Library / Probability distributions / PRD-33
Probability distributions
Laplace error: two-sided tolerance exceedance
Probability that a Laplace error lies outside a symmetric tolerance of half-width t about the location, counting both tails together.
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- PRD-33
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- v1
- Field
- Probability distributions
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- P
For a symmetric tolerance of half-width t about the location, the probability of falling outside both limits together is e^(−t/s). Each tail carries half of it. The result depends only on the ratio t/s, so a tolerance of one scale length leaves 37 % outside.
Probability that a Laplace error lies outside a symmetric tolerance of half-width t about the location, counting both tails together.
Calculator
- Probability within the tolerance, 1 − e^(−t/s)0.4866
- Probability above the upper limit only0.2567
- Half-width for 1 % outside, s ln 10013.82 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 tolerance of ±2 mm leaves 51.3 % of readings outside, 25.7 % in each tail.
- Doubling the tolerance squares the exceedance probability, which is how fast exponential tails shrink.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| t Tolerance half-width t | 0 to 15 mm | 0.006738 | 1 | 0.9933 | +10% | −6.45 % |
| s Scale s | 0.5 to 10 mm | 0.01832 | 0.8187 | 0.8004 | +10% | +6.25 % |
Sweep of Tolerance half-width t
| Tolerance half-width t (mm) | Probability outside ±t |
|---|---|
| 0 | 1 |
| 1.875 | 0.5353 |
| 3.75 | 0.2865 |
| 5.625 | 0.1534 |
| 7.5 | 0.08208 |
| 9.375 | 0.04394 |
| 11.25 | 0.02352 |
| 13.13 | 0.01259 |
| 15 | 0.006738 |
Sweep one variable
Variables and how each one contributes
Probability outside ±t
What the equation gives you. Everything else feeds it.
Scale s
The scale, in mm. The mean absolute error equals s. Widens the distribution and lowers the peak. Raising it by 10 % (to 3.3) moves the result by +6.25 %, from 0.5134 to 0.5455.
Tolerance half-width t
The half-width of the symmetric tolerance, in mm. A wider tolerance lets fewer readings fall outside. Raising it by 10 % (to 2.2) moves the result by −6.45 %, from 0.5134 to 0.4803.
Worked example with the values above
- Ratio of tolerance to scale: t/s = 2 / 3 = 0.66667.
- Negate and exponentiate: e^(−0.66667).
- P = 0.51342.
Assumptions and limits
- Errors are independent, symmetric about m and have exponential tails.
- The scale is positive and comes from data for the same measurement process.
Common mistakes
- Using a normal model when outliers are more common than it predicts, which understates the tails.
- Mixing up the scale s with the standard deviation: the Laplace standard deviation is s√2.
- Reading the density at a point as a probability.
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
- Measurement errors and residuals with more large excursions than a normal curve gives.
- The noise model behind least-absolute-deviation (L1) fitting.
Maths you need
- Absolute value: the size of a number, ignoring its sign
- Exponential function: e raised to a power