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The linear regression calculator helps you determine the line of best fit using the least squares method. It provides instant results along with a graphical representation of the regression line for your dataset.
“Linear regression is a statistical technique used to predict the value of a dependent variable based on one independent variable.” It assumes a straight-line relationship between the variables. This calculator makes it easy to compute the slope, intercept, and predicted values accurately.
The equation for the line of best fit is:
ŷ = bX + a
Where:
ŷ = predicted value of the dependent variable
b = slope of the line
X = independent variable
a = y-intercept (value of y when X = 0)
To calculate the slope (b) and intercept (a), use the following formulas:
b = SP / SSx
SP (Σ(X - Mx)*(Y - My)) = sum of the products of deviations of X and Y from their means
SSx (Σ(X - Mx)²) = sum of squared deviations of X from its mean
a = My - b * Mx
Where Mx and My are the means of X and Y respectively.
Here’s an example to illustrate how linear regression works:
Calculate the least squares regression line for the dataset:
{(1, 2), (3, 5), (4, 4), (6, 7)}
Also, predict the value of Y for X = 2 and X = 5.
Solution:
Step 1: Calculate sums and means:
Sum of X = 1 + 3 + 4 + 6 = 14
Sum of Y = 2 + 5 + 4 + 7 = 18
Mean of X = Mx = 14 / 4 = 3.5
Mean of Y = My = 18 / 4 = 4.5
Step 2: Compute deviations, squares, and products:
| X - Mx | Y - My | (X - Mx)² | (X - Mx)*(Y - My) |
|---|---|---|---|
| -2.5 | -2.5 | 6.25 | 6.25 |
| -0.5 | 0.5 | 0.25 | -0.25 |
| 0.5 | -0.5 | 0.25 | -0.25 |
| 2.5 | 2.5 | 6.25 | 6.25 |
Step 3: Calculate slope (b) and intercept (a):
SSx = 6.25 + 0.25 + 0.25 + 6.25 = 13
SP = 6.25 - 0.25 - 0.25 + 6.25 = 12
b = SP / SSx = 12 / 13 ≈ 0.923
a = My - b * Mx = 4.5 - (0.923 * 3.5) ≈ 1.37
Step 4: Form the regression equation:
ŷ = 0.923X + 1.37
Step 5: Predict Y for X = 2 and X = 5:
| X | Estimated Y |
|---|---|
| 2 | 3.22 |
| 5 | 6.485 |
The graphical plot of the line of best fit clearly shows the trend of the dataset, helping visualize the relationship between X and Y.
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