A sample has mean response 10.2 and a 95% confidence interval for the mean of 9.5–10.9. Does the next independent sample almost certainly fall inside it? Does the interval cover 95% of population observations?
No to both. A range for a mean, a range for one new observation, and a range covering most population observations answer different questions.
Are ranges for a mean, a new observation, and most of a population really the same question?
The previous unit established that a mean confidence interval targets the fixed parameter μ. This unit prevents that interval from being stretched into every possible meaning of “range.”
State the target before the interval name
Memorizing only CI, PI, and TI makes them easy to swap. Complete the question first.
Uncertainty in the estimation of a population parameter
Estimated uncertainty + natural variation in individual observations
Two levels of coverage p + confidence γ
Determination in advance, not range estimated by statistics
- “Where is the population mean μ?” → confidence interval, CI
- “Where might one next independent observation from the same process fall?” → prediction interval, PI
- “What range covers a specified proportion of population observations?” → tolerance interval, TI
All three may begin with the same x̄ and s, but they include different sources of uncertainty and therefore produce different results.
Under a common normal-model comparison at equal confidence, a mean CI is often narrower than a PI, while a TI requiring high coverage may be wider. Do not memorize width order alone: confidence, coverage, number of future observations, sidedness, and model change the width. The target defines the interval; width follows.
A confidence interval expresses uncertainty about a parameter
A 95% CI for a mean targets μ. In repeated sampling, about 95% of intervals made by the same procedure contain μ.
Because it describes precision of the estimated mean, the interval tends to narrow as independent n grows. Natural spread of individual observations does not disappear. Even a very narrow mean CI can exclude the next sample value.
A CI is not designed to cover the central 95% of individual values. Several raw points outside a narrow mean CI do not by themselves indicate a calculation error.
A prediction interval includes variation of the next observation
Predicting one next independent sample includes two uncertainties:
- uncertainty from estimating the population mean with a sample;
- natural variation of an individual value around the mean even if the mean were known.
That is why a PI for one next observation is generally wider than a mean CI from the same sample. In a representative normal model with unknown mean, a two-sided PI has structure:
x̄ ± t* × s × √(1 + 1/n)
Compared with s/√n in the mean CI, √(1+1/n) preserves individual-observation variation. Even with very large n and precise mean estimation, a PI does not collapse to zero because individual spread remains.
“Next” means a randomly obtained independent observation from the same defined process and conditions. Changes in instrument, material, operator, donor, culture condition, or time range weaken the basis for applying the original PI.
A tolerance interval targets a population proportion
A TI aims to cover a specified proportion of population observations, rather than one mean or one next observation. A complete TI statement therefore needs two percentages.
For example: “a two-sided tolerance interval containing at least 95% of the population with 95% confidence.”
- coverage
p=95%: proportion of observations in one population the interval aims to contain; - confidence
γ=95%: long-run confidence that repeated TI procedures achieve that coverage target.
You may abbreviate this as a 95%/95% TI, but identify which number is confidence and which is coverage. “A 95% tolerance interval” is incomplete.
A two-sided TI for a normal population is often written x̄ ± k·s. The factor k is not fixed at 1.96 or 3; it depends on n, confidence, coverage, and whether the limit is one- or two-sided.
If true μ and σ were known for a normal population, μ±1.96σ would cover about 95% of observations. In a real sample, μ and σ are unknown and x̄ and s vary. A TI must include this estimation uncertainty and therefore uses an n-dependent, typically larger k.
If a normal model is inappropriate, its intended coverage guarantee may fail. Distribution-free methods exist, but may require much larger samples or wider intervals for the same confidence and coverage.
A specification is not a statistical interval
Specification limits or acceptance criteria are external requirements that a product, process, or test value must meet. They should be set before analysis from design needs, scientific purpose, quality risk, clinical meaning, or regulatory and contractual context.
CI, PI, and TI are calculated from a sample and model, so they move from sample to sample.
Comparing a TI with specification lines can help investigate whether a modeled population may lie within the requirements. It does not create these equalities:
TI = specificationTI inside specification → automatic approvalpost-change values inside a pre-change TI → comparability established
Comparability is a domain judgment about what must remain comparable, which quality attributes matter, and what differences and risks are acceptable. One historical TI cannot replace that full assessment.
