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03 // Bidirectional STEM Calculators // STATISTICSOR = (p₁ / (1 - p₁)) / (p₂ / (1 - p₂))

Odds Ratio (OR) & Group Probability Calculator

The Odds Ratio (OR) compares the odds of an outcome occurring in an exposed or treatment group p₁/(1 - p₁) relative to a control baseline group p₂/(1 - p₂).

OR = (p₁ / (1 - p₁)) / (p₂ / (1 - p₂))
Pick which value you want to calculate (or type into any field):Solving for: OR — Odds Ratio
Calculated

Ratio of exposed odds to control odds (1.0 = no association)

0.01100
unit

Event probability in group 1 (0.01 to 0.99)

0.0010.999
unit

Baseline event probability in group 2 (0.01 to 0.99)

0.0010.999
unit
Calculated Result · Odds Ratio (OR)
2.25

Step-by-Step Calculation

01Canonical Law: OR = (p₁ / (1 - p₁)) / (p₂ / (1 - p₂))
02Given Inputs (Base Units): p₁ = 0.36, p₂ = 0.2
03Solve for OR (Odds Ratio): OR = 2.25

The Odds Ratio (OR) compares the odds of an outcome occurring in an exposed or treatment group p₁/(1 - p₁) relative to a control baseline group p₂/(1 - p₂).

Real-World Reference Values

Null Association BaselineOR = 1.00 (p₁ = p₂)

Exposure has zero effect on outcome odds

Logistic Regression Coefficient (β)β = ln(OR) | OR = e^β

At OR = 2.25, log-odds coefficient β = +0.8109

Relative Risk (RR = p₁ / p₂)At p₁=0.36, p₂=0.20 → RR = 1.80

OR approximates RR closely when baseline p₂ < 0.05

20-Row Comparison Table

Odds Ratio (OR) · Exposed / Group 1 Probability (p₁)

Odds Ratio (OR) & Group Probability Calculator — Exposed / Group 1 Probability → Odds Ratio
#Exposed / Group 1 Probability (p₁)Odds Ratio (OR)Δ %
010.001 0.004004 -99.82%
020.048842 0.205401 -90.87%
030.096684 0.42813 -80.97%
040.144526 0.675772 -69.97%
050.192368 0.952753 -57.66%
060.240211 1.2646 -43.79%
070.288053 1.6184 -28.07%
080.335895 2.0231 -10.08%
090.36 Current2.25 0.00%
100.431579 3.037 +34.98%
110.479421 3.6838 +63.72%
120.527263 4.4614 +98.28%
130.575105 5.4141 +140.63%
140.622947 6.6086 +193.72%
150.670789 8.1503 +262.23%
160.718632 10.2162 +354.05%
170.766474 13.1287 +483.50%
180.814316 17.542 +679.64%
190.862158 25.0187 +1011.94%
200.91 40.4444 +1697.53%
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Frequently Asked Questions

How do I convert a logistic regression odds ratio into an actual probability p₁?+

Click SOLVE FOR on p₁, enter your model's Odds Ratio OR, and enter the baseline control probability p₂.

What is the difference between Odds Ratio (OR) and Relative Risk (RR)?+

Relative Risk divides probabilities (p₁ / p₂), whereas Odds Ratio divides odds ((p₁/(1-p₁)) / (p₂/(1-p₂))). When outcomes are common, OR is further from 1.0 than RR.

What does an Odds Ratio below 1.0 indicate?+

An OR < 1.0 indicates a protective effect where the event is less likely in Group 1 than in Group 2.