Where the next measurement lands, not where the average is.
Where the next observation falls
Mean 10.0900, SD 0.2490. The confidence interval is [9.9735, 10.2065] — width 0.2331. The prediction interval is [9.5560, 10.6240] — width 1.0681, which is 4.58 times wider.
n = 20, mean 10.0900, SD 0.2490
95% prediction interval: 9.5560 to 10.6240
The 95% confidence interval on the MEAN is [9.9735, 10.2065] — width 0.2331 against the prediction interval's 1.0681, a factor of 4.58. They answer different questions: one says where the average is, the other where a single new measurement will land. The confidence interval shrinks toward zero as you collect more data; this one does not.
Prediction interval
±0.5340
k = 2.14471
Confidence interval
±0.1165
k = 0.46801
Ratio
4.58×
and it grows with n
Width as n → ∞
0.9761
the CI limit is 0
The two factors, as the sample grows
Both are exact functions of n alone. Multiply by your sample SD to get a width.
Confidence and prediction interval factors at seven sample sizes
n
Confidence k
Prediction k
Prediction ÷ confidence
5
1.24166
3.04144
2.45×
10
0.71536
2.37257
3.32×
20
0.46801
2.14471
4.58×
30
0.37341
2.07904
5.57×
50
0.28420
2.02957
7.14×
100
0.19842
1.99411
10.05×
1,000
0.06205
1.96332
31.64×
∞
0
1.95996
∞
The confidence factor goes to zero and the prediction factor goes to 1.95996 — so the ratio between them grows without limit. Collecting more data makes you certain about the mean and no more certain about the next measurement, because the next measurement is genuinely variable and no amount of history changes that.
k = t·√(1 + 1/n), never below 1.96 Covers ONE future observation, not a proportion Normality matters far more here than for a CI
What this tool shows
The confidence-interval factor falls from 1.24166 at n = 5 to 0.06205 at n = 1,000 and keeps going to zero. The prediction factor falls from 3.04144 to 1.96332 and stops there. So the prediction interval is 2.45 times as wide as the confidence interval at n = 5 and 31.64 times as wide at n = 1,000, and that ratio grows without limit. More data makes you certain about the mean and no more certain about the next measurement.
A 95% prediction interval for the next single observation
The confidence interval on the mean printed beside it, for contrast
Both k factors, which depend only on n — multiply by your SD for a width
The exact factors at seven sample sizes, and their limits as n → ∞
The ratio between the two intervals, which grows without bound
What each interval converges to, rather than what it looks like at your n
Next observation Both k factors Limits as n → ∞ Beside the CI
Normality matters much more here than for a confidence interval.
Updated 12 September 2026 · Works in any browser, no installation
A prediction interval says where a single NEW observation from the same process is likely to fall. A confidence interval says where the population MEAN is. The second shrinks toward a point as you collect data; the first cannot, because a single observation is genuinely variable however much history you have.
At a glance
Formula shown
x̄ ± t_{α/2, n−1} · s · √(1 + 1/n). The 1 under the root is the variability of the new observation itself and the 1/n is the uncertainty in the estimated mean — which is exactly the confidence interval’s term. As n grows the 1/n vanishes and the factor converges to z_{α/2} = 1.95996, while the confidence factor t/√n goes to zero.
Scenario support
Quality control limits for the next unit off a line, forecasting a single future value, clinical reference ranges from a sample, warranty and tolerance planning, and any situation where someone will ask "what will the next one be?" rather than "what is the average?".
Educational estimate
Planning support from the values you enter — not professional advice.
The one term that never goes away
The formula makes it obvious once the two sources of uncertainty are separated, and the factor table makes it exact.
Under the square root there are two terms: 1 and 1/n. The 1/n is uncertainty about where the mean is, and it is exactly what a confidence interval consists of. The 1 is the new observation’s own variability.
Only the 1/n responds to sample size. At n = 1,000 the prediction factor is 1.96332, barely above the limiting 1.95996 — the estimation uncertainty has all but vanished and what remains is the process itself.
Meanwhile the confidence factor has gone from 1.24166 to 0.06205, and it keeps going. The ratio between the two intervals is 2.45 at n = 5, 4.58 at n = 20, 10.05 at n = 100 and 31.64 at n = 1,000.
Which is why quoting a confidence interval when someone asked about a single unit is a serious error. On the shipped twenty-point sample the confidence interval spans 0.2331 and the prediction interval spans 1.0681 — a limit set from the wrong one would be violated constantly.
Normality matters far more here than for a confidence interval
This is the asymmetry that catches people who learned that the central limit theorem makes normality optional.
A confidence interval is about a MEAN, and means of n observations are approximately normal whatever the underlying distribution, for n large enough. That is the central limit theorem doing the work.
A prediction interval is about a SINGLE observation, which has whatever distribution the process has. No averaging happens, so no central limit theorem applies, and the normality assumption is load-bearing rather than a formality.
On skewed data the symmetric interval is simply wrong. It will be too wide on one side and too narrow on the other, and the achieved coverage on each side will not be 2.5%.
The non-parametric alternative is order statistics: with n observations, the range from smallest to largest covers a new value with probability (n−1)/(n+1). Twenty points gives 90.5%, and reaching 95% takes 39.
Or transform first. A log-transform is standard for right-skewed positive data, and the interval is computed on the log scale and exponentiated back — which correctly produces an asymmetric interval on the original scale.
