In classical linear regression, each data point exerts equal influence on the fitted line. Yet in many real-world scenarios, measurements come with varying degrees of reliability. Imagine combining observations from different sensors: some provide precise values, while others are noisy. If we treat them equally, the uncertain readings could distort the overall trend. Weighted least squares (WLS) solves this by allowing each point to contribute in proportion to its assigned weight. The weight might reflect the inverse of measurement variance, the importance of certain samples, or domain-specific considerations.
More formally, we model paired observations \((x_i, y_i)\) with positive weights \(w_i\). The goal is to find coefficients \(a\) and \(b\) for a line \(y = a x + b\) that minimize the weighted sum of squared residuals
.
This expression generalizes the ordinary least squares criterion by attaching a multiplier \(w_i\) to each squared deviation. The optimal solution emerges from setting the partial derivatives of \(S\) with respect to \(a\) and \(b\) to zero. The resulting formulas rely on weighted means and covariances, as shown below.
First compute the total weight \(W = \sum_i w_i\). Then the weighted means are
and .
The weighted covariance and variance become
and .
The slope minimizing \(S\) is
.
The intercept follows as
.
Enter three comma- or space-separated lists: X values, Y values, and corresponding weights. The lists must be equal in length. The calculator parses the numbers, computes weighted means, then applies the formulas above to return the slope and intercept. It also prints the total weight, which can help diagnose input errors.
If your data do not all carry equal importance, WLS offers a principled approach to emphasize those points you trust most. For example, in a physics experiment with repeated measurements, you may know the standard deviation of each reading. Setting \(w_i = 1/\sigma_i^2\) ensures that high-variance points influence the fit less than precise ones. The improved line can yield more accurate predictions and reveal trends that ordinary least squares would obscure.
Suppose we measure temperature at three stations but know that the first sensor is less accurate. We record data \((x, y) = (1,2), (2,2.9), (3,4.1)\) with weights \(w = (1,5,5)\). Running the weighted fit yields coefficients close to \(a = 1.05\) and \(b = 0.9\). The heavily weighted points drive the line so it passes near them, while the lower-weighted first point exerts little influence. A standard unweighted regression would tilt the line downward because that first reading sits below the trend.
The slope indicates the estimated change in \(y\) for a one-unit increase in \(x\), factoring in the weights. The intercept corresponds to the predicted \(y\) value when \(x=0\). If the weights reflect inverse variances, you can show that the resulting estimates are unbiased and have minimal variance among all linear unbiased estimators—a classic result in statistics.
Weighted least squares traces back to Gauss and Legendre, who pioneered the method of least squares in the early nineteenth century. Astronomers needed a way to combine noisy observations to determine planetary orbits. The principle of minimizing weighted errors elegantly solved the problem and paved the way for modern regression analysis. Today WLS appears in econometrics, experimental physics, and any discipline where data quality varies.
Choosing weights can be tricky. If the weights are inaccurate or arbitrary, the fit may be misleading. Additionally, WLS assumes that the variance of each error is proportional to \(1/w_i\). If this assumption fails, alternative approaches like robust regression or generalized least squares may be appropriate. Always examine residual plots to ensure that heteroscedasticity—the change of variance with \(x\)—is properly handled.
You can extend WLS to multiple regression with several independent variables. Matrix notation streamlines the derivation. Let \(X\) be the design matrix, \(W\) a diagonal matrix of weights, and \(y\) a column vector of observations. Then the estimated coefficient vector \(\hat{\beta}\) solves \((X^T W X)\hat{\beta} = X^T W y\). Inverting the weighted normal equations or applying QR decomposition yields the solution. This calculator focuses on the simple case of one predictor, but the ideas are the same.
By experimenting with weights, you can see how they reshape the regression line. This offers insight into the reliability of your data and provides a practical introduction to more advanced statistical modeling.
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