Confidence vs Prediction vs Tolerance Intervals: Three Different Questions

A confidence interval (CI) is a range of plausible values for an UNKNOWN PARAMETER — typically the population mean μ. A 95% CI means that if you repeated the sampling many times, 95% of such intervals would contain the true parameter. A prediction interval (PI) is a range of plausible values for the NEXT SINGLE OBSERVATION drawn from the same population. A 95% PI means that 95% of new observations will fall inside it, in the long run. A tolerance interval (TI) covers a SPECIFIED PROPORTION OF THE POPULATION with a stated confidence — a 95/95 TI is a range expected to contain at least 95% of the population, with 95% confidence. CIs shrink as sample size grows because the parameter sits at a fixed point; PIs converge to the population’s ± z·σ width because the observation itself has variance that does not shrink; TIs converge similarly. Misusing one for another is the single most common interval-related mistake in applied work.

For the mean of a normal population with unknown variance, the 95% CI is x̄ ± t_{0.025, n−1} × (s / √n). The standard error s/√n shrinks like 1/√n, so doubling the sample roughly cuts the CI width by √2. The width depends on sample size n, variability s, and the chosen confidence level. The CI does NOT tell you where new observations will fall; it tells you where μ is likely to be. A frequent misuse is computing a tight CI from n = 1000 and concluding that 95% of customers will pay within that range — the CI is for the average customer, not the next customer.

For the next single observation from a normal population, the 95% PI is x̄ ± t_{0.025, n−1} × s × √(1 + 1/n). The extra “1 +” inside the square root accounts for the observation’s own variance, which is the dominant term for large n. As n → ∞ the t-multiplier converges to 1.96 and the term √(1+1/n) converges to 1, so the PI width approaches ±1.96σ — about ±1.96 standard deviations regardless of sample size. This is the right number when you need to say “95% of next units will measure between A and B,” but it is wider than a CI and many practitioners do not realize they should be reporting it.

A two-sided (p, 1−α) tolerance interval covers at least proportion p of the population with confidence 1 − α. For a normal population, the form is x̄ ± k × s, where k is a tolerance factor that depends on n, p, and α and is read from a table or computed from the non-central t distribution. A 95/95 TI requires a larger k than the 1.96 of a CI or PI — for n = 30 the factor is approximately 2.55 and for n = 100 it drops to about 2.23 (converging to the z-quantile that captures p of a normal, namely 1.96 for p = 0.95, plus a confidence buffer). TIs are the right answer to regulatory and engineering questions like “what range will contain 99% of manufactured parts with 95% confidence?”

A sample of n = 25 AA batteries shows mean lifetime x̄ = 12.4 hours and s = 1.6 hours, with the data approximately normal. 95% CI for μ: t_{0.025, 24} = 2.064; CI = 12.4 ± 2.064 × (1.6 / √25) = 12.4 ± 0.661 = (11.74, 13.06). 95% PI for next battery: 12.4 ± 2.064 × 1.6 × √(1 + 1/25) = 12.4 ± 2.064 × 1.6 × 1.0198 = 12.4 ± 3.37 = (9.03, 15.77). 95/95 TI: k_{25, 0.95, 0.95} ≈ 2.631; TI = 12.4 ± 2.631 × 1.6 = 12.4 ± 4.21 = (8.19, 16.61). The CI is narrow because n is moderate; the PI is much wider; the TI is widest because it must contain 95% of all batteries with 95% confidence. A manager asking “how long does an average battery last?” gets the CI; an end-user asking “how long will MY next battery last?” gets the PI; a regulator asking “what range covers 95% of batteries?” gets the TI.

Twenty cured concrete cylinders have mean compressive strength 4,200 psi with s = 250 psi. A structural engineer needs to certify that the design strength will be met by at least 95% of cylinders with 95% confidence — a textbook tolerance-interval question. With n = 20 and a one-sided 95/95 TI, the tolerance factor k₁ ≈ 2.396. The one-sided lower TI is 4,200 − 2.396 × 250 = 4,200 − 599 = 3,601 psi. The engineer reports that with 95% confidence at least 95% of cylinders will exceed 3,601 psi. A CI of (4,083, 4,317) would be irrelevant — it speaks to the mean, not to individual cylinders, and would over-promise the lower bound of any single cylinder.

Three errors keep recurring. First, reporting a tight CI and treating it as a forecast for individuals — “the 95% CI for income is $48k–$52k” said as if 95% of customers earn in that range. The PI or TI is the right answer. Second, growing the sample size and trumpeting a narrower CI as evidence of more precise individual predictions — only the parameter estimate got tighter; individual variance is unchanged. Third, conflating one-sided and two-sided tolerance factors. A 95% lower one-sided TI (acceptable to spec) is not the same as a 95/95 two-sided TI; the tolerance factor and the conclusion differ. Always state which interval and which side(s) explicitly.

Provide a sample and a confidence level, plus a population proportion for tolerance, and StatsIQ produces the CI, PI, and TI side by side with the right t-multiplier and tolerance factor, choosing the one-sided or two-sided form you need. It also explains which interval answers which framing of the question — parameter, next observation, or population proportion. This content is for educational purposes only.