Logistic Regression, Sigmoid Probability & Odds Ratio Calculator

Statistical Theory of Logistic Regression & Binary Classification

While linear regression assumes continuous, unbounded dependent variables with normally distributed errors, binary classification problems (disease vs. healthy, default vs. non-default, click vs. no-click) require bounding predicted values strictly between 0 and 1:

• The Logit Transform & Sigmoid Link: Logistic regression maps the infinite real line \ of linear predictors \ to the probability domain [0, 1] using the standard logistic (sigmoid) function σ(z) = (1) / (1 + e^-z).

• Log-Odds and the Linear Scale: The natural logarithm of the odds ((P) / (1-P)) is completely linear with respect to X, allowing estimation via Maximum Likelihood Estimation (MLE).

• Interpreting Odds Ratios (e^β_1):

• OR > 1.0: Each unit increase in X multiplies the odds of the positive class.

• The predictor has zero effect on outcome odds.

• OR < 1.0: Each unit increase in X reduces outcome odds (protective factor).

• Classification Threshold: Standard classification assigns Class 1 when P 0.50 (where z 0). In high-cost false negative scenarios (medical diagnostics, fraud detection), the decision threshold is often lowered to 0.20 or 0.30 to optimize sensitivity / recall.

How to Use This Calculator

Enter your Intercept Coefficient (β₀ / Constant) (e.g. -3.5 (Log-odds baseline when X=0)), Slope Coefficient (β₁ / Log-Odds per Unit of X) (e.g. 0.8 (Positive = higher probability as X increases)), Target Predictor Value (X) (e.g. 5.0), and Binary Classification Decision Threshold (e.g. 0.50 (Standard 50% cutoff for Class 1 vs Class 0)). Results update instantly as you type — adjust any value to compare scenarios.

The Formula

Logistic Regression, Logit Link & Sigmoid equations:

Linear Predictor (Logit): z = β_0 + β_1 X

Logit Link Function: logit(P) = ( (P) / (1 - P) ) = z

Sigmoid Probability: P(Y = 1) = σ(z) = (1) / (1 + e^-z) = (e^z) / (1 + e^z)

Event Odds: Odds = (P) / (1 - P) = e^z = e^β_0 + β_1 X

Odds Ratio (OR per 1-unit Δ X = 1): OR = Odds_x+1Odds_x = e^β_1

Decision Threshold Rule: Y = cases 1 & if P(Y=1) p_threshold \\ 0 & if P(Y=1) < p_threshold cases

Worked Example

For example, evaluating a credit default logistic model with intercept \ and slope \ for a customer with risk score The logit is \ = +0.

• The predicted default probability is \ = (1) / (1 + e^-0.50) = (1) / (1 + 0.6065) = 0.6225 (62.25\%). Event odds are . The Odds Ratio per 1-unit score increase is \ (a 122.5% increase in default odds).

Understanding Your Result

Statistical results require context: a mean alone is misleading without knowing the standard deviation (spread). Normal distribution reference: 68% of values fall within ±1 SD of the mean, 95% within ±2 SD, 99.7% within ±3 SD. For small sample sizes (n < 30), use t-distributions rather than z-scores. Statistical significance (p < 0.05) does not imply practical significance — effect size (Cohen's d, R², odds ratio) measures real-world importance. Correlation coefficient r: |r| < 0.3 weak, 0.3–0.7 moderate, > 0.7 strong.