Calculator Inputs
Choose a model for the quality characteristic X, then set target, interval, and specification limits.
Example Data Table
Use these sample scenarios to test the calculator structure and export workflow.
| Scenario | Distribution | Parameters | Target x | LSL | Specification window | Estimated yield |
|---|---|---|---|---|---|---|
| Normal shaft diameter | Normal | μ = 50, σ = 0.8 | 49.0 | 51.0 | 48.5 to 51.5 | 99.38% |
| Incoming defect count | Binomial | n = 25, p = 0.04 | 1 | 0 | 0 to 2 | 98.15% |
| Scratches per panel | Poisson | λ = 2.2 | 3 | 1 | 0 to 4 | 92.65% |
| Failure waiting time | Exponential | λ = 0.35 | 2.0 | 1.0 | 0.5 to 4.0 | 75.34% |
| Uniform fill range | Uniform | a = 98, b = 102 | 100 | 99 | 99 to 101 | 50.00% |
Formula Used
This calculator estimates the behavior of a quality characteristic X under an assumed distribution.
Normal: E(X) = μ, Var(X) = σ², z = (x - μ) / σ, and P(a ≤ X ≤ b) = F(b) - F(a).
Binomial: E(X) = np, Var(X) = np(1 - p), and P(X = k) = C(n,k)pk(1 - p)n-k.
Poisson: E(X) = λ, Var(X) = λ, and P(X = k) = e-λ λk / k!.
Exponential: E(X) = 1 / λ, Var(X) = 1 / λ², and F(x) = 1 - e-λx for x ≥ 0.
Uniform: E(X) = (a + b) / 2, Var(X) = (b - a)² / 12, and f(x) = 1 / (b - a) inside the range.
Yield: Estimated yield = P(LSL ≤ X ≤ USL). Defect rate = 1 - yield. Expected defective units = sample size × defect rate.
Capability for normal data: Cp = (USL - LSL) / 6σ and Cpk = minimum of centered one-sided capability values.
How to Use This Calculator
1. Select a distribution that matches the process behavior you want to assume.
2. Enter the main parameters for that distribution, such as mean and standard deviation or count rate.
3. Add a target x value to inspect the cumulative probability and point density or point probability.
4. Enter an interval to estimate the chance that X falls within a practical operating window.
5. Add lower and upper specification limits to estimate yield, defect rate, and expected defective units.
6. Review the result table, the summary cards, and the Plotly chart to understand the assumption visually.
7. Export the result as CSV for data review or PDF for reporting and documentation.
FAQs
1. What does “assume the random variable X” mean here?
It means you choose a probability model for the quality characteristic X. The calculator then estimates probabilities, yield, variation, and defect exposure from that assumption.
2. When should I use the normal option?
Use it for measurements that cluster around a center with symmetric spread. Examples include dimensions, weights, thickness, or fill levels after stable process control.
3. When are binomial and Poisson more suitable?
Binomial fits fixed trial counts, such as pass or fail checks. Poisson fits counts per area, time, length, or unit when events happen independently.
4. Why is the point value not always a probability?
Continuous models return density at x, not direct probability. Real probability comes from an interval. Discrete models return a true point probability for each count.
5. What do yield and defect rate show?
Yield estimates the share of outcomes inside the specification window. Defect rate is everything outside that window. Together they help quantify process risk under the chosen model.
6. Why are Cp and Cpk shown only for normal data?
Those capability indices rely on the normal spread concept and standard deviation relationship. They are most interpretable when the process behaves approximately normally.
7. Can I use this for one-sided specifications?
Yes. Leave one specification limit blank. The calculator will estimate yield against the remaining side and treat the missing side as unbounded.
8. How should I validate the assumption before decisions?
Compare the assumption with real sample data, charts, and engineering knowledge. This tool supports planning, but actual release decisions should use verified process evidence.