Weighted Reciprocal Regression
Fit the reciprocal model y = a + b / x to frequency-weighted (x, y) data using weighted least squares. Returns the coefficients a and b, the coefficient of determination R², and the total weight, with a scatter chart whose point size reflects each frequency.
Input
Fit the reciprocal model y = a + b / x to frequency-weighted (x, y) data with weighted least squares. Enter one point per line as x,y or x,y,frequency.
One point per line: x,y or x,y,frequency. Separate with commas, spaces, or tabs. x must not be 0 and frequency must be positive.
Result
Regression equation
y = 0.971864 + 10.234396 / x
Constant a
0.971864
Coefficient b
10.234396
R-squared
1
Total weight
15
Data points
5
Scatter plot and fitted curve
Data and fitted values
| No. | x | y | Frequency | Fitted |
|---|---|---|---|---|
| 1 | 1 | 11.2 | 4 | 11.206 |
| 2 | 2 | 6.1 | 5 | 6.089 |
| 3 | 3 | 4.4 | 3 | 4.383 |
| 4 | 4 | 3.5 | 2 | 3.53 |
| 5 | 5 | 3 | 1 | 3.019 |
How it works
- Each data point (x, y) carries a frequency weight w, and the reciprocal model y = a + b / x is fitted. Substituting u = 1 / x turns it into a weighted straight-line regression y = a + b u.
- With weighted means ubar = Sum(w u) / Sum(w) and ybar = Sum(w y) / Sum(w), the coefficients are b = Sum(w (u - ubar)(y - ybar)) / Sum(w (u - ubar)^2) and a = ybar - b ubar.
- The coefficient of determination comes from the weighted total variation SStot = Sum(w (y - ybar)^2) and residual variation SSres = Sum(w (y - yhat)^2) as R-squared = 1 - SSres / SStot.
- At x = 0 the term 1 / x is undefined, so such rows are rejected. Frequencies must be positive.
- Enter each row as x,y or x,y,frequency. Separators can be commas, spaces, or tabs. A row without a frequency is treated as frequency 1.
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Weighted Reciprocal Regression