# Residual Calculator

## Table of Content

Feedback

Our dynamic residual calculator is designed to find residual values for each observation on the basis of a linear regression model by putting the values of the independent (X variable) and dependent (Y variable). The residual values are defined to know the error in our estimation compared to the real values in the marketplace.

## What is Residual?

“The residual is the difference between actual and predicted value by the linear regression model (y-ŷ) of a given point”.

## Evaluation Formula:

Residual is evaluated after running the regression model. The residual value calculator evaluates the residual for all observations in a linear regression model.

Residual = Observed Value - Predicted Value

Residual value is the statistical approach also known as the estimated worth after the asset has been fully undervalued. So the residuals calculator is utilized for getting the accuracy in the estimated results.

• If there is a negative residual value that means the predicted value is too high
• If there is a positive residual value that means the predicted value is too low

## Practical Demonstration:

Suppose the set of variables that are available for the residual observation in a simple linear regression model.

Dependent Variables = 7, 4, 12, 9, 10, 14
Independent Variables = 3, 5, 8, 11, 19, 4

To answer a problem use the residual calculator statistics or put the variables by hand in the formula:

#### Data Set Values:

The data set values for the dependent and independent variables are:

 Obs. X Y 1 3 7 2 5 4 3 8 12 4 11 9 5 19 10 6 4 14

#### Regression Coefficient:

Now, construct the estimated regression coefficient using the values of the predicted and response variables:

 Obs. X Y Xᵢ² Yᵢ² Xᵢ · Yᵢ 1 3 7 9 49 21 2 5 4 25 16 20 3 8 12 64 144 96 4 11 9 121 81 99 5 19 10 361 100 190 6 4 14 16 196 56 Sum = 50 56 596 586 482

The sum of the square generated from the above table are:

$$SS_{XX} = \sum^n_{i-1}X_i^2 - \dfrac{1}{n} \left(\sum^n_{i-1}X_i \right)^2$$

$$= 596 - \dfrac{1}{6} (50)^2$$

$$= 179.33$$

$$SS_{YY} = \sum^n_{i-1}Y_i^2 - \dfrac{1}{n} \left(\sum^n_{i-1}Y_i \right)^2$$

$$= 586 - \dfrac{1}{6} (56)^2$$

$$= 63.333$$

$$SS_{XY} = \sum^n_{i-1}X_iY_i - \dfrac{1}{n} \left(\sum^n_{i-1}X_i \right) \left(\sum^n_{i-1}Y_i \right)$$

$$= 482 - \dfrac{1}{6} (50) (56)$$

$$= 15.333$$

The slope of the line and y-intercept are calculated by the following formulas:

$$\hat{\beta}_1 = \dfrac{SS_{XY}}{SS_{XX}}$$

$$= \dfrac{15.333}{179.33}$$

$$= \dfrac{15.333}{179.33}$$

$$= 0.085502$$

$$\hat{\beta}_0 = \bar{Y} - \hat{\beta}_1 \times \bar{X}$$

$$= 9.3333 - 0.085502 \times 8.3333$$

$$= 8.6208$$

Hence, The regression equation is:

$$\hat{Y} = 8.6208 + 0.085502X$$

After finding the regression equation, we can gather the predicted values by inserting the independent variable in the regression equation that is also taken into account by the online statistics residual calculator:

$$\hat{Y} = 8.6208 + 0.085502X$$

#### Predicted and Residual Values:

 Obs. X Y Predicted Values ŷ Residuals(y - ŷ) 1 3 7 8.6208 + 0.085502 x 3 = 8.8773 7 - 8.8773 = -1.8773 2 5 4 8.6208 + 0.085502 x 5 = 9.0483 4 - 9.0483 = -5.0483 3 8 12 8.6208 + 0.085502 x 8 = 9.3048 12 - 9.3048 = 2.6952 4 11 9 8.6208 + 0.085502 x 11 = 9.5613 9 - 9.5613 = -0.56134 5 19 10 8.6208 + 0.085502 x 19 = 10.245 10 - 10.245 = -0.24535 6 4 14 8.6208 + 0.085502 x 4 = 8.9628 14 - 8.9628 = 5.0372

### Scattered Plot of Y and X:

Residual plot maker can enhance the understanding by sketching the graph which is as follows: ## Working of Residual Calculator:

This residual plot calculator is designed to find the residuals for each observation in a simple linear regression model by taking into account the below values:

Input:

• Enter the independent variable X data
• Put the dependent variable Y data
• Tap on the “Calculate” button

Output:

• The residual regression
• Predicted and residual values in a given table
• Scattered plot of residual and predicted values
• Step-by-step calculations

### What are the Types of Residuals in Regression?

There are the following types that reflect the difference between predicted and observed values:

• Ordinary Residual
• Pearson Residual
• Studentized Residual

### What Does this Residual Calculator Indicate?

This fun residuals calculator is an estimation tool designed to help you evaluate the unlimited income potential.

Suppose you want to predict the future stock prices by using the following model:

Stock Price = 2.1 × GDP growth + 18

If the expected GDP growth of the current year is 8%, then the stock price of Company Alpha is:

2.1 × 8 + 18 = \$34.8

## References:

Wikipedia: Introduction to residuals and least-squares regression, Calculating residual example, Calculating and interpreting residuals. ### Alan Walker

Studies mathematics sciences, and Technology. Tech geek and a content writer. Wikipedia addict who wants to know everything. Loves traveling, nature, reading. Math and Technology have done their part, and now it's the time for us to get benefits.