Simple Linear Regression Calculator

What Is Simple Linear Regression?

Simple linear regression is a method for modeling the straight-line relationship between one independent variable (X) and one continuous dependent variable (Y). The goal is to find the single line that minimizes the total squared distance between each observed data point and the line itself — a technique called ordinary least squares (OLS).

Think of how gas mileage drops as vehicle weight increases, or how monthly energy bills climb as winter temperatures fall. In both cases, one numeric variable reliably tracks another in a roughly straight-line pattern. Regression puts a precise mathematical equation on that pattern, so you can predict an unknown Y value from any given X. The resulting line is called the line of best fit, or the regression line.

According to the NIST/SEMATECH Engineering Statistics Handbook, linear regression is the most widely applied modeling technique in statistics because its outputs — slope, intercept, and R² — are interpretable without specialist training, yet the method underlies everything from econometric forecasts to clinical dose-response modeling.

The Simple Linear Regression Formula

The regression equation takes the form ŷ = β₀ + β₁x, where β₁ is the slope and β₀ is the y-intercept. Both are derived from your data using least squares formulas that no amount of eyeballing or guessing can match for accuracy.

The slope β₁ tells you how many units Y is expected to increase for each one-unit increase in X. A slope of 150 in a house-price model means each additional square foot is associated with $150 more in predicted price. The intercept β₀ is Y when X = 0 — sometimes interpretable (as a baseline cost), sometimes not (a house with zero square feet has no meaning, so β₀ is a mathematical anchor rather than a real prediction).

Step-by-Step Worked Example: Real Estate Pricing

The dataset below maps square footage to sale price. You can load this exact example into the calculator above using the "Example (Real Estate)" button.

n = 5, ΣX = 9,000, ΣY = 1,450,000, ΣX² = 17,100,000, ΣXY = 2,745,000,000

Interpreting Your Results: R², Pearson's r, and Residuals

Three outputs carry most of the interpretive weight in a simple linear regression: the equation itself, R², and the residual pattern. Getting all three right is what separates a useful model from a misleading one.

Difference Between r and R²

Pearson's r and R² both measure model fit, but they serve different questions. An r of +0.91 tells you the relationship is strong and positive. R² = 0.83 (which is 0.91²) tells you the model accounts for 83% of the variance in Y — leaving 17% unexplained by X. In simple linear regression these two always relate by R² = r². In multiple regression, R² still works the same way but r no longer applies the same connection.

Assumptions of Simple Linear Regression (LINE)

Regression results are only trustworthy when the four assumptions hold. Statisticians use the acronym LINE to remember them. Violating these doesn't always break the model — but it does change which conclusions you can draw.

Penn State's STAT 501 course, available via online.stat.psu.edu, provides a thorough walkthrough of diagnosing each assumption using residual plots, Q-Q plots, and the Durbin-Watson test for independence. Their materials are widely cited in applied statistics curricula for good reason — the worked examples track industrial and biological datasets where at least one assumption is always slightly bent.

Simple Linear Regression in Python, Excel, and R

For analysts working with larger datasets, the following code snippets replicate exactly what this calculator computes — useful for verifying outputs before integrating regression into a larger workflow.

Python (NumPy / SciPy / Scikit-Learn)

import numpy as np
from scipy import stats

x = np.array([1200, 1500, 1800, 2100, 2400])
y = np.array([200000, 245000, 290000, 335000, 380000])

slope, intercept, r, p_value, std_err = stats.linregress(x, y)

print(f"Slope (β₁):     {slope:.4f}")        # 150.0
print(f"Intercept (β₀): {intercept:.4f}")    # 20000.0
print(f"R² value:       {r**2:.4f}")         # 0.9989
print(f"Pearson r:      {r:.4f}")            # 0.9994
print(f"Predicted (1900 sqft): {intercept + slope * 1900:.0f}")  # 305000

Microsoft Excel (LINEST function)

=LINEST(B2:B6, A2:A6, TRUE, TRUE)   // Returns full stats array: slope, intercept, R², etc.

// Individual outputs:
=SLOPE(B2:B6, A2:A6)               // Slope β₁  → 150
=INTERCEPT(B2:B6, A2:A6)           // Intercept β₀ → 20000
=RSQ(B2:B6, A2:A6)                 // R² → 0.9989
=CORREL(A2:A6, B2:B6)              // Pearson r → 0.9994
=FORECAST.LINEAR(1900, B2:B6, A2:A6)  // Predicted Y → 305000

Note: Excel's LINEST vs. this calculator agrees to all decimal places on slope and intercept. The calculator above vs. =LINEST() is a useful cross-check students can run side by side for any homework dataset.

R

x <- c(1200, 1500, 1800, 2100, 2400)
y <- c(200000, 245000, 290000, 335000, 380000)

model <- lm(y ~ x)
summary(model)
# Coefficients: (Intercept) = 20000, x = 150
# R-squared: 0.9989

predict(model, newdata = data.frame(x = 1900))  # 305000
plot(x, y); abline(model, col = "blue")         # Scatter + regression line

Simple vs. Multiple Linear Regression: When to Use Which

The choice between simple and multiple regression comes down to how many independent variables you need to describe reality accurately. Simple regression works when one predictor genuinely captures most of the variation. Once a second variable adds meaningful explanatory power — and is not just noise — multiple regression is the appropriate extension.

For a full treatment of extending the model beyond one predictor, see the multiple linear regression guide on this site. When your outcome is categorical rather than continuous, logistic regression is the standard choice — it models the log-odds of a binary event rather than a continuous Y.

Related Topics on Statistics Fundamentals

Linear regression connects to probability, hypothesis testing, and visualization. These pages build the surrounding foundation.

Sources and Further Reading

Frequently Asked Questions