Three intervals, three questions
Confidence, prediction and tolerance intervals are routinely confused, and the confusion always runs in the direction of quoting something too narrow.
A confidence interval covers the population mean. “The average is between 9.97 and 10.21.” Width → 0 as n grows.
A prediction interval covers the NEXT observation. “The next unit will be between 9.56 and 10.62.” Width → 2 × 1.96σ.
A tolerance interval covers a PROPORTION of the population, with confidence. “95% of all units lie between 9.40 and 10.78, and I am 95% confident of that.” Wider still, because it carries two probabilities.
On the same twenty points the widths are 0.2331, 1.0681 and 1.3706. A factor of nearly six between the narrowest and the widest, from one dataset.
The test for which you need: what is the sentence? “The average” is a confidence interval, “the next one” a prediction interval, “most of them” a tolerance interval.
In regression the interval varies along the line
The same distinction appears in regression, where the two bands are usually plotted together and mislabelled about equally often.
The confidence band is narrowest at the mean of x and flares outward, because the fitted line pivots around that point and the pivoting costs more precision further away.
The prediction band has the same shape plus a constant, so it is always wider and flares less in relative terms — the residual variance dominates.
Both flare sharply outside the observed range of x, which is the arithmetic warning against extrapolation. The model has no information out there and the interval says so.
A regression prediction interval also assumes the residuals are homoscedastic, which is an extra assumption on top of normality — and one that fails routinely when the outcome is a count or a spend.
The factor is a property of n, not of your data
Everything the tool prints as a k factor depends only on the sample size and the confidence level. The data enters once, through the standard deviation.
k = t × √(1 + 1/n), which is 3.04144 at n = 5, 2.14471 at n = 20, 1.99411 at n = 100 and 1.96332 at n = 1,000.
Which means the factor table is reusable. Multiply by any sample’s SD and the interval follows — no recomputation needed, which is why these factors appear as published tables in quality-control handbooks.
Widening the confidence level moves it sharply. At 99% rather than 95% the limit is 2.5758 rather than 1.95996, so a 99% prediction interval is about 31% wider at large n.
And the interval scales linearly with the SD, so the hardest part of a prediction interval is estimating the spread — which at n = 5 is itself barely estimated, and is why the factor is 3.04 rather than 2.
Reporting a prediction interval
Four things, and the first is what prevents the confusion this page exists to correct.
Call it a prediction interval, explicitly. “95% interval” will be read as a confidence interval by most readers, and it is four to thirty times too narrow if they act on it.
Give n and the sample SD. They are what let a reader reconstruct the factor and check it.
Say what was assumed about the distribution. The symmetric interval is a normality claim about individual observations, not about the mean, and that is a much stronger assumption.
And say whether it covers one future value or several. A 95% interval that is expected to contain the next TEN observations is a different and wider calculation — the simultaneous version, which is a tolerance interval in all but name.
Sources and methodology
References for prediction intervals and their distinction from confidence intervals.
Method. The prediction factor is computed as t × √(1 + 1/n) and the confidence factor as t/√n from the same t quantile, so the pair differs only in the term that survives as n grows — which is what makes the limits exact rather than illustrative. The suite asserts that the prediction factor is strictly greater than the confidence factor at every n, that it never falls below the normal quantile 1.95996, and that the ratio between the two is monotonically increasing in n. The published factor table is generated by the same code that computes your interval, so the two can never drift apart. Fewer than three values, a sample with zero variation and any non-finite value return no result rather than an interval of zero width. 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:
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.
Tolerance IntervalCovers a stated proportion of the population with stated confidence: even at n = 1,000 the 95/95 factor is 2.03608, not the 1.95996 that covers 95% of a known normal.
Linear RegressionThe least-squares line with r and r² — and the regression of x on y beside it, because those are two different lines rather than one line rearranged.
Standard ErrorStandard error of a mean or proportion, printed beside the standard deviation it gets confused with — the ratio is always √n, and at n = 50 that is a factor of seven.
Bootstrap Confidence IntervalNormal, percentile, basic and BCa intervals from one seeded resample set, with measured coverage: on lognormal data at n = 20 a nominal 95% achieves 82.60% to 87.20% depending on the method.
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.
An educational tool. A prediction interval covers a single future observation and assumes those individual observations are normally distributed — an assumption the central limit theorem does not relieve, because no averaging takes place. On skewed data the symmetric interval is too wide on one side and too narrow on the other.
Published a prediction interval printed beside the confidence interval on the same data, because quoting the wrong one is the common error and it runs several-fold too narrow.
Made the distinction exact rather than verbal. The confidence factor falls from 1.24166 at n = 5 to 0.06205 at n = 1,000 and keeps going to zero; the prediction factor falls from 3.04144 to 1.96332 and stops. The ratio between the two intervals is 2.45 at n = 5, 4.58 at n = 20, 10.05 at n = 100 and 31.64 at n = 1,000 — and grows without limit.
Recorded the asymmetry that catches people who learned the central limit theorem makes normality optional: it does for a MEAN, and not at all for a single observation, which is what this interval covers.
Gave the distribution-free alternative with its exact cost: the range from smallest to largest of n observations covers a new value with probability (n−1)/(n+1), which is 90.5% at n = 20 and needs n = 39 to reach exactly 95%.
Published the factor table generated by the same code that computes the interval, so the two can never drift apart.
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