Library / Probability distributions / PRD-32
Probability distributions
Laplace error: variance
Variance of a Laplace error, in squared measurement units.
- Ref
- PRD-32
- Rev
- v1
- Field
- Probability distributions
- Result
- Var[X] [mm²]
The variance 2s² converts the scale into squared measurement units. The standard deviation is s√2, about 1.41 s, so a Laplace distribution with the same scale as a normal one's standard deviation is wider.
Variance of a Laplace error, in squared measurement units. It is twice the square of the scale.
Calculator
- Standard deviation, s√24.243 mm
- Mean absolute deviation, s3 mm
- Interquartile range, 2s ln 24.159 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 scale of 3 mm gives a standard deviation of 4.243 mm.
- The variance rises with the square of the scale: doubling s quadruples the variance.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (mm²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| s Scale s | 0.5 to 10 mm | 0.5 | 200 | 199.5 | +10% | +21 % |
Sweep of Scale s
| Scale s (mm) | Variance (mm²) |
|---|---|
| 0.5 | 0.5 |
| 1.688 | 5.695 |
| 2.875 | 16.53 |
| 4.063 | 33.01 |
| 5.25 | 55.13 |
| 6.438 | 82.88 |
| 7.625 | 116.3 |
| 8.813 | 155.3 |
| 10 | 200 |
Sweep one variable
Variables and how each one contributes
Variance
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 +21 %, from 18 to 21.78 mm².
Worked example with the values above
- Square the scale: s² = 9 mm².
- Double it.
- Var = 18 mm².
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
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
- 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