Calculator Suite

Linear Regression Calculator

Perform simple linear regression analysis with comprehensive diagnostics and assumption checking

Analysis Settings
Data Input Method
Choose how to input your regression data

Select manual entry or use a sample dataset

Manual Data Entry
Enter X,Y pairs separated by commas, one pair per line

Format: X,Y (one pair per line). Example: 1,2.5

Label for the independent variable

Label for the dependent variable

Analysis Options
Configure regression analysis settings

Confidence level for intervals and tests

Display Options

Display confidence intervals around regression line

Identify potential outlier points

Quick Start
Load sample datasets to explore regression analysis
Sales vs AdvertisingIce Cream Sales vs TemperatureStudy Hours vs Test Scores

Educational Resources

Regression Formulas
Key formulas for simple linear regression analysis

Regression Equation

y=b0+b1x+ϵy = b_0 + b_1x + \epsilon

b0b_0 = y-intercept

b1b_1 = slope coefficient

ϵ\epsilon = error term

Slope Formula

b1=∑(xi−xˉ)(yi−yˉ)∑(xi−xˉ)2b_1 = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sum(x_i - \bar{x})^2}

Y-Intercept Formula

b0=yˉ−b1xˉb_0 = \bar{y} - b_1\bar{x}

R-squared

R2=SSregSStot=1−SSresSStotR^2 = \frac{SS_{reg}}{SS_{tot}} = 1 - \frac{SS_{res}}{SS_{tot}}

Proportion of variance explained by the model

TL;DR — Linear Regression Explained

Linear regression finds the best-fit line through your data points. The equation y = b₀ + b₁x predicts Y from X. R² tells you how well the model fits (0-1, higher = better). Check residual plots to verify assumptions before trusting results.

How to Use This Calculator: Step-by-Step
1

Enter Data Points

Input X,Y pairs manually or select a sample dataset.

2

Configure Analysis

Set confidence level and display options (confidence bands, outliers).

3

Run Regression

Click "Run Regression Analysis" to calculate slope, intercept, and R².

4

Analyze Results

Review scatter plot, residuals, Q-Q plot, and assumption checks.

📊 Example: Advertising Spend vs Sales

Data: 15 months of ad spend ($k) vs revenue ($k)

Equation

y = 12.4 + 3.2x

R²

0.87

p-value

<0.001

Interpretation

Significant

For every $1k increase in ad spend, revenue increases by $3.2k. The model explains 87% of revenue variation.

⚠️ Assumptions & Limitations

Key Assumptions:

  • • Linear relationship between X and Y
  • • Independent observations
  • • Homoscedasticity (constant variance)
  • • Normally distributed residuals
  • • No autocorrelation in residuals

When to Use Alternatives:

  • • Curved relationship: Polynomial regression
  • • Multiple predictors: Multiple regression
  • • Binary outcome: Logistic regression
  • • Non-normal data: Transform variables
Frequently Asked Questions

What does R-squared actually mean?

R² measures how much of Y's variation is explained by X. R² = 0.80 means 80% of variance is predicted. Higher is generally better, but a high R² doesn't guarantee a good model.

How do I interpret the slope?

The slope is the expected change in Y per one-unit increase in X. Slope = 2.5 means Y increases by 2.5 when X increases by 1.

What are residuals and why do they matter?

Residuals = observed - predicted Y values. Analyzing residual plots helps verify assumptions and identify outliers.

How many data points do I need?

While 3+ points work technically, aim for 10-20+ observations for reliable results and stable estimates.

What if my data isn't linear?

Consider variable transformations (log, sqrt), polynomial regression, or non-linear models. Always check residual plots first.

📺 Video Tutorials