What is the formula to calculate the slope (m) of the best fit line in linear regression?
The formula to calculate the slope (m) of the best fit line in linear regression is derived from the concept of ordinary least squares (OLS) estimation. Linear regression is a widely used statistical method for modeling the relationship between a dependent variable and one or more independent variables. The best fit line represents the line that minimizes the sum of squared differences between the observed and predicted values.
To calculate the slope of the best fit line, we need to estimate the coefficients of the linear regression model. In a simple linear regression, where we have a single independent variable, the formula for the slope (m) is given by:
m = Σ((xi – x̄)(yi – ȳ)) / Σ((xi – x̄)²)
Here, xi represents the observed values of the independent variable, x̄ is the mean of the independent variable, yi represents the observed values of the dependent variable, and ȳ is the mean of the dependent variable. The summation symbol (Σ) denotes the sum of the values over the entire dataset.
Let's consider an example to illustrate the calculation of the slope. Suppose we have a dataset with the following values:
x = [1, 2, 3, 4, 5] y = [2, 3, 4, 5, 6]
First, we calculate the means of x and y:
x̄ = (1 + 2 + 3 + 4 + 5) / 5 = 3
ȳ = (2 + 3 + 4 + 5 + 6) / 5 = 4
Next, we calculate the numerator and denominator of the slope formula:
Numerator:
Σ((xi – x̄)(yi – ȳ)) = (1 – 3)(2 – 4) + (2 – 3)(3 – 4) + (3 – 3)(4 – 4) + (4 – 3)(5 – 4) + (5 – 3)(6 – 4)
= (-2)(-2) + (-1)(-1) + (0)(0) + (1)(1) + (2)(2)
= 4 + 1 + 0 + 1 + 4
= 10
Denominator:
Σ((xi – x̄)²) = (1 – 3)² + (2 – 3)² + (3 – 3)² + (4 – 3)² + (5 – 3)²
= (-2)² + (-1)² + (0)² + (1)² + (2)²
= 4 + 1 + 0 + 1 + 4
= 10
Finally, we divide the numerator by the denominator to obtain the slope:
m = 10 / 10
= 1
Therefore, the slope of the best fit line for this example is 1.
The formula to calculate the slope (m) of the best fit line in linear regression is obtained by dividing the sum of the products of the differences between the observed values and their means by the sum of the squared differences between the independent variable values and their mean. This formula allows us to estimate the slope, which represents the rate of change of the dependent variable with respect to the independent variable.
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