Tensions in Practice: Precision about a mean in tension with coverage of a new outcome¶
A stipulated normal measurement process
Assume independent normal readings with unknown mean and known standard deviation 4. In each illustrative sample the observed mean is 10. A confidence interval for the fixed mean uses the uncertainty of the sample mean, 4/√n. A prediction interval for one independent future reading must also include that reading’s own variability. Using a multiplier of 2 gives about 95.45% repeated-sampling coverage under this exact model; the two procedures cover different targets.
Locate the process mean
Communicate precision about a fixed average level.
Cover a new reading
Allow for variation in one additional outcome.
Why these aims pull against each other
More observations reduce uncertainty in the mean but do not remove the variability of a new reading. A narrow mean interval cannot replace an outcome interval.
Choose an arrangement to see what changes and what remains difficult.
Rows keep the same sample sizes, center and known individual variability. Only the target-specific half-width changes. SE is standard error; SD is standard deviation.
What this choice protects
What it costs
When it fits
Compare the arrangements
Report the mean interval
Use sample mean ± 2 × 4/√n. The table holds the observed center at 10 to isolate the role of sample size.
| Mean SE | New SD | Half-width | |
|---|---|---|---|
| n = 4 | 2 | 4 | 4 |
| n = 16 | 1 | 4 | 2 |
| n = 64 | 0.5 | 4 | 1 |
- What it protects
- The interval directly expresses precision about the fixed population mean.
- What it costs
- It is too narrow to provide the stated coverage for a single future reading.
- When it fits
- Fits inference about the stable mean under the declared independent normal model and known variance.
Illustration note: The mean standard error is 2, 1 or 0.5. Repeated-sampling coverage belongs to the procedure, not a posterior probability assigned to the realized interval.
Report the prediction interval
Use sample mean ± 2 × √(16 + 16/n). Variances add because the new reading is independent of the sample.
| Mean SE | New SD | Half-width | |
|---|---|---|---|
| n = 4 | 2 | 4 | 8.94 |
| n = 16 | 1 | 4 | 8.25 |
| n = 64 | 0.5 | 4 | 8.06 |
- What it protects
- The procedure covers one new reading at the same stated level under the model.
- What it costs
- Its wide interval does not communicate the much tighter precision of the mean estimate.
- When it fits
- Fits planning for one additional reading from the same process.
Illustration note: The half-width approaches 8 as n grows. Displayed widths are rounded to two decimals; the formula defines coverage.
What this illustration does—and does not—establish
The source supplies the structural tension; the invented example makes one relation inspectable. Costs and conditions are part of each arrangement, not exceptions to a universal recommendation.
- All samples, means and variances are invented. The exact normal and independence assumptions are essential.
- Known standard deviation 4 is not estimated from these samples. Unknown variance would require a different construction.
- This is not a tolerance interval covering a fraction of a population, nor a promise about many future readings jointly.
Source entries
Confidence Intervals
The canonical tension motivates this comparison. The setting, finite values and arrangements are declared editorial illustrations, not measured findings.
Prediction intervals versus confidence intervals for parameters
A 95% CI on a parameter (e.g., population mean) answers "where is the true mean?" and gets narrower with larger sample size (width ~ 1/√n). A 95% prediction interval answers "where will a future observation fall?" and stays roughly constant width regardless of sample size (reflecting individual observation variability, not sampling error of the mean). The two are easily confused; applied researchers sometimes report prediction intervals when confidence intervals are needed, or vice versa. The tension is between parameter uncertainty (CI) and outcome uncertainty (PI), which have different decision relevance depending on the problem.
The source operation
a confidence interval is an interval [L(X), U(X)] computed from sample data X that, under repeated sampling from the same data-generating process, covers the true unknown parameter value θ with a pre-specified long-run frequency 1−α (typically 95%, sometimes 90% or 99%) — formally, P(L(X) ≤ θ ≤ U(X)) ≥ 1−α under the assumed probability model and sampling design; the coverage probability is a property of the procedure (the construction rule for L and U), not of any particular realized interval once data are observed, and it is this long-run frequency calibration that distinguishes frequentist confidence intervals from Bayesian credible intervals (which have direct probability interpretations for the specific observed data given a prior)