Library / Probability distributions / PRD-04
Probability distributions
Uniform tolerance: variance
Variance of a value drawn uniformly between a and b, in squared measurement units.
- Ref
- PRD-04
- Rev
- v1
- Field
- Probability distributions
- Result
- Var[X] [mm²]
The variance (b − a)²/12 turns a tolerance band into the variance used in uncertainty budgets. Its square root, (b − a)/√12, is the standard uncertainty of a rectangular distribution: about 0.29 times the full width, or 0.58 times the half-width.
Variance of a value drawn uniformly between a and b, in squared measurement units. Its square root, (b − a)/√12, is the standard uncertainty of a rectangular distribution.
Calculator
- Standard uncertainty, (b − a)/√121.732 mm
- Standard uncertainty for half-width h, h/√31.732 mm
- Band width, b − a6 mm
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A band of 6 mm gives a variance of 3 mm² and a standard uncertainty of 1.732 mm.
- Doubling the band width quadruples the variance but only doubles the standard uncertainty.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (mm²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| a Lower limit a | −5 to 7.5 mm | 0.02083 | 14.08 | 14.06 | +0.25 mm | −8.16 % |
| b Upper limit b | 2.5 to 15 mm | 0.02083 | 14.08 | 14.06 | +10% | +28.4 % |
Sweep of Upper limit b
| Upper limit b (mm) | Variance (mm²) |
|---|---|
| 2.5 | 0.02083 |
| 4.063 | 0.3545 |
| 5.625 | 1.095 |
| 7.188 | 2.243 |
| 8.75 | 3.797 |
| 10.31 | 5.758 |
| 11.88 | 8.126 |
| 13.44 | 10.9 |
| 15 | 14.08 |
Sweep one variable
Variables and how each one contributes
Variance
What the equation gives you. Everything else feeds it.
Lower limit a
The lower limit of the band, in mm. Every value below it has zero probability. Moves the whole band; the width b − a is what sets the spread. Raising it by 10 % (to 2.2) moves the result by −6.56 %, from 3 to 2.803 mm².
Upper limit b
The upper limit of the band, in mm. It must be larger than a. Widening the band lowers the density and raises the variance. Raising it by 10 % (to 8.8) moves the result by +28.4 %, from 3 to 3.853 mm².
Worked example with the values above
- Band width: b − a = 8 − 2 = 6 mm.
- Square it: 6² = 36 mm².
- Divide by 12: Var = 3 mm².
Assumptions and limits
- Every value between a and b is equally likely, and nothing outside the band can occur.
- The limits are known exactly and a < b.
Common mistakes
- Using a uniform model when the values really bunch near the centre, which understates how often they are near nominal and overstates the extremes.
- Treating the density as a probability: a density of 0.17 per mm does not mean a 17 % chance.
- Forgetting that the variance has squared units, so (b − a)²/12 is in mm², not mm.
Statistics and uncertainty
A mean, variance or percentile is calculated from fitted parameters, and the uncertainty in those parameters passes straight through. A distribution fitted to a small sample can have parameters that are far from the truth, especially in the tails and for the variance, so quote the result together with the sample size it rests on.
Where it is used
- Standard uncertainty of a tolerance or resolution limit: u = (b − a)/(2√3) for a band of half-width (b − a)/2.
- Rounding and quantisation errors, which are close to uniform over one step.
Maths you need
- Probability density: area under the curve gives probability
- Cumulative probability: chance of being at or below a value