The regression residual is the distinction between the discovered and expected values in the regression version of the dataset. The residual calculator affords accuracy and precision of the anticipated consequences. without a doubt, the residual discover the margin of error of the dataset values with the aid of drawing the distinction between the real and forecasted values.
The formula for the residual in statistics is given below:
Residual = Observed Value − Predicted Value
Understand the concept of the residual with this practical example: Suppose we have a set of independent variables 2, 4, 6, 8, 10 and dependent variables 3, 7, 5, 15, and 12. Now, the residuals for each observation in a simple linear regression model are calculated as follows:
Solution:
Dependent and Independent Variables:
The dataset values for the dependent and independent variables are:
| Obs. | X | Y |
|---|---|---|
| 1 | 2 | 3 |
| 2 | 4 | 7 |
| 3 | 6 | 5 |
| 4 | 8 | 15 |
| 5 | 10 | 12 |
The Regression Coefficient:
Now, calculate the regression coefficients using the given values:
| Obs. | X | Y | Xᵢ² | Yᵢ² | Xᵢ × Yᵢ |
|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 9 | 6 |
| 2 | 4 | 7 | 16 | 49 | 28 |
| 3 | 6 | 5 | 36 | 25 | 30 |
| 4 | 8 | 15 | 64 | 225 | 120 |
| 5 | 10 | 12 | 100 | 144 | 120 |
| Sum = | 30 | 42 | 220 | 452 | 304 |
Sum of Squares Values:
The calculated sum of squares is as follows:
\[SS_{XX} = 44\]
\[SS_{YY} = 90\]
\[SS_{XY} = 58\]
The Slope of the Line:
\(\hat{\beta}_1 = \dfrac{SS_{XY}}{SS_{XX}} = \dfrac{58}{44} = 1.32\)
\(\hat{\beta}_0 = \bar{Y} - \hat{\beta}_1 \cdot \bar{X} = 5.76\)
The regression equation is:
\(\hat{Y} = 5.76 + 1.32X\)
Predicted Values:
By substituting the independent variable values into the regression equation, the predicted values are:
| Obs. | X | Y | Predicted Values | Residuals (Y − P.V) |
|---|---|---|---|---|
| 1 | 2 | 3 | 5.76 + 1.32 × 2 = 8.40 | 3 − 8.40 = -5.40 |
| 2 | 4 | 7 | 5.76 + 1.32 × 4 = 11.04 | 7 − 11.04 = -4.04 |
| 3 | 6 | 5 | 5.76 + 1.32 × 6 = 13.68 | 5 − 13.68 = -8.68 |
| 4 | 8 | 15 | 5.76 + 1.32 × 8 = 16.32 | 15 − 16.32 = -1.32 |
| 5 | 10 | 12 | 5.76 + 1.32 × 10 = 18.96 | 12 − 18.96 = -6.96 |
the online records residual calculator requires the values of the “X” and “Y” variables: let’s discover how!
Enter:
Output:
The residual values are an excellent way to know the high-quality of the sample facts. the principle cause is that you are comparing the actual values with the predicted values of certain phenomena. the web statistics residuals calculator will increase the satisfactory of the regression evaluation.
