A frequency polygon carries the same information as a histogram but replaces the bars with a line joining the top centre of each bar. The line is brought down to zero one class width before the first class and one after the last, so the polygon encloses the same area as the histogram. Because it is just a line, several polygons can be drawn on one chart to compare groups – boys and girls, this year and last year – which is awkward with overlapping bars.
For ungrouped data the points sit at each distinct value, and the zero ends are placed one step (the smallest gap between values) outside the data. For grouped data the points sit at class midpoints. A relative frequency polygon uses percentages instead of counts, which lets you compare groups of different sizes.
The cumulative version, called an ogive, plots running totals against upper class boundaries. It always rises from zero to n, and reading across from n/2 gives an estimate of the median; from n/4 and 3n/4, the quartiles.
How to use the Frequency Polygon Calculator
- Data set
- Type of data
- Start of the first class — Lower limit of the first class; must not be above the smallest value.
- Class width
- Graph
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How it works
The data are counted as in a frequency table (ungrouped values or equal-width classes). For a polygon the counts are plotted against values or class midpoints and joined, with a zero point added at each end; for an ogive the cumulative counts are plotted against the upper class boundaries starting from zero at the first lower boundary.
Worked example
Thirty exam scores are grouped in classes of width 10 starting at 50: 50–<60 has 2 scores, 60–<70 has 5, 70–<80 has 10, 80–<90 has 8 and 90–<100 has 5. The midpoints are 55, 65, 75, 85 and 95, so the polygon passes through (45, 0), (55, 2), (65, 5), (75, 10), (85, 8), (95, 5) and (105, 0). The ogive joins (50, 0), (60, 2), (70, 7), (80, 17), (90, 25) and (100, 30); reading across from 15 gives a median a little under 80 – the exact median of the raw scores is 77.5.
When to use each graph
| Graph | Best for |
|---|---|
| Frequency polygon | Seeing the shape of one distribution and overlaying several |
| Relative frequency polygon | Comparing groups of different sizes |
| Ogive | Reading medians, quartiles and “how many are below x” |
Formula
Point i = (class midpoint mᵢ, frequency fᵢ), with extra points (m₁ − w, 0) and (mₖ + w, 0) · Midpoint = (lower + upper) ÷ 2 · Ogive point = (upper boundary, cumulative frequency)
Worked example
Inputs — Data set: 52, 58, 61, 64, 67, 68, 70, 71, 72, 74, 75, 75, 77, 78, 79, 80, 81, 82, 83, 85, 86, 88, 89, 90, 91, 93, 95, 97, 63, 76; Type of data: Grouped – equal-width classes; Start of the first class: 50; Class width: 10; Graph: Frequency polygon.
Results — Plotted points (x, y) = (45, 0) (55, 2) (65, 5) (75, 10) (85, 8) (95, 5) (105, 0); Frequency table = Class Midpoint f Rel. % Cum. f
───────────────────────────────────────────
50 – <60 55 2 6.67% 2
60 – <70 65 5 16.67% 7
70 – <80 75 10 33.33% 17
80 – <90 85 8 26.67% 25
90 – <100 95 5 16.67% 30; Mean = 77.3333; Median = 77.5; Mode = 75 (2 times); Sample standard deviation = 11.3846.
Key terms
- Frequency polygon
- A line graph of frequencies plotted at class midpoints.
- Class midpoint
- The average of a class's lower and upper limits.
- Ogive
- A cumulative frequency polygon.
- Upper class boundary
- The value where one class ends and the next begins.
Frequently asked questions
How do you draw a frequency polygon?
Find the midpoint of each class, plot each midpoint against its frequency, join the points with straight lines, and add a zero point one class width before the first and after the last class.
What is the difference between a frequency polygon and a histogram?
They show the same counts; the histogram uses bars, the polygon a line through the bar tops. Polygons are easier to overlay for comparisons.
What is an ogive?
A cumulative frequency polygon: running totals plotted against upper class boundaries. It rises from 0 to the total number of observations.
How do I find the median from an ogive?
Go across from n/2 on the vertical axis to the curve and down to the horizontal axis. With 30 scores, read across from 15.
Why does the polygon start and end at zero?
Closing it at zero one class outside the data makes the area under the line equal the area of the histogram and shows there are no observations there.
When should I use a relative frequency polygon?
When comparing groups of different sizes – percentages put a class of 30 students and a class of 120 on the same scale.