The classic 95/95: 95% of the population, with 95% confidence. Interval [9.4047, 10.7753], width 1.3706 — against a prediction interval of 1.0681 and a confidence interval of 0.2331.
n = 20, 95.0% coverage at 95% confidence
9.4047 to 10.7753
At least 95.0% of the population lies inside that range, and we are 95% confident of the statement. The factor is 2.75228 standard deviations, against 2.14471 for a prediction interval and 0.46801 for a confidence interval on the mean — widths of 1.3706, 1.0681 and 0.2331 from the same twenty-odd numbers.
Tolerance factor k
2.75228
95.0% / 95%
Prediction factor
2.14471
one future observation
Confidence factor
0.46801
the mean only
Limit as n → ∞
1.95996
approached from above
The three intervals on the same data, with what each one claims
Interval
Range
Width
The claim it makes
Confidence
9.9735 – 10.2065
0.2331
the population MEAN is in here
Prediction
9.5560 – 10.6240
1.0681
the NEXT observation is in here
Tolerance
9.4047 – 10.7753
1.3706
95.0% of ALL of them are in here
The tolerance factor, as the sample grows
95% coverage at 95% confidence. Multiply by your sample SD for a half-width.
Tolerance, prediction and confidence factors at seven sample sizes
n
Tolerance k
Prediction k
Confidence k
Excess over 1.95996
5
5.09353
3.04144
1.24166
+159.9%
10
3.38191
2.37257
0.71536
+72.5%
20
2.75228
2.14471
0.46801
+40.4%
30
2.54964
2.07904
0.37341
+30.1%
50
2.37877
2.02957
0.28420
+21.4%
100
2.23280
1.99411
0.19842
+13.9%
1,000
2.03608
1.96332
0.06205
+3.9%
Even a thousand observations leave the factor 3.9% above the 1.95996 that covers 95% of a KNOWN normal. That residue is the price of the word “confidence”, and it never quite disappears.
Two probabilities, both in the label Howe’s factor, within about 1% of exact for n ≥ 5 Assumes normality of individual observations
What this tool shows
The 95/95 tolerance factor is 5.09353 at n = 5, 2.75228 at n = 20, 2.23280 at n = 100 and 2.03608 at n = 1,000. Even a thousand observations leave it 3.9% above the 1.95996 that would cover 95% of a normal whose parameters you actually knew. That residue is the price of the word “confidence”, and it is what separates this interval from the other two the tool prints beside it.
Two-sided normal tolerance intervals at 90%, 95%, 99% and 99.9% coverage
The confidence and prediction intervals on the same data, for contrast
The tolerance factor, which depends only on n, coverage and confidence
How far above the known-parameter limit the factor still sits, at each n
The 95/95 convention, with both numbers named rather than assumed
Howe’s closed-form factor, stated as the approximation it is
Two probabilities Howe factor All three intervals Coverage selectable
“95%” alone does not identify a tolerance interval.
Updated 12 September 2026 · Works in any browser, no installation
A tolerance interval covers a stated PROPORTION of the population, with a stated confidence. “95% of all units lie between 9.40 and 10.78, and I am 95% confident of that.” A confidence interval covers one parameter and a prediction interval covers one future observation; this is the only one of the three that makes a claim about the population as a whole.
At a glance
Formula shown
k₂ = √[(n−1)(1 + 1/n)·z²_{(1+p)/2} / χ²_{1−γ, n−1}], where p is the coverage and γ the confidence — Howe’s approximation, which is within about 1% of the exact non-central factor for n ≥ 5. The numerator is the coverage requirement and the denominator is the confidence requirement, expressed as how badly the sample might have underestimated the variance.
Scenario support
Manufacturing specification limits, batch release and acceptance sampling, clinical reference ranges, environmental compliance limits, pharmaceutical content uniformity, and any requirement phrased as "at least X% of units must fall within these limits".
Educational estimate
Planning support from the values you enter — not professional advice.
Two probabilities, and both belong in the label
Every other interval on this site carries one probability. This one carries two, and leaving either out makes the statement unreadable.
