94.6, 105.4]
How it works
A confidence interval (CI) for the mean provides a range of plausible values for the true population mean, based on a sample. A 95% CI means: if you repeated the sampling procedure many times, 95% of the resulting intervals would contain the true population mean. It does NOT mean "there is a 95% probability the true mean is in this interval" — the true mean is fixed; the interval is random.
**Formula and calculation** For large samples (n ≥ 30) or known σ: CI = x̄ ± z × (σ / √n), where z = 1.645 (90%), 1.960 (95%), 2.576 (99%). For small samples with unknown σ: CI = x̄ ± t × (s / √n), where t is the t-distribution critical value with n−1 degrees of freedom. As n → ∞, the t-distribution converges to the normal distribution.
**Interpreting width** Narrow CI → high precision (large sample, low variance). Wide CI → low precision (small sample, high variance). To halve the CI width, quadruple the sample size (width ∝ 1/√n). Reporting a 95% CI alongside an effect size communicates both the direction and precision of your estimate.
**Common misinterpretations** "95% of individual observations fall within the 95% CI" — FALSE (that would be a prediction interval, which is wider). "A non-significant result means no effect" — FALSE (it means the effect, if real, is indistinguishable from zero at the chosen sample size — underpowered studies miss real effects).
Frequently Asked Questions
- No — this is the most common misinterpretation of confidence intervals. The true population mean is a fixed (unknown) value, not a random variable. What '95% confidence' means: if you repeated the experiment many times and computed a CI each time, 95% of those intervals would contain the true mean. Your specific interval either contains the true mean or it doesn't — you just don't know which. The 95% is a property of the procedure, not of any single interval.
- Use z-interval (z critical value from normal distribution) when: the population standard deviation σ is known, or when sample size n ≥ 30 (by CLT, the sample mean is approximately normal). Use t-interval (t critical value with n−1 degrees of freedom) when: σ is unknown AND n < 30. In practice, σ is almost never known, so t-intervals are used by default. For n ≥ 30, the t and z critical values are very similar (1.96 vs 2.00 for 95% CI at n=30).
- CI width = 2 × z × (σ/√n). To halve the width: quadruple the sample size (n ∝ 1/width²). Increase n from 100 to 400 to halve CI width. Alternatively: reduce variability by controlling extraneous factors, using more precise measurement instruments, or stratifying the sample. Decreasing confidence level (95% → 90%) also narrows the interval but at the cost of lower confidence. The width cannot be reduced without either more data or less confidence.
- A confidence interval for the mean estimates where the true population mean lies. A prediction interval estimates where a single new observation will fall. The prediction interval is always wider because it includes both uncertainty about the mean AND natural variability of individual observations. Formula: PI = x̄ ± t × s × √(1 + 1/n). For large n, the prediction interval converges to x̄ ± 1.96 × s (approximately ±2 standard deviations), regardless of sample size — you cannot predict individual outcomes more precisely just by having more data.