Linear Regression calculator

What is linear regression?

In a simple Linear Regression model, the relationship between the dependent variable (Y) and the independent variable (X) is represented by a straight line:

Where:

• “Y” is the dependent variable (output/response)
• “X_i” is the independent variable
• “f” is a linear function
• “β” is an unknown parameter
• “e_i” are the error terms

Important Considerations:

While Linear Regression is a powerful and widely-used statistical technique, it's essential to consider its assumptions and limitations:

Linearity: 

The relationship between the independent and dependent variables must be linear. If the relationship is nonlinear, other methods may be more appropriate.

Independence: The observations should be independent of each other. In cases of time series or spatial data, other techniques may be more suitable.

Homoscedasticity: 

The variance of the error terms should be constant across all levels of the independent variable.

Normality: 

The error terms should be normally distributed.

When using Linear Regression, always validate the assumptions and evaluate the model's performance using appropriate metrics, such as the coefficient of determination (R-squared), residual analysis, and cross-validation.

Examples:

In this section, we’ll describe the method of calculating the linear regression between any two data sets.

Example 1:

Calculate the regression equation if:

X = 5, 22, 19, 8, 33, 10

Y = 9, 71, 31, 12, 44, 28

Solution:

Step 1: Calculate the mean of the data sets

Mean of X = μ_x = ΣX / n

μ_x = (5 + 22 + 19 + 8 + 33 + 10) / 6

μ_x = 97 / 6

μ_x = 16.67

Mean of Y = μ_y = ΣY / n

μ_y = (9 + 71 + 31 + 12 + 44 + 28) / 6

μ_y = 195 / 6

μ_y = 32.5

Step 2: Make a table:

Step 3: Calculate slope “m”

m = {(n * ∑ X_i​ *Y_i​) − (∑X_i​ * ∑Y_i​)​} / {n * ∑X_i² − (∑X_i)²}

m = {(6 * 4024) – (97 – 195)} / {(6 *2123) - (9409)}

m = 5229 / 3329

m = 1.5707

Step 4: Calculate Y-intercept

b = {(∑Y_i​) − (m * ∑X_i​)} / n ​

b = {(195) − (1.5707 * 97)} / 6

b = 7.107

Step 5: Make the linear equation

Y = (slope)x + (y-intercept)