Linear Regression Calculator

Regression and Prediction Models

Enter your data points as (x, y) pairs to calculate the line of best fit and other statistical metrics.

Examples

Use these examples to see how the calculator works.

Study Hours vs. Exam Score

Simple Positive Correlation

A simple dataset showing a positive correlation between hours studied and exam scores.

Data Points:

1, 65 2, 70 3, 75 4, 85 5, 90

Age of Car vs. Value

Simple Negative Correlation

A dataset illustrating that as a car gets older, its value tends to decrease.

Data Points:

1, 20000 2, 18000 3, 16500 5, 12000 8, 7000

Shoe Size vs. IQ

No Correlation

A dataset showing two variables that are not expected to have any correlation.

Data Points:

8, 110 9, 95 10, 120 11, 105 12, 100

House Size vs. Price

Real Estate Data

A dataset of house size (in sq. ft.) and its market price (in thousands).

Data Points:

1400, 245 1600, 312 1700, 279 1875, 308 2100, 405 2500, 450
Other Titles
Understanding Linear Regression: A Comprehensive Guide
Explore the principles of linear regression, its applications, and how to interpret the results from this calculator.

What is Linear Regression?

  • Definition of Linear Regression
  • The Line of Best Fit
  • Key Components of the Regression Equation
Linear regression is a fundamental statistical and machine learning technique used to model the relationship between a dependent variable and one or more independent variables. The goal is to find a linear equation that best predicts the value of the dependent variable based on the value of the independent variable(s). In its simplest form, simple linear regression, we use a single independent variable (X) to predict a single dependent variable (Y).
The Line of Best Fit
The core of linear regression is finding the 'line of best fit'. This is a straight line that passes through a scatter plot of data points in a way that minimizes the distance between the line and each data point. The most common method to determine this line is the 'Least Squares Method', which aims to minimize the sum of the squared vertical distances (residuals) of the points from the line.
The Regression Equation: y = mx + c
The output of a linear regression analysis is a linear equation of the form y = mx + c, where:
y: The predicted value of the dependent variable.
x: The value of the independent variable.
m (Slope): Represents the change in y for a one-unit change in x. A positive slope means y increases as x increases, while a negative slope means y decreases as x increases.
c (Y-Intercept): The value of y when x is 0. It's the point where the regression line crosses the y-axis.

Conceptual Examples

  • Predicting a student's final exam score based on the number of hours they studied.
  • Estimating a house's price based on its square footage.
  • Forecasting a company's sales for the next quarter based on its advertising budget.

Step-by-Step Guide to Using the Calculator

  • Inputting Your Data
  • Making Predictions
  • Interpreting the Results
1. Inputting Your Data

In the 'Data Points (x, y)' text area, enter your paired data. Each pair should be on a new line. You can separate the x and y values with either a comma or a space. For example, to input the points (1, 2), (3, 5), and (4, 7), you would type: 1, 2 3, 5 4, 7

2. Making Predictions (Optional)
If you want to predict a y value for a specific x value that is not in your original dataset, enter that x value into the 'Predict Y for a given X' field. The calculator will use the generated regression equation to compute the predicted y.
3. Interpreting the Results
After clicking 'Calculate', you will see several key metrics:
Regression Equation: The formula for the line of best fit.
Slope (m) & Y-Intercept (c): The core components of your equation.
Correlation Coefficient (r): A value between -1 and 1 that measures the strength and direction of the linear relationship. A value near 1 or -1 indicates a strong relationship, while a value near 0 indicates a weak or no linear relationship.
Coefficient of Determination (R²): A value between 0 and 1 that represents the proportion of the variance in the dependent variable that is predictable from the independent variable. For example, an R² of 0.75 means that 75% of the variation in y can be explained by the linear relationship with x.

Real-World Applications of Linear Regression

  • Economics and Finance
  • Medical Research
  • Business and Marketing
Linear regression is not just an academic concept; it's a powerful tool used across many industries.
Economics and Finance
It's used to model relationships between economic variables. For example, predicting consumer spending based on disposable income or analyzing the impact of interest rates on stock market prices.
Medical Research
Researchers use it to analyze the relationship between a risk factor and a health outcome, such as modeling the effect of dosage of a new drug on blood pressure reduction.
Business and Marketing
Companies use regression to forecast sales based on advertising spend, predict employee performance based on training hours, or understand how customer satisfaction impacts loyalty.

Industry Use Cases

  • A real estate agent using regression to price a house based on its features (size, location, etc.).
  • An insurance company predicting the claim amount for a policyholder based on their age and driving history.
  • A farmer estimating crop yield based on the amount of rainfall and fertilizer used.

Mathematical Derivation and Formulas

  • The Least Squares Method
  • Formula for Slope (m)
  • Formula for Y-Intercept (c)
The calculator finds the line of best fit by using the least squares method. The formulas to calculate the slope (m) and y-intercept (c) for a set of n data points (x, y) are derived from this method.
Formula for Slope (m)
m = (nΣ(xy) - ΣxΣy) / (nΣ(x²) - (Σx)²)
Formula for Y-Intercept (c)
c = (Σy - mΣx) / n
Formula for Correlation Coefficient (r)
r = (nΣ(xy) - ΣxΣy) / √[(nΣ(x²) - (Σx)²)(nΣ(y²) - (Σy)²)]
Where Σx is the sum of all x-values, Σy is the sum of all y-values, Σxy is the sum of the products of corresponding x and y values, Σx² is the sum of squared x-values, and Σy² is the sum of squared y-values.

Common Pitfalls and Best Practices

  • Correlation vs. Causation
  • The Danger of Extrapolation
  • Checking for Linearity
Correlation is Not Causation
A common mistake is to assume that because two variables are strongly correlated, one must cause the other. Linear regression can only show the strength of a relationship; it cannot prove causality. There might be a third, unobserved variable (a lurking variable) that is influencing both.
The Danger of Extrapolation
Extrapolation means making predictions outside the range of your original data. For example, if your data for house sizes is between 1000 and 3000 sq. ft., using your model to predict the price of a 6000 sq. ft. mansion can be highly inaccurate. The linear relationship might not hold for values far outside the observed range.
Always Visualize Your Data
Before performing regression, it's crucial to create a scatter plot of your data. This helps you visually confirm if a linear relationship is appropriate. Data might have a non-linear pattern (e.g., a curve), or there might be significant outliers that could heavily skew the results. Anscombe's quartet is a famous example demonstrating why visualizing data is critical.