The coverage is how much of the population is inside. 95% coverage means at least 95% of all units lie within the limits.
The confidence is how sure of that you are. 95% confidence means the procedure produces a genuinely-95%-covering interval in 95% of samples. It might over-cover; it should rarely under-cover.
“95/95” names both, and is the industry convention precisely because a bare “95% tolerance interval” is ambiguous between the two.
They move the interval in different ways. Raising the coverage from 95% to 99% takes the factor on the shipped data from 2.75228 to 3.61712 — 31.4% wider — because the normal quantile moves. Raising the confidence widens it through the chi-square term instead, which matters most at small n.
The factor never reaches 1.95996
If you KNEW the mean and standard deviation, 95% of a normal population would lie within ±1.95996σ. Every tolerance factor above that is the cost of estimating them.
At n = 5 the factor is 5.09353 — 159.9% above the known-parameter value. Five points barely constrain a standard deviation, and the confidence requirement has to absorb how badly it might have been underestimated.
At n = 20 it is 2.75228, still 40.4% above; at n = 50, 2.37877 and 21.4%; at n = 100, 2.23280 and 13.9%.
At n = 1,000 it is 2.03608, and still 3.9% above. The convergence is real and it is slow, because the chi-square term shrinks as 1/√n rather than 1/n.
Which is why spec limits set from a small pilot are so often too narrow. Using x̄ ± 2s from five units and calling it a 95% range understates the true factor by more than a factor of two.
Against the confidence and prediction intervals
The tool prints all three from one dataset because the choice between them is the most common error in this area, and it always runs toward quoting something too narrow.
On the shipped twenty points the widths are 0.2331, 1.0681 and 1.3706. Confidence, prediction, tolerance — a factor of nearly six between the first and the last.
The confidence interval is the wrong tool for any spec limit. It describes the average, and a limit set from it would be violated by roughly half the units.
The prediction interval covers the next ONE unit. It is the right tool for a single forecast and the wrong one for a batch, because the probability that all of a hundred units fall inside a 95% prediction interval is far below 95%.
The tolerance interval is the one that scales to a population. That is why it appears in acceptance sampling and release criteria, and why it is the widest of the three.
But the ordering is not unconditional, and that surprises people. It holds when the coverage matches the prediction level. Ask for only 90% of the population and the tolerance interval is wider than a 95% prediction interval up to about n = 40 — 2.051606 against 2.047818 — and NARROWER beyond it: 1.996326 against 2.029572 at n = 50. The estimation penalty shrinks while the coverage target stays permanently lower.
The test is the sentence. “The average” is confidence, “the next one” is prediction, “95% of them” is tolerance.
Howe’s factor is an approximation, and says so
The exact two-sided factor has no closed form — it requires inverting a non-central chi-square — so every practical tool uses an approximation, and which one it uses is worth stating.
Howe’s 1969 approximation is the standard, and it is within about 1% of the exact factor for n ≥ 5, converging from there.
Wald-Wolfowitz is the other common one and is slightly less accurate at small n, which is exactly where accuracy matters most.
The ONE-sided factor is a genuinely different quantity, computed from a non-central t rather than by symmetry — so halving a two-sided interval to get a one-sided bound is wrong, and gives a bound that is too tight.
For small n the difference between approximations is dwarfed by the interval itself. At n = 5 the factor is above 5, and a 1% error in it is invisible next to the fact that five observations cannot pin down a population.
Normality is load-bearing, and there is a way out
Like a prediction interval and unlike a confidence interval, this is a statement about individual observations — so no central limit theorem helps.
A skewed population makes the symmetric interval wrong on both sides, too wide on the short tail and too narrow on the long one, and the coverage failure concentrates exactly where the spec limit matters.
Heavy tails are worse still, because the factor is built on a normal quantile and heavy-tailed populations put far more than 5% beyond ±1.96σ.
The distribution-free alternative uses order statistics. The range from the smallest to the largest of n observations is a tolerance interval with a coverage and confidence that depend only on n — no distribution assumed at all.
