Frequency Polygon Calculator

Background.

A grouped frequency table tells you how many observations fall in each interval: 10 to 20, 20 to 30 and so on. A frequency polygon turns that table into a line graph, with one point per class placed above the middle of the interval at a height equal to its count, each joined to its neighbours by straight lines. The shape that emerges shows where the data bunch up, how quickly they thin out on either side, and whether there is more than one peak.

Each point needs two numbers. The vertical one is simply the frequency. The horizontal one is the class midpoint, the average of the lower and upper boundaries, which stands in for every value in the class because the table no longer records where inside the interval each observation fell. That substitution is the polygon's built-in approximation, and it is the same one used to estimate a mean from grouped data.

The page works out one point at a time: enter a class's boundaries and count and it returns the coordinates to plot, so you repeat it for every row of the table. Two extra points finish the graph properly, a class of zero frequency one width below the first class and another one width above the last, so the line starts and ends on the horizontal axis. Closed that way, with equal class widths, the polygon encloses the same area as the histogram of the same data.

Polygons earn their place over histograms when you want to compare distributions. Two sets of bars on one chart hide each other; two lines in different colours do not. For groups of different sizes, plot relative frequencies, each count divided by its group's total, so the shapes share a scale.

What is frequency polygon calculator?

A frequency polygon is a line graph of a grouped frequency distribution. Each class is represented by the point (midpoint, frequency), where the midpoint lies halfway between the class's lower and upper boundaries, and consecutive points are joined with straight segments. It is different from an ogive, or cumulative frequency polygon, which plots running totals against the upper class boundaries rather than individual counts against midpoints.

How to use this calculator.

1. 01Enter the Class lower boundary and Class upper boundary. For whole-number data grouped as 10–19, 20–29, you can use the stated limits or the true boundaries 9.5 and 19.5; the midpoint comes out at 14.5 either way.
2. 02Enter the Class frequency, the number of observations in that class. To compare groups of different sizes, enter the relative frequency (count ÷ total) instead; decimals are accepted.
3. 03If your classes have different widths, enter frequency divided by class width so that wide classes do not look artificially tall.
4. 04Plot the returned x and y, repeat for each class, then add a zero-frequency point one class width beyond each end and join the points in order.

The formula.

Each class becomes the point (x, y) with x = (lower boundary + upper boundary) ÷ 2 and y = frequency. The midpoint is used because, once data are grouped, every observation in the class is treated as if it sat at the centre of the interval. If the observations in a class are spread evenly that is a fair summary; if they cluster towards one end, the polygon is shifted slightly from the true shape of the data. The frequency passes through unchanged, so whatever scale you enter, raw counts, relative frequencies or densities, is the scale plotted. The upper boundary must be greater than the lower one and the frequency cannot be negative, while negative boundaries are allowed for data such as temperatures or profit and loss.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. Introductory Statistics 2e, 2023. Chapters 10–13. Retrieved 2026-08-06.
2. [2]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.
3. [3]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.