Library / Probability distributions / PRD-50
Probability distributions
Batch variation: variance of the successes
Variance of the success count, which includes the extra variation caused by the success probability changing between batches.
- Ref
- PRD-50
- Rev
- v1
- Field
- Probability distributions
- Result
- Var[X]
The variance n a b (a + b + n)/[(a + b)²(a + b + 1)] includes the extra variation caused by the success probability changing from batch to batch. It exceeds the plain binomial variance by the factor (a + b + n)/(a + b + 1), which is greater than 1 whenever n > 1.
Variance of the success count, which includes the extra variation caused by the success probability changing between batches. It exceeds the plain binomial variance.
Calculator
- Plain binomial variance at the same mean p2.4
- Overdispersion factor, (a + b + n)/(a + b + 1)2.5
- Standard deviation of the count2.449
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.
- The variance of 6 is 2.5 times the plain binomial variance of 2.4.
- The smaller a + b is, the more batches differ from each other and the larger the extra variance.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| n Trials per batch n | 1 to 40 | 0.24 | 72 | 71.76 | +1 | +17.3 % |
| b Shape b | 0.2 to 10 | 2.35 | 7.359 | 5.008 | +10% | −4.9 % |
| a Shape a | 0.2 to 10 | 1.842 | 6 | 4.158 | +10% | −0.267 % |
Sweep of Trials per batch n
| Trials per batch n | Variance of the successes |
|---|---|
| 1 | 0.24 |
| 5.875 | 2.556 |
| 10.75 | 6.772 |
| 15.63 | 12.89 |
| 20.5 | 20.91 |
| 25.38 | 30.83 |
| 30.25 | 42.65 |
| 35.13 | 56.38 |
| 40 | 72 |
Sweep one variable
Variables and how each one contributes
Variance of the successes
What the equation gives you. Everything else feeds it.
Trials per batch n
The number of trials in a batch (a whole number, at least 1). More trials per batch raise the mean count and the variance. Raising it by 1 (to 11) moves the result by +17.3 %, from 6 to 7.04.
Shape a
The first shape of the beta distribution of batch probabilities (a plain number). Raises the average success probability and narrows its spread. Raising it by 10 % (to 2.2) moves the result by −0.267 %, from 6 to 5.984.
Shape b
The second shape of the beta distribution of batch probabilities (a plain number). Lowers the average success probability and narrows its spread. Raising it by 10 % (to 3.3) moves the result by −4.9 %, from 6 to 5.706.
Worked example with the values above
- Numerator: n a b (a + b + n) = 900.
- Denominator: (a + b)²(a + b + 1) = 150.
- Var = 6.
Assumptions and limits
- Within a batch the trials are independent with a common probability p; across batches p follows a beta distribution with shapes a, b > 0.
- n and x are whole numbers, and 0 ≤ x ≤ n.
Common mistakes
- Using a plain binomial when batches differ in quality: it understates the variance and the chance of extreme batches.
- Treating a and b as probabilities: they are shape parameters, and a/(a + b) is the mean probability.
- Using the formula for x outside 0 to n: the probability is zero there.
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
- Defect counts in batches of parts when batch quality varies.
- Test pass counts across lots, and any count with more spread than a binomial.
Maths you need
- Binomial distribution: successes in independent trials
- Beta distribution: a model for a probability that varies