Library / Probability distributions / PRD-02
Probability distributions
Uniform tolerance: probability below a limit
Probability that a value drawn uniformly between a and b lies at or below a test value x.
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- PRD-02
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- v1
- Field
- Probability distributions
- Result
- F
F rises in a straight line from 0 at a to 1 at b, so it is just the fraction of the band that lies below x. It is clipped to 0 below a and 1 above b. Use it to estimate the share of parts expected to be at or under a limit when the spread is flat.
Probability that a value drawn uniformly between a and b lies at or below a test value x. It is zero below a, one above b, and rises in a straight line between.
Calculator
- Fraction above x, 1 − F0.6667
- Distance of x from the lower limit2 mm
- Value with F = 0.95, a + 0.95(b − a)7.7 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 test value of 4 mm sits 33.3 % of the way along a band from 2 to 8 mm, so about that share of items lie at or below it.
- The relationship is linear, so each extra millimetre of x adds the same 16.7 % until x reaches b.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| x Test value x | 0 to 10 mm | 0 | 1 | 1 | +10% | +20 % |
| b Upper limit b | 2.5 to 15 mm | 0.1538 | 1 | 0.8462 | +10% | −11.8 % |
| a Lower limit a | −5 to 7.5 mm | 0 | 0.6923 | 0.6923 | +0.25 mm | −8.7 % |
Sweep of Test value x
| Test value x (mm) | Probability at or below x |
|---|---|
| 0 | 0 |
| 1.25 | 0 |
| 2.5 | 0.08333 |
| 3.75 | 0.2917 |
| 5 | 0.5 |
| 6.25 | 0.7083 |
| 7.5 | 0.9167 |
| 8.75 | 1 |
| 10 | 1 |
Sweep one variable
Variables and how each one contributes
Probability at or below x
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 −6.9 %, from 0.3333 to 0.3103.
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 0.3333 to 0.2941.
Test value x
The test value, in mm: the limit you want the probability below. Moves the test point along the band. Raising it by 10 % (to 4.4) moves the result by +20 %, from 0.3333 to 0.4.
Worked example with the values above
- Distance above the lower limit: x − a = 4 − 2 = 2 mm.
- Band width: b − a = 6 mm.
- F = 2 / 6 = 0.33333.
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