Calculator Input
Example Data Table
| X | Observed Y | Predicted Y | Residual | Squared Error |
|---|---|---|---|---|
| 1 | 2 | 2.8 | -0.8 | 0.64 |
| 2 | 4 | 3.4 | 0.6 | 0.36 |
| 3 | 5 | 4.0 | 1.0 | 1.00 |
| 4 | 4 | 4.6 | -0.6 | 0.36 |
| 5 | 5 | 5.2 | -0.2 | 0.04 |
Formula Used
The main least squares error formula is:
SSE = Σ(y - ŷ)²
Here, y is the observed value. ŷ is the predicted value.
The calculator also uses these related measures:
MSE = SSE / n
RMSE = √MSE
MAE = Σ|y - ŷ| / n
R² = 1 - SSE / SST
How to Use This Calculator
- Enter X values in the first box.
- Enter observed Y values in the second box.
- Select linear fit for automatic regression.
- Select manual mode when predicted values are known.
- Press the calculate button.
- Review SSE, MSE, RMSE, MAE, and R squared.
- Download the results as CSV or PDF.
Understanding Least Squares Error
What It Measures
Least squares error measures how far predictions are from observed values. Each residual is squared. Then all squared residuals are added. This total is called the sum of squared errors. A smaller value usually means the model fits the data better. The method is widely used in regression, forecasting, calibration, and experimental analysis.
Why Squared Errors Matter
Squaring makes every error positive. It also gives more weight to large mistakes. This helps reveal models that miss important patterns. A model with one large residual can have a high least squares error. That warning is useful when accuracy is important.
Linear Regression Review
This calculator can build a linear model from paired X and Y data. It estimates the best straight line using the least squares method. The line gives predicted values for every X value. The calculator compares observed values against those predictions. It then reports residuals and error metrics.
Manual Prediction Mode
You can also enter your own predicted values. This is helpful when testing another model. You may compare forecasts, machine learning outputs, lab predictions, or fitted equations. The calculator checks each prediction against the actual value and builds a full residual table.
Interpreting the Output
SSE shows total squared error. MSE gives average squared error. RMSE converts error back toward the original scale. MAE shows average absolute error. R squared explains how much variation is captured by the model. These values should be reviewed together.
Best Practice
Do not rely on one metric only. Check residuals for patterns. Large residuals may show outliers, missing variables, or curved relationships. If residuals grow with X values, another model may be better. Use the table to inspect every point clearly. Save the result for reporting or comparison.
FAQs
What is least squares error?
Least squares error is the sum of squared differences between observed and predicted values. It shows how far a model misses the actual data.
Is a lower SSE always better?
A lower SSE usually means a better fit for the same dataset. Compare models only when they use the same observed values.
What is a residual?
A residual is the observed value minus the predicted value. It shows the error for one data point.
What is the difference between SSE and MSE?
SSE adds all squared errors. MSE divides SSE by the number of observations, giving the average squared error.
Why use RMSE?
RMSE is easier to read than MSE because it returns error toward the original data scale.
Can I enter predicted values manually?
Yes. Select manual predicted values mode. Then enter one prediction for each observed Y value.
What does R squared mean?
R squared estimates how much variation is explained by the model. Higher values often show stronger fit.
Can this compare forecasting models?
Yes. Enter actual values and each model’s predictions separately. Then compare SSE, RMSE, MAE, and R squared.