StatsCalculators

Simple Linear Regression

Created:November 15, 2024

Last Updated:February 28, 2025

This Simple Linear Regression Calculator helps you analyze the relationship between two variables. It provides comprehensive analysis including model summary statistics, coefficient estimates, confidence intervals, and diagnostic tests. The calculator also generates a regression plot with the fitted line and confidence bands. To learn about the data format required and test this calculator, click here to populate the sample data.

Calculator

1. Load Your Data

2. Select Columns & Options

Independent Variable (X):

Dependent Variable (Y):

Exclude Outliers

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Simple Linear Regression

Definition

Simple Linear Regression models the relationship between a predictor variable (X) and a response variable (Y) using a linear equation. It finds the line that minimizes the sum of squared residuals.

Key Formulas

Regression Line:

\hat{Y} = b_0 + b_1X

Slope:

b_1 = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sum(x_i - \bar{x})^2}

Intercept:

b_0 = \bar{y} - b_1\bar{x}

R-squared:

R^2 = 1 - \frac{\sum(y_i - \hat{y}_i)^2}{\sum(y_i - \bar{y})^2}

Key Assumptions

Linearity: Relationship between X and Y is linear

Independence: Observations are independent

Homoscedasticity: Constant variance of residuals

Normality: Residuals are normally distributed

Practical Example

Step 1: Data

$X$	$Y$	$(X-\bar{X})$	$(Y-\bar{Y})$	$(X-\bar{X})^2$	$(X-\bar{X})(Y-\bar{Y})$
1	2.1	-2	-3.82	4	7.64
2	3.8	-1	-2.12	1	2.12
3	6.2	0	0.28	0	0
4	7.8	1	1.88	1	1.88
5	9.3	2	3.38	4	6.76
$\Sigma=15$	$\Sigma=29.2$	$\Sigma=0$	$\Sigma=0$	$\Sigma=10$	$\Sigma=18.4$

Means: $\bar X = 3$ , $\bar Y = 5.84$

Step 2: Calculate Slope ( $b_1$ )

b_1 = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sum(x_i - \bar{x})^2} = \frac{18.4}{10} = 1.84

Step 3: Calculate Intercept ( $b_0$ )

b_0 = \bar{y} - b_1\bar{x} = 5.84 - 1.84(3) = 0.32

Step 4: Regression Equation

\hat{Y} = 0.32 + 1.84X

Step 5: Calculate $R^2$

$R^2 = 0.986$ (98.6% of variation in Y explained by X)

Code Examples

R

library(tidyverse)

# Example data
data <- tibble(x = c(1, 2, 3, 4, 5), 
               y = c(2.1, 3.8, 6.2, 7.8, 9.3))

# Fit linear model
model <- lm(y ~ x, data=data)

# Print summary
summary(model)

# Get confidence intervals
confint(model)

# plot with ggplot2
ggplot(data, aes(x = x, y = y)) +
  geom_point() +
  geom_smooth(method = "lm", se = FALSE) +
  theme_minimal()

Python

import numpy as np
import pandas as pd
import statsmodels.api as sm

# Example data
X = [1, 2, 3, 4, 5]  # predictor variable
y = [2.1, 3.8, 6.2, 7.8, 9.3]  # response variable

# Add constant to X for intercept
X = sm.add_constant(X)

# Fit model
model = sm.OLS(y, X).fit()

# Print summary
print(model.summary())

# Get coefficients
print(f'Intercept: {model.params[0]:.4f}')
print(f'Slope: {model.params[1]:.4f}')
print(f'R-squared: {model.rsquared:.4f}')

Verification

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