Linear Regression Calculator Graph

Linear Regression Equations:

\[ m = \frac{n \sum(xy) - \sum x \sum y}{n \sum(x^2) - (\sum x)^2} \] \[ b = \frac{\sum y - m \sum x}{n} \]

Slope (m):

Intercept (b):

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1. What is Linear Regression?

Linear regression is a statistical method that models the relationship between a dependent variable (y) and one or more independent variables (x) by fitting a linear equation to observed data.

2. How Does the Calculator Work?

The calculator uses the ordinary least squares method to find the best-fitting line:

\[ m = \frac{n \sum(xy) - \sum x \sum y}{n \sum(x^2) - (\sum x)^2} \] \[ b = \frac{\sum y - m \sum x}{n} \]

Where:

\( m \) — Slope of the line
\( b \) — Y-intercept
\( n \) — Number of data points
\( x \) — Independent variable values
\( y \) — Dependent variable values

Explanation: The equations minimize the sum of the squares of the vertical distances between the observed points and the fitted line.

3. Importance of Linear Regression

Details: Linear regression is widely used for prediction and forecasting, understanding relationships between variables, and testing scientific hypotheses.

4. Using the Calculator

Tips: Enter comma-separated values for x and y variables. Both lists must have the same number of values. Example: "1,2,3,4" and "2,4,5,4".

5. Frequently Asked Questions (FAQ)

Q1: What does the slope (m) represent?
A: The slope indicates how much y changes for each unit change in x. A positive slope means y increases as x increases.

Q2: What does the intercept (b) represent?
A: The intercept is the predicted value of y when x equals zero (though this may not always have practical meaning).

Q3: How many data points do I need?
A: At least two points are required to calculate a line, but more points provide more reliable results.

Q4: What are the assumptions of linear regression?
A: Key assumptions include linear relationship, independence, homoscedasticity, and normal distribution of residuals.

Q5: Can I use this for non-linear relationships?
A: No, this calculator is for linear relationships only. Non-linear relationships require different regression methods.