Calculator Inputs
Example Data Table
This sample shows a case where Distribution A has more mass on larger outcomes, so first-order dominance is expected.
| Support Value | Distribution A Probability | Distribution B Probability |
|---|---|---|
| 0 | 0.05 | 0.15 |
| 1 | 0.10 | 0.20 |
| 2 | 0.20 | 0.20 |
| 3 | 0.25 | 0.20 |
| 4 | 0.20 | 0.15 |
| 5 | 0.20 | 0.10 |
Formula Used
The calculator assumes two discrete distributions defined on the same ordered support values.
Step 1: Cumulative distribution for each support point
F_A(x_k) = Σ p_A(x_i) for all x_i ≤ x_k
F_B(x_k) = Σ p_B(x_i) for all x_i ≤ x_k
Step 2: First-order stochastic dominance
A dominates B in first order when F_A(x_k) ≤ F_B(x_k) for every tested support point, with strict inequality at least once.
Step 3: Second-order stochastic dominance
A dominates B in second order when the cumulative area H(x_k) = ∫[−∞ to x_k] (F_A(t) − F_B(t)) dt never becomes positive, and is strictly negative at least once.
Discrete area update used here
H(x_k) = H(x_(k−1)) + (F_A(x_(k−1)) − F_B(x_(k−1))) × (x_k − x_(k−1))
How to Use This Calculator
- Enter the ordered support values in one list.
- Enter probabilities for Distribution A and Distribution B in matching positions.
- Choose labels, tolerance, decimal precision, and the desired analysis mode.
- Keep normalization checked if your probability totals are approximate or unscaled.
- Click Run Dominance Test to display the decision, metrics, chart, and detailed table.
- Use the CSV or PDF buttons to save the current output.
Frequently Asked Questions
1. What does first-order stochastic dominance mean?
It means one distribution gives outcomes that are never worse in cumulative probability terms and are strictly better at least once. Decision-makers who prefer more to less will generally prefer the dominating distribution.
2. What does second-order stochastic dominance mean?
It means the cumulative area under one CDF stays weakly below the other. This is useful when first-order dominance fails but one option is still preferred by risk-averse decision-makers.
3. Do the support values need to match for both distributions?
Yes. This page expects one common support list for both probability series. If your source data differ, first map them onto a shared ordered support before testing.
4. Why does the calculator offer normalization?
Normalization rescales each probability list so it sums to one. It helps when you paste rounded or weighted values that are proportionally correct but do not total exactly one.
5. What does the tolerance setting control?
Tolerance reduces false failures caused by tiny rounding differences. A small positive tolerance treats near-equal values as practically equal during the dominance checks.
6. Can I use this for continuous distributions?
Yes, approximately. First discretize the continuous distribution into support points with associated probabilities. Then test dominance on that discrete representation.
7. What does the integrated difference column show?
It shows the running cumulative area between the two CDFs. That quantity is the key object used for checking second-order dominance in the discrete setting.
8. Why might neither distribution dominate the other?
Dominance fails when the cumulative curves cross or when the cumulative area rule breaks. In those cases, preference depends on stronger assumptions or different decision criteria.