What a frequency polygon is
A frequency polygon is a line graph that displays a grouped frequency distribution. Instead of drawing a bar for each class the way a histogram does, you plot a single point per class — its class midpoint on the horizontal axis against its frequency on the vertical axis — and join consecutive points with straight line segments. The result is a jagged line whose shape reveals the distribution: where data clusters, whether it is symmetric or skewed, and whether it has one peak or several.
Because each class is reduced to one point, frequency polygons are ideal for comparing two or more distributions on the same axes — something that overlapping histograms make visually cluttered.
How the calculation works
The calculator turns your grouped table into plot points in three steps:
- Class midpoint (class mark): for each interval, midpoint x = (lower boundary + upper boundary) / 2. A class of 20–30 has a midpoint of 25. These midpoints are the x-coordinates of the polygon.
- Frequency: the count f you recorded for that class becomes the y-coordinate.
- Total frequency: Σf is the sum of all the class counts — the size of your dataset.
The calculator also estimates the mean of the grouped data using the standard formula mean ≈ Σ(f·x) / Σf, where each class is represented by its midpoint. This is an estimate: the individual values inside a class are unknown, so every observation in a class is treated as if it sat exactly at the midpoint.
Worked example
Take the classes 0–10, 10–20, 20–30, 30–40, 40–50 with frequencies 3, 7, 12, 8, 5. The midpoints are 5, 15, 25, 35, 45. Total frequency Σf = 3 + 7 + 12 + 8 + 5 = 35. The weighted sum Σ(f·x) = 3·5 + 7·15 + 12·25 + 8·35 + 5·45 = 15 + 105 + 300 + 280 + 225 = 925, so the estimated mean = 925 / 35 ≈ 26.43. The plot points are (5, 3), (15, 7), (25, 12), (35, 8) and (45, 5), with the modal class 20–30.
Closing the polygon
By convention a frequency polygon is anchored to the horizontal axis at both ends. You add an imaginary class with zero frequency one interval below the first midpoint and one interval above the last, then run the line down to those points. In the example above that means extra points at (−5, 0) and (55, 0). This closes the figure and makes the area under the polygon equal to the area of the corresponding histogram.
Frequency polygon vs. histogram vs. ogive
A histogram uses bars whose heights are the frequencies; a frequency polygon connects the midpoints of those bar tops with lines. An ogive (cumulative frequency polygon) plots cumulative frequency against the upper class boundary instead of raw frequency against the midpoint, and always rises. Use a frequency polygon when you want to see the shape of a distribution or compare several distributions; use an ogive when you want percentiles, medians, or "how many fall below this value" answers.