Statistical Confidence Calculator


It is tempting to take the rate of an event as determined in a single test as the nominal rate. If you see only one failure out of a batch of 100 devices, is it safe to say that the failure rate is 1%? No. The range that the actual rate may fall in to a certain level of probability is known as the confidence interval.

The normal approximation (shown below) is reasonably accurate for determining the confidence interval for trials where the probability of an event is not close to zero or one. In single event radiation testing, the probability of a single particle causing a particular event is often less than one in a million, so this approximation is often not particularly accurate for such testing.

\[p \approx \hat{p} ± z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\]

\(\hat{p}\) is the number of successes (or failures if that's what you're counting) divided by trials for a given experiment. n is the number of trials. z is the quantile of the standard normal distribution corresponding to the allowable error rate. For a confidence interval of 95%, the allowable error rate is 5%, and using the probit (probability unit) function, the corresponding z value of 1.96 can be calculated.

A better approximation is the Wilson Interval shown below:

\[p \approx \frac{1}{1+\frac{z^2}{n}}\left(\hat{p}+\frac{z^2}{n}\right) ± \frac{z}{1+\frac{z^2}{n}} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}+\frac{z^2}{4n^2}}\]

The Wilson approximation is implemented in javascript in the below calculator which is tailored for typical values found in radiation testing. The default case is for 10 events observed in 1e11/cm^2 fluence. As a general note, fluence is almost always expressed in terms of number of protons or ions per unit area. This allows the probability of an interaction to be expressed as a physical cross-section area.

If you enter 0 events, Probit will be invalid and the math behind the rule of three will be applied.

The wikipedia article on Binomial Proportion Confidence Intervals states "Of the approximations listed above, Wilson score interval methods (with or without continuity correction) have been shown to be the most accurate and the most robust, though some prefer the Agresti–Coull approach for larger sample sizes. Wilson and Clopper–Pearson methods obtain consistent results with source significance tests, and this property is decisive for many researchers." So I will stick with the Wilson interval on this page. This calculator also works for failure rates; enter the number of observed failures and the total accumulated operating time to get error bounds on the failure rate.

In single event radiation effects testing, you often encounter parts that have no observable upsets over the tested fluence (of heavy ions or protons). What is the confidence interval when you see no events? Generally, this follows the Rule of Three; The 95% confidence interval when no events are observed in n trials is 3/n. In single event radiation testing, each proton or ion constitutes a trial. If a part has no upsets in 3e9 protons, the 95% confidence interval for its upset cross-section is 0 to 1e-9 cm^2. A deeper explanation of the rule of three from a medical statistics perspective is given here. The Wilson approximation is a bit over conservative here; it estimates an interval of 0 to 3.83/n rather than 1 to 3/n. This page includes an if statement that applies the math behind the rule of three when zero failures are entered:

\[p \approx \frac{-ln(1-confidence)}{n} \]

Side note: For practical purposes, parts are sometimes considered to be immune to a particular single event effect if it is not observed over a large enough fluence. JESD234 lists 1e10 p+/cm^2 as a typical target fluence for soft (commercial) devices and 1e12 p+/cm^2 as a typical target fluence for rad-hard devices. JESD57A similarly lists 1e7 ions/cm^2 as a heavy ion target fluence for integrated devices.

Interestingly, section 5.2.11 of JESD57A states that the equation for a 95% confidence interval of actual rate when no events are observed is \(\sigma = \frac{3.7}{\text{fluence of no errors} * cos(\theta)}\) where \(\theta\) is the angle from normal of particles upon the die of the device under test. I think this equation is in error. The value 3.7 appears to be based on an approximation function listed in annex A.10 of the same document which is based on the Chi-squared distribution. Approximation functions (including the Wilson approximation on this page) will have significant error when there are no observed events, and the calculation is the same every time, so there is no need to use a function — you can just use the rule of three.