Frequency polygon

A frequency polygon plots one point per bin, at the bin's midpoint horizontally and the bin's frequency vertically, then joins consecutive points with straight segments. The bins have to be equal in width, because a height stands for a count inside a fixed-width interval and counts from intervals of different sizes are not comparable. The line is usually anchored down to zero at one empty bin below the first and one above the last, which closes the polygon onto the axis and makes the area it encloses equal to the total area of the histogram bars.

Take 40 exam scores binned by tens: 3 in the 50s, 8 in the 60s, 14 in the 70s, 10 in the 80s, 5 in the 90s. The midpoints are 55, 65, 75, 85 and 95, so the plotted points are $(55, 3)$, $(65, 8)$, $(75, 14)$, $(85, 10)$ and $(95, 5)$, anchored at $(45, 0)$ and $(105, 0)$.

Here is the sentence to avoid: "the line passes through 11 at a score of 70, so 11 students scored 70." Nothing was measured at 70. The segment from $(65, 8)$ to $(75, 14)$ is drawn, not observed, and halfway along it the height is simply $(8 + 14) / 2 = 11$. Even the plotted points claim less than they look like they claim: $(75, 14)$ says 14 scores fell somewhere between 70 and 80, and says nothing about any score equalling 75.

Overlaying two groups on one set of axes is the main reason to draw a polygon instead of a histogram, and it only works when both groups use identical bins. If the groups differ in size, plot relative frequency instead of count, so the 70s point above becomes $14 / 40 = 0.35$.

Everything the polygon shows is inherited from the binning underneath it, so bin width still decides the shape you end up describing.