Charts & Visual Data Modeling
Frequency Polygon Chart Formula: Class Midpoints, Zero Anchor Bins & Grouped Mean
Construct a true closed Frequency Polygon line/scatter chart: calculate Class Midpoints x_m = (Lower + Upper) / 2, add zero-frequency boundary anchor bins so polygon area equals histogram area, and compute Grouped Mean.
Why a true Frequency Polygon requires Class Midpoints AND two zero-frequency outer anchor bins!
Unlike a column histogram that spans full bin widths [L_k, U_k), a Frequency Polygon connects vertices plotted at each bin’s exact Class Midpoint x_m,k = (L_k + U_k) / 2! Crucially, if you omit an empty lower anchor bin before the first populated class and an empty upper anchor bin after the last populated class, the line hangs floating in mid-air instead of closing against the horizontal axis!
Closing both ends at Frequency = 0 guarantees a remarkable geometric theorem: for equal bin widths w, the total area enclosed under the Frequency Polygon equals N * w — identical to the underlying histogram’s total bar area! Furthermore, SUMPRODUCT((Lower + Upper) / 2, Frequency) / SUM(Frequency) yields the exact grouped-data sample mean directly from the polygon table.
| Bin Role | Lower Bound L_k | Upper Bound U_k | Class Midpoint x_m,k | Frequency f_k | Relative Frequency (%) |
|---|---|---|---|---|---|
| Left Zero Anchor Bin | 0 | 10 | 5.0 | 0 | 0.00% |
| Class 1 | 10 | 20 | 15.0 | 8 | 20.00% |
| Class 2 (Modal Peak) | 20 | 30 | 25.0 | 18 | 45.00% |
| Class 3 | 30 | 40 | 35.0 | 14 | 35.00% |
| Right Zero Anchor Bin | 40 | 50 | 45.0 | 0 | 0.00% |
Class Interval Midpoint x_m = (Lower + Upper) / 2
Computes the horizontal X-coordinate vertex for class [A4, B4) after verifying Upper > Lower.
=IF(B4 <= A4, "Upper Bound Must Exceed Lower Bound", ROUND((A4 + B4) / 2, 2))Grouped Sample Mean from Class Midpoints & Frequencies
Evaluates SUMPRODUCT(((A2:A6 + B2:B6) / 2), C2:C6) / SUM(C2:C6) across all polygon vertices (zero-frequency anchor bins contribute 0 automatically).
=IF(OR(MIN(C2:C6) < 0, SUM(C2:C6) <= 0), "Requires Positive Total Frequency", ROUND(SUMPRODUCT((A2:A6 + B2:B6) / 2, C2:C6) / SUM(C2:C6), 2))Try the example
Use the original values or change them to test your understanding.