Logistic Regression Probability Curve Visualizer

You trained a logistic model to predict whether a lead will convert. The output for a specific lead is 0.73. That number is not a score or a rank — it is a probability estimated by a sigmoid function that squashes the linear predictor into the 0–1 range. A logistic regression probability curve visualiser plots that S-shaped mapping across the full input range so you can see exactly where the transition from “almost certainly no” to “almost certainly yes” happens, and how steeply it occurs.

The mistake that confuses most first-time users: reading the linear predictor (log-odds) as though it were the probability. A log-odds value of 2.0 does not mean “200% chance” — it maps to a probability of about 0.88 through the sigmoid. The curve visualiser makes this distinction concrete by showing both the straight line (log-odds) and the S-curve (probability) side by side.

The probability curve does not classify anything on its own — you need a threshold to convert probabilities into yes/no decisions. Draw a horizontal line at threshold = 0.5 across the S-curve. Every data point whose predicted probability falls above that line is classified positive; every point below is negative. The resulting counts of correct and incorrect decisions fill the four cells of the confusion matrix: TP, FP, TN, FN.

Sliding the threshold line up or down shifts where the S-curve crosses it, which changes the x-value boundary between the two classes. Move the threshold from 0.5 to 0.3 and you classify more observations as positive — TP goes up, but so does FP. Move it to 0.7 and FP drops, but FN rises because you are now demanding higher confidence before predicting positive. The curve visualiser lets you see this mechanically: the threshold line intersects the S-curve at a single x-value, and everything on one side gets one label.

This connection matters because stakeholders often ask “why did the model miss that case?” The answer is usually that the predicted probability was just below the threshold — visible on the curve as a point sitting barely under the horizontal line. Adjusting the threshold by a small amount would have caught it, at the cost of more false positives elsewhere.

The default threshold of 0.5 treats false positives and false negatives as equally costly. In most real problems they are not. A disease screening tool should minimise missed cases (high recall), even if that means more healthy patients get follow-up tests (lower precision). A fraud alert system that pages an analyst at 2 a.m. should avoid false alarms (high precision), accepting that some low-confidence fraud slips through (lower recall).

To find the right threshold, compute precision and recall at several candidate values — 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8 — and build a table. Each row is a different confusion matrix. The operating point that best matches your cost structure is the one to deploy. There is no formula for this choice; it requires knowing the dollar cost (or human cost) of each error type.

A common trap: optimising the threshold on the training set. The sigmoid curve fit to training data is slightly overconfident, so a threshold tuned there will underperform on new data. Always tune on a held-out validation set, and report final metrics on a separate test set that was never used for any decision.

The logistic model operates in three linked scales. The linear predictor z = β₀ + β₁x lives on the real line (−∞ to +∞). Exponentiate it and you get odds: eᵖ = P/(1−P), always positive. Apply the sigmoid and you get probability: P = 1/(1+e⁻ᵖ), bounded 0–1. Each scale answers a different question.

Odds ratios are how researchers report logistic regression results in journals: “each additional year of experience multiplies the odds of promotion by 1.15” means eᵝ₁ = 1.15. But decision-makers think in probabilities: “this candidate has a 68% chance of promotion.” The curve visualiser bridges the gap — you set the coefficients, and it shows the probability at any x-value, while the underlying odds ratio explains how much each unit of x shifts the odds.

One subtlety: the odds ratio is constant across all x-values (each unit increase in x multiplies odds by the same factor), but the probability change per unit of x is not constant. Near the midpoint of the S-curve, a 1-unit increase in x produces the largest probability jump. Near the tails, the same 1-unit increase barely moves the probability because the sigmoid flattens. This is why reporting “a 5-percentage-point increase in probability” requires specifying the baseline — the same coefficient produces different probability shifts at different starting points.

The slope β₁ is 0.5. Does that mean a 50% increase in probability per unit of x?
No. β₁ is in log-odds units, not probability units. A slope of 0.5 means each unit of x adds 0.5 to the log-odds — the equivalent probability change depends on where you are on the curve. At the midpoint it is roughly β₁/4 ≈ 12.5 percentage points; near the tails it is much smaller.

I set a negative intercept and the curve starts below 0.5. Is the model biassed?
Not necessarily. A negative intercept (β₀ < 0) means the baseline log-odds at x = 0 are negative, so the probability at x = 0 is below 50%. This is expected when the positive class is uncommon at the reference point. The midpoint of the curve simply shifts to x = −β₀/β₁, which may be far from zero.

Two models have different curves but similar AUC. Which is better?
AUC measures overall ranking ability, not calibration. One model might assign well-separated probabilities (steep curve) while the other assigns probabilities bunched near 0.5 (flat curve). If you need well-calibrated probabilities for risk scoring, the steeper curve is more useful even if AUC is identical. Check calibration plots alongside AUC.

Can I use this single-feature curve for a multi-feature model?
Only as a partial-effect visualisation. In a multi-feature model, the curve for one feature holds all other features at fixed values (often their means). Changing those held-out values shifts and reshapes the curve. The visualiser shows one slice of a higher-dimensional surface, not the full model.

Three equations define the logistic model:

Units note: β₀ and β₁ are in log-odds units. x can be in any measurement unit — the sigmoid converts everything to a dimensionless probability. The maximum slope of the probability curve occurs at the midpoint and equals β₁/4.

Scenario: A SaaS company models trial-to-paid conversion using a single feature: number of actions taken during the 14-day trial. The fitted logistic model has β₀ = −3.0 and β₁ = 0.12. Decision threshold is 0.5.

Step 1 — Midpoint.
Midpoint = −(−3.0) / 0.12 = 25 actions. At 25 actions the predicted conversion probability is exactly 50%. Below 25 the model predicts “will not convert”; above 25 it predicts “will convert.”

Step 2 — Probability at key points.
At 10 actions: z = −3.0 + 0.12×10 = −1.8, P = 1/(1+e¹⋅⁸) ≈ 0.14 (14%). At 40 actions: z = −3.0 + 0.12×40 = 1.8, P ≈ 0.86 (86%). The curve rises steeply between roughly 15 and 35 actions and flattens outside that range.

Step 3 — Odds ratio interpretation.
eᵝ₁ = e⁰⋅¹² ≈ 1.127. Each additional trial action multiplies the odds of conversion by about 1.13 — a 13% increase in odds per action, regardless of starting point. But the probability increase per action is largest near 25 actions (about 0.12/4 = 3 percentage points) and smaller near the extremes.

Step 4 — Business use.
The sales team targets leads with 15–25 actions — the steepest part of the curve where a nudge (webinar invite, feature walkthrough) could push the probability above the threshold. Leads below 10 actions are too cold; leads above 35 are likely converting on their own.