It costs sample size. A distribution-free 95/95 two-sided interval needs 93 observations to use the min and max; the normal-theory version manages it with a handful, which is what the normality assumption is buying.
Check normality before relying on this, with a normality test or a probability plot — and treat a transform as the first response rather than the last.
Reporting a tolerance interval
Four things, and the first two are the ones that make it a tolerance interval rather than an unlabelled range.
Give both probabilities. “A 95/95 tolerance interval” or “95% coverage with 95% confidence”. One number alone is ambiguous between two very different claims.
Give n. The factor depends on it far more strongly than most readers expect — 5.09 at five observations against 2.23 at a hundred.
Say whether it is one-sided or two-sided. They use different factors, and a one-sided bound taken as half a two-sided interval is too tight.
And state the normality evidence. The interval is a claim about individual units, so the distributional assumption is doing real work and a reader is entitled to know what supports it.
Sources and methodology
References for tolerance intervals and their factors.
Method. The tolerance factor uses Howe’s approximation with the chi-square quantile found by bisection on the survival function to 200 iterations rather than from a table, so it is exact to the approximation’s own accuracy at any n and coverage rather than only at tabulated values. All three intervals come from one pass over the data, which is what makes their side-by-side widths a property of the methods rather than of rounding: on the shipped twenty points they are 0.2331, 1.0681 and 1.3706. The suite asserts the ordering — prediction wider than confidence at every sample size, and tolerance wider than prediction whenever the coverage matches the prediction level — and that the tolerance factor never falls below the normal quantile for the requested coverage, which is the theoretical floor it approaches from above. Fewer than three values, zero variation, and a coverage or confidence outside (0, 1) all return no result. That engine is verified on every change against 95 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Prediction IntervalFor the next single observation, printed beside the confidence interval: the confidence factor goes to zero as n grows while this one converges to 1.95996 and stops.
Confidence IntervalIntervals for a mean or a proportion using t at every sample size and Wilson rather than the textbook Wald formula — with both methods shown, because Wald returns [0,0] at zero successes.
CpkCp, Cpk, Pp and Ppk with the defect rates they predict and the rate actually observed — including the built-in case where Cp is 2.05, Cpk is 0.57 and a quarter of the sample is already out of spec.
Normality TestShapiro-Wilk, Anderson-Darling and Jarque-Bera with a Q-Q plot, plus a resampled sweep answering the question the tests cannot: was your sample size big enough to detect anything?
Empirical RuleGives the exact 68.27/95.45/99.73 at any k, plus the multiplier for a round 95% — and prints Chebyshev's distribution-free floor beside each figure so the cost of assuming normality is a number.
Standard DeviationSample and population standard deviation, plus variance, mean, median, quartiles, z-scores, outliers, and confidence intervals.
An educational tool. Normal tolerance intervals assume individual observations are normally distributed — an assumption the central limit theorem does not relieve — and use Howe’s approximation to the exact factor, which is within about 1% for n ≥ 5. A one-sided bound is a different calculation and is not half a two-sided interval.
Published the interval that carries TWO probabilities — a stated proportion of the population, with stated confidence — printed beside the confidence and prediction intervals on the same data.
Quantified what the second probability costs: the 95/95 factor is 5.09353 at n = 5, 2.75228 at n = 20, 2.23280 at n = 100 and 2.03608 at n = 1,000. Even a thousand observations leave it 3.9% above the 1.95996 that would cover 95% of a normal whose parameters you actually knew.
Found and documented a caveat the first draft got wrong: 'confidence < prediction < tolerance' is only true when the coverage matches the prediction level. A 90%-coverage tolerance interval is wider than a 95% prediction interval up to about n = 40 (2.051606 against 2.047818) and NARROWER by n = 50 (1.996326 against 2.029572). The assertion suite failed on exactly those two cases, which is how the caveat reached the page.
Computed the chi-square quantile by bisection rather than from a table, so the factor is exact to Howe's own accuracy at any n and coverage rather than only at tabulated values.
Stated that the one-sided factor is a different quantity, not half a two-sided interval — which is a common and consequential error in specification setting.
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