This calculator finds the **Total Sum of Squares (TSS)**, **Explained Sum of Squares (ESS)**, and **Residual Sum of Squares (RSS)** for a linear regression model, and verifies that \( TSS = ESS + RSS \). The fitted line and TSS as a dashed line are also displayed.

To use the calculator, provide a list of values for the predictor and the response, ensuring they are the same length, and then click the "Calculate and Plot" button.

**Total Sum of Squares (TSS):**

**Explained Sum of Squares (ESS):**

**Residual Sum of Squares (RSS):**

**Verification (TSS = ESS + RSS):**

## Total Sum of Squares (TSS), Explained Sum of Squares (ESS), and Residual Sum of Squares (RSS) Explanation

The **Total Sum of Squares (TSS)** represents the total variation in the response variable \(Y\). It is calculated as the sum of the squared differences between each observed value \(Y_i\) and the mean of the observed values \(\bar{Y}\).

### Key Components

**Predictor Variable (\(X\))**: The independent variable used to predict the response.**Response Variable (\(Y\))**: The dependent variable that is being predicted.**Fitted Value (\(\hat{Y}\))**: The predicted value of \(Y\) for a given \(X\), based on the linear regression model.**Mean Value (\(\bar{Y}\))**: The average of the observed values of \(Y\).

### Total Sum of Squares (TSS), ESS, and RSS

The relationship between these quantities is:

TSS represents the total variation in \(Y\), ESS represents the part explained by the model, and RSS represents the part that remains unexplained. The verification checks if \( TSS = ESS + RSS \).

### Caveats and Conditions

**Linear Assumption:**These calculations assume a linear relationship between the predictor \(X\) and the response \(Y\). If the relationship is non-linear, the model may not fit well, and the sums of squares may not provide meaningful insights.**Outliers:**Outliers can significantly impact the values of TSS, ESS, and RSS. Large outliers may cause the model to fit poorly, even if \( TSS = ESS + RSS \) holds true.**Overfitting:**If the model is too complex (e.g., too many predictors), it may explain the data too well, leading to high ESS but a small RSS. This could result in overfitting, where the model performs well on the given data but fails to generalize to new data.

### Further Reading

Suf is a senior advisor in data science with deep expertise in Natural Language Processing, Complex Networks, and Anomaly Detection. Formerly a postdoctoral research fellow, he applied advanced physics techniques to tackle real-world, data-heavy industry challenges. Before that, he was a particle physicist at the ATLAS Experiment of the Large Hadron Collider. Now, he’s focused on bringing more fun and curiosity to the world of science and research online.