What does this calculator do?

Compute statistical results from your inputs.

Enter values in the input fields and view results.

What are the formulas?

See the educational content section for formulas.

For statistical analysis and computation.

Check the related calculators section below.

Results follow standard statistical methods.

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STATISTICSDescriptive StatisticsStatistics Calculator

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Constant of Proportionality Calculator

Free constant of proportionality calculator. Find k in y=kx or y=k/x from data pairs. Direct, invers

Run CalculatorExplore data analysis and statistical calculations

Why This Statistical Analysis Matters

Why: Statistical calculator for analysis.

How: Enter inputs and compute results.

STATISTICSDescriptive Statistics

Constant of Proportionality — y = kx

Find k from data. Direct, inverse, and power proportion. R² goodness of fit, scatter plot with fit line, k consistency chart.

Linear Regression →Correlation →

Real-World Scenarios — Click to Load

Relationship Type

Data Pairs (x, y)

x	y

prop_results.sh

CALCULATED

$ prop_fit --type="direct" --pairs=5

Equation

y = 2.0000x

Constant k

2.000000

k (avg y/x)

2.000000

R²

1.0000

Constant of Proportionality

y = 2.0000x

k = 2.0000

R² = 1.0000

numbervibe.com/calculators/statistics/constant-of-proportionality-calculator

y vs x with Fitted Line

k Consistency Across Data Points

Points show k computed per data pair; dashed line = mean k. Flat line indicates a strong proportional relationship.

Calculation Breakdown

COMPUTATION

k (least-squares)

2.000000

k = Σ(xy)/Σ(x²) = 110.00/55.00

k (avg y/x)

2.000000

k_avg = Σ(y/x)/n = 10.0000/5

SS_res

0.0000

\text{Sigma} (yᵢ - ŷᵢ)^{2}

SS_tot

40.0000

\text{Sigma} (yᵢ - ȳ)^{2}

R²

1.0000

1 - SS_res/SS_tot

For educational and informational purposes only. Verify with a qualified professional.

Key Takeaways

• Direct proportion: y = kx — doubling x doubles y. The ratio y/x is constant.
• Inverse proportion: y = k/x — doubling x halves y. The product xy is constant.
• Power proportion: y = kx^n — linear in log-log space: log(y) = log(k) + n·log(x).
• R² (coefficient of determination) measures how well the model fits: R² = 1 − SS_res/SS_tot.
• For direct proportion, k = Σ(xy)/Σ(x²) from least-squares; k_avg = Σ(y/x)/n is an alternative.

Did You Know?

📏Distance = speed × time is the classic direct proportion. At constant speed, d/t is always the same — that's the constant of proportionality.Source: Physics

⚡Ohm's Law V = IR: voltage is directly proportional to current when resistance is constant. The constant k is R.Source: Electrical Engineering

🌡️Boyle's Law PV = k: pressure and volume are inversely proportional at constant temperature. Double P, halve V.Source: Chemistry

📐Circle circumference C = πd. The constant of proportionality is π ≈ 3.14159 — the same for every circle.Source: Geometry

🏗️Hooke's Law F = kx: the spring constant k depends on the material and geometry. Stiffer springs have larger k.Source: Mechanics

🌍Newton's law of gravitation F ∝ 1/r² is a power law with exponent −2. Doubling distance quarters the force.Source: Astrophysics

How It Works

Direct (y = kx): k = y/x for each pair; ideally constant. Least-squares: k = Σ(xy)/Σ(x²).

Inverse (y = k/x): k = xy for each pair; ideally constant. Use k_avg = Σ(xy)/n.

Power (y = kx^n): Take logs: log(y) = log(k) + n·log(x). Fit a line in log-log space to get n and k.

R²: 1 − SS_res/SS_tot. R² = 1 means perfect fit; R² < 1 means scatter around the model.

Expert Tips

Check k Consistency

If k values vary widely across points, the relationship may not be purely proportional. Use the k consistency chart to visualize.

Zero Values

For direct proportion, x = 0 ⇒ y = 0. For inverse, x and y must be > 0 (division by zero).

Residual Patterns

If residuals show a pattern (e.g., U-shaped), try a different model type (power instead of direct).

More Data Points

At least 5–10 pairs improve reliability. Outliers can skew k; inspect the scatter plot.

Formulas Reference

Direct: k = Σ(xᵢyᵢ) / Σ(xᵢ²)

Inverse: k = Σ(xᵢyᵢ) / n

Power: log(y) = log(k) + n·log(x) → linear regression in log space

R² = 1 − SS_res / SS_tot

Step-by-Step Calculation (Direct Proportion)

Step 1: For each pair (xᵢ, yᵢ), compute yᵢ/xᵢ. If the relationship is perfectly direct, all ratios equal k.

Step 2: Least-squares minimizes Σ(yᵢ − kxᵢ)². Setting derivative to zero: k = Σ(xᵢyᵢ) / Σ(xᵢ²).

Step 3: Compute fitted values ŷᵢ = kxᵢ and residuals eᵢ = yᵢ − ŷᵢ.

Step 4: SS_res = Σeᵢ², SS_tot = Σ(yᵢ − ȳ)². R² = 1 − SS_res/SS_tot.

Interpreting R²

R² = 1: Perfect fit — all points lie exactly on the model line.
R² > 0.99: Excellent fit — proportional model is very appropriate.
R² 0.9–0.99: Good fit — minor scatter, model is reasonable.
R² < 0.9: Consider whether a proportional model is appropriate; residuals may show patterns.

Frequently Asked Questions

What is the constant of proportionality?

It is the factor k that relates two variables in a proportional relationship. For y = kx, k is the constant; for y = k/x, k is the product xy.

When is R² close to 1?

R² = 1 means the model explains 100% of the variance — a perfect fit. R² > 0.95 is often considered excellent for proportional models.

Direct vs inverse proportion?

Direct: y increases with x (y/x constant). Inverse: y decreases as x increases (xy constant).

When to use power proportion?

Use when the relationship is multiplicative: area ∝ radius², volume ∝ radius³, gravitational force ∝ 1/r². Linear in log-log space.

Real-World Examples

Hooke's Law: F = kx (spring force vs extension)
Ohm's Law: V = IR
Boyle's Law: PV = k
Speed/distance: d = vt
Weight on spring: mg = kx
Gravitational force: F ∝ 1/r²
Flow rate: Q = Av

Each preset in the calculator demonstrates a real proportional relationship. Try loading them to see how k is computed and how R² indicates fit quality.

Power Proportion Details

For y = kx^n, taking natural log of both sides: ln(y) = ln(k) + n·ln(x). This is linear in (ln(x), ln(y)) space. We fit a line to get slope n and intercept ln(k), then k = exp(intercept). Use power proportion when the relationship is multiplicative.