What the calculator is for
Prior-use claims, Route 2H and plant-specific failure data all start from the same question: given the failures you have seen over the operating time you have, what failure rate can you defend? The number of failures n over a cumulative time T gives a point estimate n / T, but with few failures that estimate is uncertain. The chi-square distribution turns the same observation into an upper bound that the true rate stays below with a chosen confidence.
- Point estimate: λ = n / T
- Upper bound at confidence CL: λ = χ²(CL; 2n + 2) / (2T)
- T is the cumulative operating time: number of devices × time each was in service.
Worked example
A site has 200 pressure transmitters of one type, each in service for 5 years: T = 200 × 5 × 8,760 = 8.76 × 10⁶ device-hours. Proof tests and trips revealed 2 dangerous undetected failures. The point estimate is 2 / 8.76 × 10⁶ = 228.3 FIT.
At 90 % confidence, χ²(0.9; 6) = 10.645, so λDU ≤ 10.645 / (2 × 8.76 × 10⁶) = 607.6 FIT. At 70 % confidence the bound is 412.7 FIT. Use the bound, not the point estimate, in the verification.
Multiplier table
| Failures n | 70 % | 90 % | 95 % |
|---|---|---|---|
| 0 | 1.204 | 2.303 | 2.996 |
| 1 | 2.439 | 3.890 | 4.744 |
| 2 | 3.616 | 5.322 | 6.296 |
| 3 | 4.762 | 6.681 | 7.754 |
| 4 | 5.890 | 7.994 | 9.154 |
| 5 | 7.006 | 9.275 | 10.513 |
| 6 | 8.111 | 10.532 | 11.842 |
| 7 | 9.209 | 11.771 | 13.148 |
| 8 | 10.301 | 12.995 | 14.435 |
| 9 | 11.387 | 14.206 | 15.705 |
| 10 | 12.470 | 15.407 | 16.962 |
Divide the multiplier by the cumulative operating hours to get the upper-bound failure rate per hour.
Common mistakes
- Counting the wrong failures. For λDU count only dangerous undetected failures, found by proof tests, demands or inspection, not every repair order.
- Reporting n / T. With few failures the point estimate is optimistic; IEC 61508 and IEC 61511 ask for an upper confidence bound.
- Treating zero failures as zero rate. No failures in 10⁶ hours still means λ ≤ 1,204 FIT at 70 % confidence.
- Pooling unlike devices. Combine operating time only for the same device type in similar service and environment.
Basis
For a constant failure rate, failures in a fixed (time-terminated) observation follow a Poisson distribution; the one-sided upper bound uses chi-square with 2n + 2 degrees of freedom, as in IEC 61508-6 and IEC 60605-4. IEC 61508-2 Route 2H calls for failure data with at least 90 % confidence; IEC 61511 expects reliability data at an upper confidence limit of at least 70 %. The chi-square values are computed exactly, not read from a table.