Let’s say we have a population and we are interested in the mean (average) of that population’s life. We select a sample (at random if at all possible) and measure a value, like time to failure, for each selected item in the sample.
We calculate the mean life of the sample by summing the sample values and dividing by the number of items in the sample.
Because we are only using a subset of the population it is possible the sample items are from one part of the population, say the tall part only. It may not be likely, yet it is possible to have selected samples that do not represent the range of values in the population.
It is this possibility that the sample statistic expected to represent the population parameter doesn’t actually even come close is the notion of statistical confidence. In a positive manner, we say there is a 95% confidence that the true unknown population parameter falls within a range of values, also called the confidence interval or bounds. That means there is a 5% chance that the actual and unknown population parameter is outside that range. In other words we are 95% confident that the sample is ‘this’ close to the actual value.
For MTBF confidence intervals we are often only interested in the lower limit (one-sided). We expect with said confidence that the true unknown value is above this lower confidence value, with Type II – predetermined number of failures terminates the test.
- θ is the calculated mean life (MTBF)
- T is the total time the samples operated before failing (or the test was ended)
- χ2 is the Chi-squared distribution
- α is the level of risk (1 – confidence)
- r is the number of failures, 2r is then the degrees of freedom for the chi-squared
Now an example. Given the MTBF for a test with 2 failures is 1525 hours. The total time, T, is 3050 hours and there were 2 failures, r. Calculate the 90% lower confidence interval for the estimated MTBF.
This means there is a 90% chance that the true and unknown population MTBF is greater than 784 hours. And, there is a 10% chance that it is less. Unless we determine the population mean (measure ever unit in the population) we won’t know.
For fun, consider we are willing to take more risk of the sample not representing the population. The same sample, just change the confidence. Let’s go from 90% to 60% and we find
Which has a higher value. Interesting. Same data, more risk, smaller confidence range. In other words by accepting more risk, we are saying there is now a greater chance that the true unknown parameter falls outside the range described by the confidence interval. The true value doesn’t change, the sample statistic doesn’t change. And, we’re saying the lower confidence value is higher than before.
Without careful consideration it appears the population lasts twice as long, 784 to 1,508 hours. Nothing actually has changed, just the increase in risk that the sample represents the true value.
Check out the next datasheet that crosses your desk – too many use a 60% confidence – why is that?