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.
It is the statistical measure of variability that shows the difference between the observed and predicted values of a given set of data.
This relation is calculated by the sum of squared residuals, which are the differences between the observed and predicted values.
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:
The residual sum of squares is the sum of the squared deviations between the observed values and the predicted values in a regression model.
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.
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:
Results Summary:
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:
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
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
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.
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.
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.
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.
There are some points that are given below which you can follow to improve the SSE of your model:
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.
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