Confidence and Tolerance Intervals
Normal Tolerance Interval Factors (k)
Factors k for tolerance intervals x̄ ± k·s that contain a stated proportion of the population with a stated confidence.
- n = 2 to 1000
- Exact values
- Any coverage and confidence
Tolerance Factors (k)
Two tables: two-sided intervals and one-sided bounds. Rows: sample size n · Columns: confidence level and coverage P · Cells: factor k.
Two-sided tolerance intervals: x̄ ± k·s
| 95% confidence · coverage P | 99% confidence · coverage P | |||||
|---|---|---|---|---|---|---|
| n | 90% | 95% | 99% | 90% | 95% | 99% |
| 2 | 31.092 | 36.519 | 46.944 | 155.57 | 182.72 | 234.88 |
| 3 | 8.306 | 9.789 | 12.647 | 18.782 | 22.131 | 28.586 |
| 4 | 5.368 | 6.341 | 8.221 | 9.416 | 11.118 | 14.405 |
| 5 | 4.291 | 5.077 | 6.598 | 6.655 | 7.870 | 10.220 |
| 6 | 3.733 | 4.422 | 5.758 | 5.383 | 6.373 | 8.292 |
| 7 | 3.390 | 4.020 | 5.241 | 4.658 | 5.520 | 7.191 |
| 8 | 3.156 | 3.746 | 4.889 | 4.189 | 4.968 | 6.479 |
| 9 | 2.986 | 3.546 | 4.633 | 3.860 | 4.581 | 5.980 |
| 10 | 2.856 | 3.393 | 4.437 | 3.617 | 4.294 | 5.610 |
| 11 | 2.754 | 3.273 | 4.282 | 3.429 | 4.073 | 5.324 |
| 12 | 2.670 | 3.175 | 4.156 | 3.279 | 3.896 | 5.096 |
| 13 | 2.601 | 3.093 | 4.051 | 3.156 | 3.751 | 4.909 |
| 14 | 2.542 | 3.024 | 3.962 | 3.054 | 3.631 | 4.753 |
| 15 | 2.492 | 2.965 | 3.885 | 2.967 | 3.529 | 4.621 |
| 16 | 2.449 | 2.913 | 3.819 | 2.893 | 3.441 | 4.507 |
| 17 | 2.410 | 2.868 | 3.761 | 2.828 | 3.364 | 4.408 |
| 18 | 2.376 | 2.828 | 3.709 | 2.771 | 3.297 | 4.321 |
| 19 | 2.346 | 2.793 | 3.663 | 2.720 | 3.237 | 4.244 |
| 20 | 2.319 | 2.760 | 3.621 | 2.675 | 3.184 | 4.175 |
| 21 | 2.294 | 2.731 | 3.583 | 2.635 | 3.136 | 4.113 |
| 22 | 2.272 | 2.705 | 3.549 | 2.598 | 3.092 | 4.056 |
| 23 | 2.251 | 2.681 | 3.518 | 2.564 | 3.053 | 4.005 |
| 24 | 2.232 | 2.658 | 3.489 | 2.534 | 3.017 | 3.958 |
| 25 | 2.215 | 2.638 | 3.462 | 2.506 | 2.984 | 3.915 |
| 26 | 2.199 | 2.619 | 3.437 | 2.480 | 2.953 | 3.875 |
| 27 | 2.184 | 2.601 | 3.415 | 2.456 | 2.925 | 3.838 |
| 28 | 2.170 | 2.585 | 3.393 | 2.434 | 2.898 | 3.804 |
| 29 | 2.157 | 2.569 | 3.373 | 2.413 | 2.874 | 3.772 |
| 30 | 2.145 | 2.555 | 3.355 | 2.394 | 2.851 | 3.742 |
| 35 | 2.094 | 2.495 | 3.276 | 2.314 | 2.756 | 3.618 |
| 40 | 2.055 | 2.448 | 3.216 | 2.253 | 2.684 | 3.524 |
| 45 | 2.024 | 2.412 | 3.168 | 2.205 | 2.627 | 3.450 |
| 50 | 1.999 | 2.382 | 3.129 | 2.166 | 2.580 | 3.390 |
| 60 | 1.960 | 2.335 | 3.068 | 2.106 | 2.509 | 3.297 |
| 70 | 1.931 | 2.300 | 3.023 | 2.062 | 2.457 | 3.228 |
| 80 | 1.908 | 2.274 | 2.988 | 2.028 | 2.416 | 3.175 |
| 90 | 1.890 | 2.252 | 2.959 | 2.001 | 2.384 | 3.133 |
| 100 | 1.875 | 2.234 | 2.936 | 1.978 | 2.357 | 3.098 |
| 150 | 1.826 | 2.176 | 2.859 | 1.906 | 2.271 | 2.985 |
| 200 | 1.798 | 2.143 | 2.816 | 1.866 | 2.223 | 2.921 |
| 300 | 1.767 | 2.106 | 2.767 | 1.820 | 2.169 | 2.850 |
| 500 | 1.737 | 2.070 | 2.721 | 1.777 | 2.117 | 2.783 |
| 1000 | 1.709 | 2.036 | 2.676 | 1.736 | 2.068 | 2.718 |
One-sided tolerance bounds: x̄ + k·s or x̄ − k·s
| 95% confidence · coverage P | 99% confidence · coverage P | |||||
|---|---|---|---|---|---|---|
| n | 90% | 95% | 99% | 90% | 95% | 99% |
| 2 | 20.581 | 26.260 | 37.094 | 103.03 | 131.43 | 185.62 |
| 3 | 6.155 | 7.656 | 10.553 | 13.995 | 17.370 | 23.896 |
| 4 | 4.162 | 5.144 | 7.042 | 7.380 | 9.083 | 12.387 |
| 5 | 3.407 | 4.203 | 5.741 | 5.362 | 6.578 | 8.939 |
| 6 | 3.006 | 3.708 | 5.062 | 4.411 | 5.406 | 7.335 |
| 7 | 2.755 | 3.399 | 4.642 | 3.859 | 4.728 | 6.412 |
| 8 | 2.582 | 3.187 | 4.354 | 3.497 | 4.285 | 5.812 |
| 9 | 2.454 | 3.031 | 4.143 | 3.240 | 3.972 | 5.389 |
