KEVOS® MAKEOS · Maintenance
MTBF Confidence Bounds (Field or Test Failures, Chi-Square Limits, Requirement Check)
From the operating hours and failures I have recorded, what is the mean time between failures, what lower bound can I claim at my confidence level, and does it meet the requirement?
Confidence bounds on the mean time between failures (MTBF) for repairable equipment or components with a constant failure rate, from total operating hours and the number of failures in a time-terminated record: the MTBF and failure rate estimates, the one-sided lower bound at your confidence, the two-sided interval, the bound with no failures, and the further failure-free hours needed to demonstrate the requirement.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Total operating hours | h | Sum of operating hours over all units in the record (not calendar hours), up to the end of the record. |
| Number of failures | failures | Failures recorded in those hours (each repair counts once). |
| Confidence for the lower bound | % | Confidence for the one-sided lower bound; the two-sided interval uses twice the remainder (95 % gives a 90 % interval). |
| Required MTBF | h | MTBF the equipment must demonstrate, from a contract or specification. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Total operating hours
- What it is
- The total operating hours of all units in the record added together, counted to the end of the record rather than to the last failure.
- Units and format
- Enter a number in h: hours. The form accepts 0.000001 h or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 800 h as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- More hours with the same failures raise the MTBF and both bounds in proportion.
- Get it right
- Use hour meters or run logs; exclude idle and standby time unless failures happen then too.
What this means: Number of failures
- What it is
- The number of failures recorded in those operating hours, counting each failure that needed a repair once.
- Units and format
- Enter a number in failures: failures. The form accepts 0 failures to 10000 failures. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2 failures as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- More failures lower the MTBF but narrow the interval around it.
- Get it right
- Use a written failure definition and count only failures of the equipment itself, not operator errors.
What this means: Confidence for the lower bound
- What it is
- The confidence for the one-sided lower bound; the two-sided interval uses twice the remainder, so 95 % gives a 90 % interval.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 50.1% to 99.9%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 95% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A higher confidence lowers the lower bound.
- Get it right
- Use the confidence named in the contract or specification; 80 %, 90 % and 95 % are common.
What this means: Required MTBF
- What it is
- The mean time between failures the equipment must demonstrate, from a contract, specification or maintenance plan.
- Units and format
- Enter a number in h: hours. The form accepts 0.000001 h or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 100 h as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Only moves the pass or fail line and the further hours needed.
- Get it right
- Make sure it uses the same failure definition and operating-hour basis as the record.
Method and formulas
- MTBF = T ÷ r and λ = r ÷ T, with T the total operating hours and r the failures (NIST e-Handbook 8.4.5.1).
- Lower bound = 2T ÷ χ²(C; 2(r + 1)) and upper bound = 2T ÷ χ²(1 − C; 2r), where χ²(P; ν) is the chi-square value with cumulative probability P; together they form a two-sided interval at 2C − 1.
- With no failures the lower bound is T ÷ (−ln(1 − C)); further hours needed = required MTBF × χ²(C; 2(r + 1)) ÷ 2 − T, with no more failures.
- Example: MTBF lower bound 127.07 h at the confidence chosen, against 100 h required.
Limitations
- Assumes a constant failure rate (exponential or HPP model): check there is no trend of improving or worsening reliability first, for example with a reliability growth plot.
- The bounds are exact for records that end at a fixed time; for a record that ends at a failure they are approximate (NIST 8.4.5.1).
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Report a defensible MTBF from field or test records
- Check whether a machine or fleet meets an MTBF requirement
- Work out how much more failure-free running would demonstrate it
Who it is for
- Maintenance and reliability engineers
- Test engineers
- Asset managers
What you get
- MTBF and failure rate estimates
- One-sided lower bound
- Two-sided interval
- Further failure-free hours needed
Interpreting the result
- NIST’s example: 800 hours with 2 failures gives an MTBF of 400 h with a 90 % interval of 127 to 2,251 h, so 127 h is the 95 % lower bound.
References
- NIST/SEMATECH e-Handbook of Statistical Methods, 8.4.5.1 Constant repair rate (HPP/exponential) model — https://www.itl.nist.gov/div898/handbook/apr/section4/apr451.htm
Learn the Concepts Behind This Tool
These KEVOS® Library readings explain the method this tool uses, with worked examples.
Method explained here · KEVOS® original book
Verify with a pilot fleet
In Connected Products by the Numbers
What it covers: Plan a pilot fleet that verifies requirements and validates the product, read failure rates with honest confidence bounds, and let field data replace the assumptions behind the design.
Method explained here · KEVOS® original book
Qualification testing: run length, feed envelope and yield basis
In First-of-a-Kind Technology by the Numbers
What it covers: Size a qualification run with chi-square bounds (1,500 h for MTBF 500 h at 80%), test across a 5–13% contamination envelope, and correct a claimed 84% yield to 78.9%.
Method explained here · KEVOS® original book
Inverter heat, reliability and spares
In Rooftop Solar for Factories by the Numbers
What it covers: How to ventilate an inverter room so the inverters do not derate, read inverter failure records with confidence bounds, and decide whether a spare inverter is worth holding.
Method explained here · KEVOS® original book
Rare events
In Simulating Noisy Systems by the Numbers
What it covers: Estimating events that happen once in years: what zero events in a run can show, extrapolating along the tail, splitting runs at an intermediate level, and why a fitted normal tail misleads.
Method explained here · KEVOS® original book
Proving the settings
In Trips and Alarms by the Numbers
What it covers: Turning calculated trip and alarm settings into evidence: end-to-end response tests, a planned demonstration of the nuisance-trip rate with confidence bounds, and an auditable register entry.
Useful alongside · KEVOS® original book
Up and down: the two-state model
In Breakdowns and Buffers by the Numbers
What it covers: A repairable machine as a two-state Markov process: failure and repair rates from a stoppage log, steady-state availability, settling after a restart, and why stop definitions matter.
Also in Are the stops random? Poisson counts and memoryless clocks
KEVOS® tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.