Logistic Regression Calculator

Free logistic curve calculator. Fit S-curve growth y = L/(1+e^(-k(x-x0))) with step-by-step solutions, R², and graphs. Private browser-based.

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Logistic Regression Calculator

Free logistic curve calculator. Fit S-curve growth y = L/(1+e^(-k(x-x0))) with step-by-step solutions, R², and graphs. Private browser-based.

Enter your data points

#	X	Y

Carrying Capacity (L) Leave blank to auto-estimate

Results

Equation

Growth Rate (k)

Midpoint (x₀)

Carrying Capacity (L)

R²

Predicted Y

For X = —

Statistics

Statistic	Value
Standard Error
Sample Size (n)
Degrees of Freedom

Chart

Step-by-Step Solution

Jak tego używać Logistic Regression Calculator

Wzrost i zanik

Model S-curve growth with carrying capacity L, growth rate k, and midpoint x₀.

Dane wejściowe

Enter paired X and Y values. Optionally set carrying capacity L or let the calculator auto-estimate it.

Wyniki statystyczne

Get growth rate k, midpoint x₀, carrying capacity L, R², and prediction diagnostics.

Logistic curve fitting requires 0 < Y < L. L is auto-estimated from max(Y) × 1.05 but can be overridden.

What Is Logistic Curve Fitting?

📐 Logistic curve fitting models growth processes that start exponentially but eventually slow down as they approach a maximum limit called the carrying capacity (L). The classic formula is y = L / (1 + e^{−k(x − x₀)}), where k controls the steepness of the curve and x₀ is the midpoint (the x-value where y reaches L/2).

📊 Real-world examples include: (1) Population biology — bacterial colonies that slow as nutrients run out, (2) Marketing — product adoption that saturates as market share fills, (3) Epidemiology — infection curves that flatten as herd immunity builds, and (4) Machine learning — training accuracy that plateaus as the model converges.

📐 Logistic curves are S-shaped (sigmoid). They are fundamentally different from exponential curves because they have an upper bound. This makes them essential whenever unlimited growth is physically impossible.

How Logistic Curve Fitting Works

1
Determine L: Identify the carrying capacity — the maximum possible Y value. The calculator auto-estimates L as max(Y) × 1.05, but you can override it.
2
Linearize via logit: Transform each data point using z = ln(y / (L − y)). This converts the logistic model into a linear equation z = −k·x + k·x₀.
3
Perform linear regression: Apply ordinary least squares to (x, z). The slope equals −k and the intercept equals k·x₀.
4
Recover parameters: Compute k = −slope and x₀ = intercept / k. The final equation is y = L / (1 + e^{−k(x − x₀)}).
5
Assess fit on original scale: Compute R² using the original (non-transformed) data to verify the S-curve actually fits the observations.

When to Use Logistic Curve Fitting

Growth has a natural upper limit or saturation point
Early data shows exponential-like growth that eventually slows
You need to estimate carrying capacity, midpoint, or growth rate
Modeling population dynamics, product adoption, or learning curves

When to Avoid Logistic Curve Fitting

When Y values can exceed any reasonable upper bound
When growth is strictly exponential without any slowing
When data is monotonically decreasing without an asymptote
When you have fewer than 5 data points (unreliable parameter estimates)

Logistic Regression Calculator