| 10 | 2.355 | 2.911 | 3.981 | 3.048 | 3.738 | 5.074 |
| 11 | 2.275 | 2.815 | 3.852 | 2.898 | 3.556 | 4.829 |
| 12 | 2.210 | 2.736 | 3.747 | 2.777 | 3.410 | 4.633 |
| 13 | 2.155 | 2.671 | 3.659 | 2.677 | 3.290 | 4.472 |
| 14 | 2.109 | 2.614 | 3.585 | 2.593 | 3.189 | 4.337 |
| 15 | 2.068 | 2.566 | 3.520 | 2.521 | 3.102 | 4.222 |
| 16 | 2.033 | 2.524 | 3.464 | 2.459 | 3.028 | 4.123 |
| 17 | 2.002 | 2.486 | 3.414 | 2.405 | 2.963 | 4.037 |
| 18 | 1.974 | 2.453 | 3.370 | 2.357 | 2.905 | 3.960 |
| 19 | 1.949 | 2.423 | 3.331 | 2.314 | 2.854 | 3.892 |
| 20 | 1.926 | 2.396 | 3.295 | 2.276 | 2.808 | 3.832 |
| 21 | 1.905 | 2.371 | 3.263 | 2.241 | 2.766 | 3.777 |
| 22 | 1.886 | 2.349 | 3.233 | 2.209 | 2.729 | 3.727 |
| 23 | 1.869 | 2.328 | 3.206 | 2.180 | 2.694 | 3.681 |
| 24 | 1.853 | 2.309 | 3.181 | 2.154 | 2.662 | 3.640 |
| 25 | 1.838 | 2.292 | 3.158 | 2.129 | 2.633 | 3.601 |
| 26 | 1.824 | 2.275 | 3.136 | 2.106 | 2.606 | 3.566 |
| 27 | 1.811 | 2.260 | 3.116 | 2.085 | 2.581 | 3.533 |
| 28 | 1.799 | 2.246 | 3.098 | 2.065 | 2.558 | 3.502 |
| 29 | 1.788 | 2.232 | 3.080 | 2.047 | 2.536 | 3.473 |
| 30 | 1.777 | 2.220 | 3.064 | 2.030 | 2.515 | 3.447 |
| 35 | 1.732 | 2.167 | 2.995 | 1.957 | 2.430 | 3.334 |
| 40 | 1.697 | 2.125 | 2.941 | 1.902 | 2.364 | 3.249 |
| 45 | 1.669 | 2.092 | 2.898 | 1.857 | 2.312 | 3.180 |
| 50 | 1.646 | 2.065 | 2.862 | 1.821 | 2.269 | 3.125 |
| 60 | 1.609 | 2.022 | 2.807 | 1.764 | 2.202 | 3.038 |
| 70 | 1.581 | 1.990 | 2.765 | 1.722 | 2.153 | 2.974 |
| 80 | 1.559 | 1.964 | 2.733 | 1.688 | 2.114 | 2.924 |
| 90 | 1.542 | 1.944 | 2.706 | 1.661 | 2.082 | 2.883 |
| 100 | 1.527 | 1.927 | 2.684 | 1.639 | 2.056 | 2.850 |
| 150 | 1.478 | 1.870 | 2.611 | 1.566 | 1.971 | 2.740 |
| 200 | 1.450 | 1.837 | 2.570 | 1.524 | 1.923 | 2.679 |
| 300 | 1.417 | 1.800 | 2.522 | 1.476 | 1.868 | 2.608 |
| 500 | 1.385 | 1.763 | 2.475 | 1.430 | 1.814 | 2.540 |
| 1000 | 1.354 | 1.727 | 2.430 | 1.385 | 1.762 | 2.475 |
Tip: click any value to highlight its row and column.
How to read this table
- Choose the coverage P (the proportion of the population the interval should contain) and the confidence level.
- Find k for your sample size n.
- The two-sided tolerance interval is x̄ − k·s to x̄ + k·s. A one-sided bound is x̄ + k·s (upper) or x̄ − k·s (lower), using the one-sided table.
Worked example
Twenty batches of a drug product have a mean assay of 100.2% and SD 1.4%. For an interval containing 99% of batches with 95% confidence, row 20 of the two-sided table gives k = 3.621, so the interval is 100.2 ± 3.621 × 1.4 = 95.1% to 105.3%. If this falls inside the specification limits, the process can be expected to produce conforming batches.
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Calculator
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Formula and how the values were computed
Two-sided factors are exact solutions of the Odeh-Owen integral equation; one-sided factors are exact quantiles of the noncentral t distribution. Many printed tables use Howe's or Wald and Wolfowitz's approximations, which differ slightly (for example 3.379 against the exact 3.393 for n = 10, 95% coverage, 95% confidence).
Frequently asked questions
Tolerance interval or confidence interval?
A confidence interval covers a parameter such as the mean. A tolerance interval covers a proportion of individual values, which is what matters for specifications and reference ranges.
Is normality important?
Yes. Tolerance intervals depend on the tails of the distribution, so check normality first or use a transformation or a nonparametric tolerance interval.