# Confidence Intervals for MTBF

### EDITED BY JOHN HEALY

As with other point estimates, we often want to calculate the confidence interval about the estimate. The intent is to determine the range of reasonable values for the true and unknown population parameter. For MTBF, this no different.

Keep in mind that when calculating MTBF, we are using time to failure data that is often censored in some manner. For example, if we have 100 systems operating and have experienced only 12 failures, the bulk of the systems have never failed.

The discussion here is for either an exponential distribution based point estimate of the $\theta$ parameter or for a homogeneous Poisson process (HPP). If the system is best described by a non homogeneous Poisson process (NHPP), then the confidence intervals described below are not appropriate as the intervals well depend on the specific NHPP model.

When conducting a test to estimate MTBF, we may run the systems in the test for a specific amount of time, or until we experience some number of failures. When the data ends at a point in time that does not correspond to a time of failure, the data is said to be time censored. If the ending time corresponds with a failure, then we have failure censoring.

I bring up the nature of the censoring as it changes the formula for the confidence interval for the MTBF estimate.

## χ2 Distribution

MTBF is commonly associated with the exponential distribution, so when either assuming or deliberately using the exponential distribution and the statistic MTBF the confidence intervals are described in part by the χ2 distribution. Keep in mind that the chi-squared distribution is not symmetrical, like the normal or t distributions, thus we need to find the appropriate lower and/or upper χ2 value to complete the calculation. It also means the confidence intervals are likely not symmetrical either.

## Lower Confidence Limit for Type I Censoring

Type I censoring is time terminated. For example, when the data collection period ends, say at 2,000 hours, there was not a failure at 2,000 hours. Lower confidence is often of interest as it indicates the lower range of the MTBF value, or how bad might the true result actually be.

The formula for the Type I lower confidence interval is $\displaystyle \theta \ge \frac{2T}{\chi _{\left( \alpha ,2r+2 \right)}^{2}}$

Where,

• 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+2 is then the degrees of freedom for the χ2 distribution table

## Lower Confidence Limit for Type II Censoring

In this case the censoring is failure based, meaning the test ended with a failure. The lower value is $\displaystyle \theta \ge \frac{2T}{\chi _{\left( \alpha ,2r \right)}^{2}}$

Note the small change in the calculation of the degrees of freedom.

## Two Sided Confidence Interval Formulas

Again the censoring matters. For time censored data or Type I the equations are $\displaystyle \frac{2T}{\chi _{\left( \frac{\alpha }{2},2r+2 \right)}^{2}}\le \theta \le \frac{2T}{\chi _{\left( 1-\frac{\alpha }{2},2r \right)}^{2}}$

The 2r+2 degrees of freedom is only for the lower bound.

For the Type II or failure censored data the equations are $\displaystyle \frac{2T}{\chi _{\left( \frac{\alpha }{2},2r \right)}^{2}}\le \theta \le \frac{2T}{\chi _{\left( 1-\frac{\alpha }{2},2r \right)}^{2}}$

Again the lower bound does not have the extra two degrees of freedom. The key when calculating these confidence intervals is to know if the data is time or failure censored, then use the correct formula for degrees of freedom.

You may also be interested in how setting the confidence level alters the result. You may see data sheets reporting MTBF with a 60% confidence, well what does the mean? See the article Lower Confidence for more information.

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