# SSE Calculator

## Table of Content

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Our SSE calculator helps you to determine the variability of a set of values across the regression line. So in order to improve the accuracy of any model prediction use a sum of squared errors calculator that identifies and removes outliers from the data.

## What is SSE in Statistics?

It is the statistical measure of variability that shows the difference between the observed and predicted values of a given set of data.

• A lower SSE indicates that the model is a better fit for the data.
• A higher SSE indicates a poorer fit of the model to the data.

This relation is calculated by the sum of squared residuals, which are the differences between the observed and predicted values.

### Types of Sum of Squares:

There are two types of SSE. The sum of squares error calculator finds these two types based on values for a predictor variable and a response variable. So, look at these:

#### 1. Residual Sum of Square:

The residual sum of squares is the sum of the squared deviations between the observed values and the predicted values in a regression model.

#### 2. Regression Sum of Square:

The regression sum of squares is the sum of the squared deviations between the mean of the observed values and the predicted values in a regression model.

## How SSE Calculator Function well?

To determine the dispersion of data points in the regression analysis, a sum of squared errors calculator is used. It is determined by adding the squared differences of each data point.

Inputs:

• Independent and dependent variables

Results Summary:

• Total Sum of Squared Errors: Get the computed value to understand how to find sse difference between observed and predicted values.
• Residual Analysis: Provide insights into the residual errors, showcasing how well the model fits the actual data.
• Step-by-Step Calculation: Complete calculations in the form of steps and draw a table for dependent and independent values.

## SSE Formula:

In statistics, the sum of squares formula describes how well the data is indicated by a model, and the residual sum of squares calculator is utilized to determine the variability of the data values across the regression line.

It is evaluated by determining the square distance between each point and summing all the values.

$$SSE = \sum^n_{i=1}(X_i - \bar X)^2$$

Where:

• xi are the individual points of x
• x̄ is the mean of all points

## Practical Example:

Independent variable X sample data = 7, 7, 23, 8, 2, 14, 17, 16, 21, 19
Dependent variable Y sample data = 12, 5, 15, 18, 19, 13, 12, 14, 11, 8

#### Solution:

The data represent the dependent and the independent variable:

 Obs. X Y 1 7 12 2 7 5 3 23 15 4 8 18 5 2 19 6 14 13 7 17 12 8 16 14 9 21 11 10 19 8

So, to identify the effect of the dependent variable on the independent variable or its correlation, take the help of an SSE calculator to understand this relation in the form of a table.

 Obs. X Y Xᵢ² Yᵢ² Xᵢ · Yᵢ 1 7 12 49 144 84 2 7 5 49 25 35 3 23 15 529 225 345 4 8 18 64 324 144 5 2 19 4 361 38 6 14 13 196 169 182 7 17 12 289 144 204 8 16 14 256 196 224 9 21 11 441 121 231 10 19 8 361 64 152 Sum = 134 127 2238 1773 1639

The sum of all the squared values from the table is given by:

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

$$= 2238 - \dfrac{1}{10} (134)^2$$

$$= 442.4$$

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

$$= 1773 - \dfrac{1}{10} (127)^2$$

$$= 160.1$$

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

$$= 1639 - \dfrac{1}{10} (134) (127)$$

$$= -62.8$$

The given formulas calculate the slope of the line and the y-intercepts:

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

$$= \dfrac{-62.8}{442.4}$$

$$= -0.14195$$

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

$$= 12.7 - -0.14195 \times 13.4$$

$$= 14.602$$

Then, the regression equation is:

$$\hat{Y} = 14.602 -0.14195X$$

Now, The total sum of the square is:

$$SS_{Total} = SS_{YY} = 160.1$$

Also, the regression sum of the square is calculated as:

$$SS_{R} = \hat{B}_1 SS_{XY}$$

$$= -0.14195 \times -62.8$$

$$= 8.9146$$

As we know that:

$$SS_{Total} = SS_{Regression} + SS_{Error}$$

We can calculate the sum of squares as follows:

$$SS_{E} = SS_{Total} - SS_{R}$$

$$= 160.1 - 8.9146$$

$$= 151.19$$

Thus, the Residual Sum of Square(RSS) or Sum of Square Error is = 151.19

### Why is it Important to Measure SSE?

It is important to measure the sum of squared error because it is used to evaluate the performance of the model. It also helps to analyze whether the model is a better fit for data or not.

### What is a Good SSE?

A good SSE is one that is small relative to the total sum of squares (TSS). So we say that the lower SSE is better. It indicates that the model is a better fit for the data.

### Can SSE be Negative?

No, It is always non-negative, so the sum of squared residuals must also be non-negative. If you get a negative value, it means that there is an error in your calculation.

### How do I Interpret a lower SSE Value?

A lower SSE value shows that the model is a better fit for the data. So, the SSE calculator can be used to recognize the most important features in the data, which can be used to build more efficient and accurate models. This means that the model is better able to predict the observed values.

### How can SSE be Improved?

There are some points that are given below which you can follow to improve the SSE of your model:

• Gather more accurate data
• Use some complex models
• Change the data and remove their outliers

## Citations:

Cuemath: What is the Sum of Squares?, Sum of Squares Formula, Steps to Find Sum of Squares, Example, Practical questions.

Investopedia: Understanding the Sum of Squares, Sum of Squares Formula, How to calculate?, Types, Limitations of Using the Sum of Squares, Example. ### Alan Walker

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