Let’s assume you would like to estimate the upper confidence limit of a failure rate. As an example, assume you have 21 samples of 1000 that fail. What can you say, with 95% confidence, is the upper limit of the fails? See figure 1 for the derivation of the estimate for the 95% upper limit. This techniques uses the confidence interval on proportions as discussed in a recent post.
Figure 1. The calculations to determine the upper confidence limit of failure rates
For this example, substituting n = 1000, x = 21, we see from Figure 2 that with 95% confidence, the upper limit on the estimate of the defect rate is about 3% or 30 defects per 1,000.
Figure 2. The upper confidence limit when there are 21 fails out of 1,000.
This approach assumes common cause fails that are normally distributed. When failure rates are extremely low, say 1 in 100,000 or less, the fails may be special cause and not normally distributed. If this is the case, this approximation is not valid.