Widening specifications after seeing data so every observation fits removes an independent decision criterion. Record the basis, decision time, and change history of every limit.
Compare three intervals from the same sample
The Lab draws independent samples from a normal process with μ=10 and σ=3. All three intervals begin with the same x̄ and s but target different objects.
- At n=20, compare the mean CI, the PI for one next observation, and a 95% confidence/95% coverage TI.
- Press
New sampleand watch the center and intervals move together. - Compare n=5 with n=40 to see the role of estimation uncertainty.
If this is your first time: What should I press?
- 1. Read the question firstIn the Lab title, check the one thing you will compare this time.
- 2. Change just one conditionInitially, change only one of the inputs: n, effect, or spread.
- 3. New sample pressureNew synthetic data is created. The same conditions may vary depending on the sample.
- 4. targets and widths of three intervals CompareWrite in one sentence what moves and what stays the same before and after the change.
If it gets stuckresetGo back to see the default results and change just one condition. This Lab is not a correct answer tester but a pattern observation tool.
Answer three different questions with the same sample
We use the normal generation process μ=10, σ=3. Three objects are compared on the same axis: the mean, the next observation, and 95% of the population observations.
Three intervals from the same sample
this sample
95%/95% TI is a sample-based approximation of the two-sided normal tolerance interval. If the conditions for normality and independent sampling are not met, another method is needed.
CI: Two-tailed Student t mean interval · PI: Two-tailed t prediction interval for the next normal observation of the unknown mean · TI: Two-sided 95% confidence/95% coverage normal tolerance coefficient based on NIST Table 9.6. This is a synthetic material for educational purposes and cannot be used to determine approval, specifications, or comparability.
The Lab reveals μ and σ only because we defined the generating process. In actual analysis they are unknown, and the model and sampling scope must be justified.
The three JMP outputs begin with different input questions
Current JMP Distribution documentation provides separate results for Confidence Intervals, Prediction Intervals, and Tolerance Intervals for continuous variables. Read their targets, not a click sequence.
Displays confidence limits for parameters such as Mean and Std Dev separately.
Aims at summarizing the next one random observation or the next sample.
Confidence Level and Proportion to Cover are treated as separate inputs.
For a Prediction Interval, distinguish one future observation from a summary of a future sample; confidence level and future sample count affect interpretation. Tolerance Interval output separately asks for confidence level and proportion to cover—the two percentages in this unit.
Before and after reading JMP output, ask:
- Is my target a parameter, one future observation, or a population proportion?
- What are the independent units and population scope?
- Is the distributional model appropriate?
- Is a one-sided limit or two-sided interval needed?
- For a TI, did I record both confidence and coverage?
- Was the specification or acceptance criterion defined independently of these data?
A result sentence must include the target
Bad:
The 95% interval was 6.2–14.1.
Better:
From 20 independent samples, the two-sided 95% Student t confidence interval for the population mean was 8.8–11.6.
Under the same normal generating process, the two-sided 95% prediction interval for one next independent observation was 3.5–16.9.
Under a normal model, the two-sided tolerance interval calculated to contain at least 95% of population observations with 95% confidence was 1.7–18.7.
The numbers change across samples, but the sentence structure remains. Preserve the target, level, sidedness, model, and independent n.
Five statements to check before finishing
- A CI targets a parameter such as a mean.
- A PI includes natural variation of one next independent observation.
- A TI covers a specified population proportion with specified confidence.
μ±1.96σis not the same as a sample-based 95%/95% TI.- Statistical intervals do not automatically set or approve specifications or comparability criteria.
The next unit moves from comparing ranges to asking whether an observed difference could arise from chance alone. We connect null and alternative hypotheses, p-values, α, and the errors created by a decision rule.
Official supplementary resources
- NIST/SEMATECH · Tolerance intervals for a normal distribution
- NIST/SEMATECH · Prediction and tolerance intervals
- JMP Help · Prediction Intervals
- JMP Help · Confidence Interval for One Sample Mean
The numbers, figures, and In-Silico Lab in this article are synthetic material for explaining statistics. They cannot support real research, clinical, quality, regulatory, or comparability decisions.