Find the constant of proportionality k from data pairs. Supports direct, inverse, squared, square root, and custom power models with R², residuals, and fit analysis.
The constant of proportionality (k) defines the scaling factor in a proportional relationship: y = kx for direct proportion, y = k/x for inverse proportion, or y = kxⁿ for power-law relationships. Finding k from data is fundamental in physics (F = kx for springs), chemistry (reaction rates), economics (supply/demand), and engineering.
This calculator fits the best proportionality constant from your data using least-squares regression through the origin. It supports five models: direct (y = kx), inverse (y = k/x), squared (y = kx²), square root (y = k√x), and custom power (y = kxⁿ). For each fit, it computes R², RMSE, individual k values, and residuals.
Enter paired x,y data and select the proportionality model. The calculator determines the best k, shows how well the model fits, and displays predicted vs actual comparisons. Use the preset examples to load common values instantly, or type in custom inputs to see results in real time. The output updates as you type, making it practical to compare different scenarios without resetting the page.
Determining proportionality constants manually requires careful computation, especially for non-linear models. This calculator handles the regression mathematics automatically, computes goodness-of-fit statistics, and shows residual analysis to help you evaluate whether your chosen model is appropriate. The visual comparison makes it easy to spot systematic deviations. This tool is designed for quick, accurate results without manual computation. Whether you are a student working through coursework, a professional verifying a result, or an educator preparing examples, accurate answers are always just a few keystrokes away.
Best-fit k (through origin): k = Σ(x^n × y) / Σ(x^n)² Individual k values: kᵢ = yᵢ / xᵢⁿ R² (coefficient of determination): R² = 1 − SS_res / SS_tot SS_res = Σ(yᵢ − k×xᵢⁿ)² SS_tot = Σ(yᵢ − ȳ)² Models: Direct: y = kx (n=1) Inverse: y = k/x (n=−1) Squared: y = kx² (n=2) Square root: y = k√x (n=0.5)
Result: k = 2.0218, R² = 0.9994
The data closely follows y ≈ 2.02x. R² = 0.9994 indicates an excellent fit. Individual k values range from 1.97 to 2.04, with minimal variation (std dev = 0.029), confirming a strong direct proportional relationship.
In direct proportionality (y = kx), as x doubles, y doubles. The graph is a straight line through the origin. In inverse proportionality (y = k/x), as x doubles, y halves. The graph is a hyperbola. The product xy remains constant at k. Power-law models (y = kxⁿ) generalize these: n = 1 is direct, n = −1 is inverse, and fractional or larger exponents describe other relationships.
The R² value alone doesn't tell the full story. A high R² with patterned residuals indicates a misspecified model. Always examine residuals: they should scatter randomly around zero. Systematic curvature in residuals means a different power or model form would fit better. The individual k values should also be reasonably consistent.
In physics, proportionality constants define fundamental relationships (spring constants, resistivity, thermal conductivity). In chemistry, rate constants determine reaction speed. In economics, elasticity measures proportional responsiveness. In data science, identifying proportionality helps simplify models and extract meaningful parameters from raw data.
It's the multiplier k in a proportional relationship. In y = kx, k is the ratio y/x that stays constant. It tells you how much y changes per unit change in x (for direct proportion) or the product xy (for inverse proportion). Understanding this concept helps you apply the calculator correctly and interpret the results with confidence.
The calculator uses least-squares regression through the origin, minimizing the sum of squared residuals. This gives k = Σ(xⁿ × y) / Σ(xⁿ)², which is the optimal k for minimizing prediction error.
R² measures how well the model explains the data variability. R² = 1.0 means perfect fit, R² = 0.95 means 95% of variability is explained. Below 0.9, the model may not capture the true relationship.
Best-fit k comes from regression (minimizing total squared error). Average k is the mean of individual kᵢ = yᵢ/xᵢⁿ values. They're similar when the model fits well; they diverge when data is noisy or the model is inappropriate.
This calculator assumes y passes through the origin (no intercept). If your data has a non-zero y-intercept, the proportionality model is inappropriate — use linear regression instead.
Hooke's law spring constant (F = kx), gravitational constant G, Coulomb's. Understanding this concept helps you apply the calculator correctly and interpret the results with confidence. constant, tax rate (tax = rate × income), speed (distance = speed × time), and many engineering coefficients.