This is just an Excerpt from a larger document, click here to view the entire document.Reliability Data Analysis
Reliability data analysis includes three components: the raw data, the statistic used to synthesize it and the underlying statistical distribution of the variable. In this START Sheet, such distribution is assumed Exponential (how to test for this distribution has been discussed in Reference 6).
The raw data consists of the device lives, or of the test times with the number of failures that occurred. Data are then synthesized into two statistics: Total Test Time, denoted "T", and Total Number of Failures, denoted "n". Statistic T, however, can be implemented as, and interpreted in, different ways, according to how the original test times were collected.
For example, in reliability testing we can place "n" devices on operation and then observe them for some pre-specified time (T0) or until "k" devices fail, where k n. The failure times (when known) are denoted T1, T2, Tk and the Total Test Time statistic is then T = Σ Ti. If k = n then we have tested the entire sample, or until all n devices have failed. This is the situation treated in this START Sheet.
On other occasions, we test a set of devices for a pre-specified time T0 and then count the number "n" of devices that have failed during this time (but we do not know the exact times when the failures occurred).
The importance of knowing which of the two life testing schemes have been implemented is that the choice determines the "degrees of freedom" (denoted DF) that the corresponding Chi-Square statistic will use. This statistic, in turn, depends on having an underlying Exponential distribution for the life data of the devices.
When the underlying distribution of the lives is indeed Exponential and the failures are independent, the distribution of the statistic "twice the Total Test Time divided by the mean life" i.e., (2*T/θ) is distributed as a Chi-Square with γ = 2n DF. The Exponential distribution then allows us (given a pre-specified probability α) to find the Chi-Square percentile (via a ChiSquare table value: χ2α/2,γ) that defines a relation between the statistic T and the Exponential mean θ (see Figure 1).
For, if 2T/θ is distributed Chi-Square with DF = 2n then the α- percentile χ2α/2,γ,
allows us to obtain a probability bound for the unknown device mean life (or the reliability or the failure rate). We can estimate or bound θ, using Equation 1.
Figure 1. The Chi-Square Distribution of 2*Total Test Time/θ = 2*T/θ (Click to Zoom)
The Chi-Square distribution is readily tabulated and its parameter is the corresponding DF. For a time terminated test, the DF used are γ = 2n + 2 (twice the number of failures plus two). In the case of failure terminated tests, the DF used are γ = 2n (twice the number of failures observed during this time).
For example, assume we place ten devices on test and want to construct a 95% CI for the unknown mean θ. Assume we know the exact failure times of all ten devices (n = 10). We need to look up the upper (1 - α/2) and lower (α/2) percentiles corresponding to DF = 2n, for a confidence 1 - α = 0.95 (95%). Since we seek a (two-sided) CI, we need to split the 5% error (0.95 =
1 - = 1 - 0.05) into two halves (α/2 = 0.025 or 2.5%) on each extreme of the Chi-Square distribution (see Figure 1). To obtain the Chi-Square table values, enter the Chi-Square table and find the columns for 0.025 (lower) and 0.975 (upper) percentiles. Go down these two columns until reaching the desired DF: γ = 2n =
2 x 10 = 20. In the present example we obtain the following (upper/lower) results (see Figure 1).
If, instead of knowing the exact failure times we had only the Total Test Time (T) and Total Number of Failures (n) we should use DF = 2n + 2 instead. Therefore, in the same example above, now for DF = γ = 2n + 2 = 20 + 2 = 22, we obtain:
All the other Exponential parameters of interest (reliability and failure rate) are obtained directly from the Exponential mean. We next illustrate how to obtain CIs for the failure rate and the device reliability, from the mean, using a large set of life data.