Library / Probability distributions / PRD-01
Probability distributions
Uniform tolerance: probability density
Height of the flat probability density when every value between a lower limit a and an upper limit b is equally likely.
- Ref
- PRD-01
- Rev
- v1
- Field
- Probability distributions
- Result
- f [1/mm]
The density is the constant 1/(b − a) between the limits, so that height times width equals one. It is a probability per unit of x (here per mm), not a probability: multiply it by a small interval to get one. Outside the band the density is zero.
Height of the flat probability density when every value between a lower limit a and an upper limit b is equally likely. The density is zero outside the band, and the area under it is one.
Calculator
- Band width, b − a6 mm
- Standard deviation, (b − a)/√121.732 mm
- Probability of a value within a 1 mm slice inside the band0.1667
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 from 2 to 8 mm has a flat density of 0.1667 per mm, so any 1 mm slice inside it holds 16.7 % of the probability.
- Halving the band width doubles the density: the same unit of probability is squeezed into a narrower band.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (1/mm) | Local step | Change in result |
|---|---|---|---|---|---|---|
| a Lower limit a | −5 to 7.5 mm | 0.07692 | 2 | 1.923 | +0.25 mm | +4.35 % |
| b Upper limit b | 2.5 to 15 mm | 0.07692 | 2 | 1.923 | +10% | −11.8 % |
Sweep of Upper limit b
| Upper limit b (mm) | Probability density inside the band (1/mm) |
|---|---|
| 2.5 | 2 |
| 4.063 | 0.4848 |
| 5.625 | 0.2759 |
| 7.188 | 0.1928 |
| 8.75 | 0.1481 |
| 10.31 | 0.1203 |
| 11.88 | 0.1013 |
| 13.44 | 0.08743 |
| 15 | 0.07692 |
Sweep one variable
Variables and how each one contributes
Probability density inside the band
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 +3.45 %, from 0.1667 to 0.1724 1/mm.
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 −11.8 %, from 0.1667 to 0.1471 1/mm.
Worked example with the values above
- Band width: b − a = 8 − 2 = 6 mm.
- Take the reciprocal: 1/6.
- f = 0.16667 per mm everywhere inside the band.
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
The inputs are parameters, usually fitted to data, and their estimation error carries into the result. The probability or density is a smooth function of the parameters, so the sweep chart shows how sensitive it is: where the curve is steep, a small error in a parameter makes a large difference. The formula treats the parameters as exact, so it gives no confidence interval on its own.
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