Result
Regression equation: y = 0.913429 + 1.031546x
| Slope | 1.031546 | Intercept | 0.913429 |
|---|---|---|---|
| R squared | 0.993457 | Adjusted R squared | 0.992366 |
| SSE | 0.614777 | MSE | 0.102463 |
| RMSE | 0.320098 | Weighted RMSE | 0.177559 |
| Weighted MAE | 0.154420 | Weighted MAPE | 2.784200% |
| Standard error of slope | 0.034177 | Standard error of intercept | 0.198609 |
| AIC | -16.527492 | BIC | -16.368609 |
| Degrees of freedom | 6 | Total used weight | 19.500000 |
Prediction
| Prediction x | 9.000000 | Predicted y | 10.197346 |
|---|---|---|---|
| Fit standard error | 0.142502 | Prediction standard error | 0.350385 |
| 95% confidence interval | 9.850533 to 10.544159 | ||
| 95% prediction interval | 9.344600 to 11.050092 | ||
Residual Detail Table
| x | y | Original weight | Used weight | Predicted y | Residual | Weighted residual square | Leverage |
|---|---|---|---|---|---|---|---|
| 1.000000 | 2.100000 | 1.000000 | 1.000000 | 1.944975 | 0.155025 | 0.024033 | 0.273020 |
| 2.000000 | 2.900000 | 1.400000 | 1.400000 | 2.976521 | -0.076521 | 0.008198 | 0.257410 |
| 3.000000 | 4.200000 | 1.800000 | 1.800000 | 4.008068 | 0.191932 | 0.066308 | 0.211517 |
| 4.000000 | 4.800000 | 2.200000 | 2.200000 | 5.039614 | -0.239614 | 0.126313 | 0.162701 |
| 5.000000 | 6.100000 | 2.600000 | 2.600000 | 6.071160 | 0.028840 | 0.002162 | 0.138322 |
| 6.000000 | 6.900000 | 3.000000 | 3.000000 | 7.102707 | -0.202707 | 0.123270 | 0.165741 |
| 7.000000 | 8.400000 | 3.500000 | 3.500000 | 8.134253 | 0.265747 | 0.247175 | 0.280327 |
| 8.000000 | 9.100000 | 4.000000 | 4.000000 | 9.165799 | -0.065799 | 0.017318 | 0.510962 |
Example Data Table
| x | y | weight | Meaning |
|---|---|---|---|
| 1 | 2.1 | 1.0 | Lower confidence observation |
| 4 | 4.8 | 2.2 | Medium confidence observation |
| 8 | 9.1 | 4.0 | Higher confidence observation |
Formula Used
For a weighted line, the model is y = a + bx. Each point has x, y, and weight w.
Sw = Σw, Sx = Σwx, Sy = Σwy, Sxx = Σwx², and Sxy = Σwxy.
Slope b = (SwSxy - SxSy) / (SwSxx - Sx²).
Intercept a = (Sy - bSx) / Sw.
Predicted value is ŷ = a + bx. Residual is e = y - ŷ.
SSE = Σw(y - ŷ)². MSE = SSE / degrees of freedom.
Weighted R² = 1 - SSE / Σw(y - weighted mean y)².
When the zero intercept option is used, b = Σwxy / Σwx².
How To Use This Calculator
Enter one observation per line. Place x first, y second, and weight third. You may separate values with commas, spaces, semicolons, or tabs.
Use larger weights for observations you trust more. Use smaller weights for noisy observations. Keep every weight positive.
Add a prediction x value if you want a fitted y value. Select the confidence level for intervals. Choose decimal places for cleaner reporting.
Press Calculate to view results above the form. Press Download CSV for spreadsheet work. Press Download PDF for a compact report.
Weighted Linear Regression Guide
What This Calculator Does
Weighted linear regression fits a straight line when observations do not have equal reliability. Ordinary regression treats every point the same. Weighted regression lets stronger points guide the line more. This is useful in lab work, calibration, finance, engineering, surveys, and mathematical modeling.
Why Weights Matter
A weight changes the influence of a row. A large weight pulls the fitted line more strongly. A small weight still contributes, but it has less control. This helps when measurements have different precision levels. It also helps when grouped data points represent different sample sizes.
Main Output
The calculator returns the slope, intercept, fitted equation, residuals, and model quality values. The slope shows the expected change in y for one unit of x. The intercept shows the fitted value when x is zero. R squared shows how much weighted variation is explained by the model.
Error Measures
SSE measures the weighted squared error. MSE adjusts this value by degrees of freedom. RMSE converts error back to y units. Weighted MAE gives an average absolute error. MAPE shows average percentage error when y values allow it. These measures make the model easier to judge.
Prediction And Intervals
Enter a new x value to predict y. The calculator also estimates a confidence interval for the fitted mean. It estimates a prediction interval for a future observation. The prediction interval is wider because a new observation includes extra random error.
Advanced Options
You may normalize weights to the sample size. This keeps the relative weight pattern while changing the total scale. You may also force the line through zero. This option is useful only when theory requires no intercept. Do not use it only to improve appearance.
Good Data Practice
Check residuals after calculation. Random residuals suggest a better linear fit. Patterns may show curvature, outliers, or missing variables. Use weights carefully. They should come from measurement precision, frequency, trust, variance, or a clear modeling rule.
FAQs
What is weighted linear regression?
It is linear regression where each observation has its own importance. Higher weights give points more influence. Lower weights give points less influence. The method is useful when data precision is unequal.
What should I enter as weight?
Enter a positive value. Use larger weights for more reliable observations. Use smaller weights for uncertain observations. If all points are equally reliable, use the same weight for every row.
Can weights be zero?
No. This calculator requires positive weights. A zero weight means the row does not contribute. Remove that row instead, or use a very small positive weight only when justified.
What does the slope mean?
The slope shows how much y changes when x increases by one unit. In weighted regression, that change is estimated after applying the selected weights to all observations.
What does weighted R squared mean?
Weighted R squared estimates how much weighted variation in y is explained by the line. A higher value often means a better fit, but residual patterns should still be checked.
When should I force the intercept through zero?
Use it only when theory says y must be zero when x is zero. Do not use it just because it gives a higher fit score or a simpler equation.
Why normalize weights?
Normalization keeps relative weights the same but rescales their total. Coefficients usually stay the same. Standard errors and interval estimates may change because the weight scale changes.
Can I export my results?
Yes. Use the CSV button for spreadsheet analysis. Use the PDF button for a simple report. Both exports use the current form values and selected options.