Calculator Library / Finance
Size an attribute sample for testing an internal control, then evaluate the deviations you find with an upper confidence limit instead of the raw sample rate.
Tool
Zero-deviation size: n = ln(1 − confidence) / ln(1 − tolerable rate)
This is a statistical illustration of attribute sampling. Your audit or testing methodology, and applicable standards, define the sample sizes and evaluation rules you must use.
Sample sizes
| Expected deviations | Sample size needed |
|---|
Evaluation
Instructions
This calculator sizes an attribute sample for testing an internal control and evaluates the result. Given a confidence level, a tolerable deviation rate, and the number of deviations you expect, it gives the sample size. After testing, you enter the deviations found and it gives the upper limit of the deviation rate at your confidence level, and whether it is within tolerance.
Use it to explain why a small sample with one deviation may not support reliance, and to see how much larger a sample would need to be.
| Measure | Method | Meaning |
|---|---|---|
| Sample size | Smallest n where the chance of at most the expected deviations, if the true rate equals the tolerable rate, is at most 1 − confidence | For zero expected deviations, n = ln(1 − conf) / ln(1 − tolerable rate). |
| Upper limit | One-sided binomial (Clopper-Pearson) upper confidence limit for the deviation rate | The highest true rate consistent with the sample at the chosen confidence. |
| Conclusion | Upper limit compared with the tolerable rate | Whether the evidence supports reliance as tested. |
| Population adjustment | n / (1 + (n − 1) / N) | A reduction for small populations; not used by every methodology. |
The pre-loaded values are 95% confidence, a 5% tolerable rate, and no expected deviations, which gives a sample of 59. If 59 items are tested and one deviation is found, the sample rate is 1.69%, but the 95% upper limit is 7.79%, above 5%, so the result does not support reliance as tested. The tool also shows that about 93 items in total would be needed for one deviation to fit within tolerance, and that finding no deviations in 59 gives an upper limit of 4.95%.
Change the tolerable rate to 10% and the sample falls to 29, and change confidence to 90% and it falls further. The trade-off between confidence, tolerance, and sample size is the point of the exercise.
For zero expected deviations it is the natural log of one minus the confidence level divided by the natural log of one minus the tolerable rate, rounded up, which gives 59 at 95% confidence and a 5% tolerable rate. With expected deviations the calculator finds the smallest sample in which that many or fewer deviations would be unlikely if the true rate equaled the tolerable rate.
Because the evaluation uses an upper confidence limit, not the sample rate. One deviation in 59 is a 1.7% sample rate, but at 95% confidence the true rate could be as high as about 7.8%, which exceeds a 5% tolerable rate, so the evidence does not support reliance at that tolerance.
No. It illustrates the statistics behind attribute sampling. Your organization's methodology and the standards that apply to you define the sample sizes, selection methods, and evaluation rules you must follow, and your auditors may have their own requirements.
No. The calculator runs in your browser and nothing is sent to a server. It does not save your entries between visits.