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Enter the independent and dependent variables in the tool and the calculator will find the SSE value.

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The SSE calculator is a statistical tool to estimate the variability of the data values across the regression line. The sum of squared residuals calculator calculates the dispersion of the points around the mean and how much the dependent variable deviates from predicted values in the regression analysis.

The sum of Square Error(SSE) is the difference between the observed and the predicted values. The SEE is also represented as RSS (residual sum of squares). The SEE is the spread of the data set values and it is an alternative to the standard deviation or absolute deviation.

Consider the data sample of the independent variables 6, 7, 7, 8, 12, 14, 15, 16, 16, 19, and the dependent variable 14, 15, 15, 17, 18, 18, 16, 14, 11, and 8. Find the sum of squared residuals or SEE values.

The data represent the dependent and the independent variable:

Obs. |
X |
Y |

1 | 6 | 14 |

2 | 7 | 15 |

3 | 7 | 15 |

4 | 8 | 17 |

5 | 12 | 18 |

6 | 14 | 18 |

7 | 15 | 16 |

8 | 16 | 14 |

9 | 16 | 11 |

10 | 19 | 8 |

Now by the predicted and the response variable, we construct the following table

Obs. |
X |
Y |
Xᵢ² |
Yᵢ² |
Xᵢ · Yᵢ |

1 | 6 | 14 | 36 | 196 | 84 |

2 | 7 | 15 | 49 | 225 | 105 |

3 | 7 | 15 | 49 | 225 | 105 |

4 | 8 | 17 | 64 | 289 | 136 |

5 | 12 | 18 | 144 | 324 | 216 |

6 | 14 | 18 | 196 | 324 | 252 |

7 | 15 | 16 | 225 | 256 | 240 |

8 | 16 | 14 | 256 | 196 | 224 |

9 | 16 | 11 | 256 | 121 | 176 |

10 | 19 | 8 | 361 | 64 | 152 |

Sum = | 120 | 146 | 1636 | 2220 | 1690 |

Our SSE calculator also calculates the deviation of the distances by summing all the mean points.

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

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

\(\ = 1636 - \dfrac{1}{10} (120)^2\)

\(\ = 196\)

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

\(\ = 2220 - \dfrac{1}{10} (146)^2\) \(= 88.4\)

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

\(\left(\sum^n_{i-1}Y_i \right)\)

\(\ = 1690 - \dfrac{1}{10} (120) (146)\)

\(\ = -62\)

The slope of the line and the y-intercepts are calculated by the given Formulas:

\(\hat{\beta}_1 = \dfrac{SS_{XY}}{SS_{XX}}\)

\(\ = \dfrac{-62}{196}\)

\(\ = -0.31633\)

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

\(\ = 14.6 - -0.31633 \times 12\)

\(\ = 18.396\)

Then, the regression equation is:

\(\hat{Y} = 18.396 -0.31633X\)

Now, The total sum of the squared values is:

\(\ SS_{Total} = SS_{YY} = 88.4\)

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

\(\ SS_{R} = \hat{B}_1 SS_{XY}\)

\(\ = -0.31633 \times -62\)

\(\ = 19.612\)

Now:

\(\ SS_{E} = SS_{Total} - SS_{R}\)

\(\ = 88.4 - 19.612\)

\(\ SS_{E} = 68.788\)

SSE sum of squared residuals error explains how closely the independent variable is related to the dependent variable.

The higher SSE means the variable is deviated from the expected value.

No, the standard deviation (SD) will always be larger than the standard error (SE).

No, it can never be negative because it is calculated by taking the sum of deviations between predicted and actual values. The square makes the values positive.

**References:**

From the source of 365datascience.com: Sum of Squares.

From the source of sixsigmastudyguide.com: SSE.

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