A frequency polygon is a line graph used to show the shape of grouped data. It is built from the same information as a histogram, but instead of drawing bars, it connects points at the class midpoints. This makes patterns such as peaks, spread, and symmetry easier to see.
Frequency polygons matter because they help compare data distributions clearly on one set of axes.
To make a frequency polygon, find the midpoint of each class interval and plot it against that class frequency. Then connect the plotted points with straight line segments, usually adding zero-frequency points before the first class and after the last class to close the shape. A faint histogram behind the polygon can show how the line graph is related to the grouped data.
When two or more frequency polygons are drawn together, differences in center, variability, and overall shape become easier to compare.
Understanding Statistics: Frequency Polygons
Grouped data gives a useful summary, but it hides the individual values. If a class covers marks from sixty to sixty nine, the graph does not show where the students scored within that range. The plotted point represents the whole class, not every value in it.
This is why a frequency polygon is an approximation of the distribution. It is most reliable when the class intervals have equal widths and are chosen sensibly.
Very wide classes can hide important features, such as two separate groups of results. Very narrow classes can make a small data set look jagged simply because of random variation.
The overall shape can tell a story about the data. A high point shows the class containing the greatest number of observations, often called the modal class. A shape that falls away at similar rates on both sides suggests approximate symmetry.
A long tail towards larger values suggests that a few unusually large values may be present. A long tail towards smaller values suggests the reverse. These clues are not proof of a cause.
A peak might result from the way classes were chosen, or from a limited sample. The straight segments are only links between summaries. They do not mean that measurements actually occurred at every position along the line.
Students meet this kind of graph when comparing test marks from two classes, daily temperatures in different months, heights of age groups, or journey times on two routes. The comparison is fair only when the graphs use the same class boundaries and the same axis scale. Sample size matters too.
A class of forty students will usually have larger frequencies than a class of twenty students, even if their patterns are similar. In that case, relative frequency is often better than raw frequency. Relative frequency shows the share of the group in each class, which makes different group sizes easier to compare.
Careful setup prevents most errors. First check that classes do not overlap and have no gaps unless a gap is intended. The labels on a histogram may show class limits, while the polygon points must sit at the centres of those classes.
Keep the horizontal spacing consistent with the class widths. The added end points at zero are drawing aids. They do not represent real observations outside the recorded range.
Label both axes clearly, including units such as minutes, centimetres, or marks. When reading the graph, focus on the location of the peak, the width of the spread, and the tails before making claims about the data. Grouped graphs are powerful summaries, but they should not be used to identify an exact individual value or an exact median.
Key Facts
- Class midpoint = (lower class limit + upper class limit) / 2
- A frequency polygon plots points in the form (class midpoint, frequency).
- Connect consecutive midpoint-frequency points with straight line segments.
- Add one zero-frequency point before the first class and one after the last class to bring the polygon down to the x-axis.
- A frequency polygon uses the same grouped data as a histogram, but it shows the distribution with a connected line.
- Frequency polygons are useful for comparing two or more distributions on the same coordinate axes.
Vocabulary
- Frequency polygon
- A frequency polygon is a line graph that displays grouped data by connecting class midpoint points plotted against their frequencies.
- Class interval
- A class interval is a range of values used to group data, such as 10 to 19 or 20 to 29.
- Class midpoint
- A class midpoint is the center value of a class interval, found by averaging the lower and upper class limits.
- Frequency
- Frequency is the number of data values that fall within a particular class interval.
- Histogram
- A histogram is a graph of grouped numerical data that uses adjacent bars to show frequencies for class intervals.
Common Mistakes to Avoid
- Plotting class limits instead of class midpoints is wrong because a frequency polygon represents each interval by its center point.
- Forgetting to add zero-frequency endpoints is wrong because the graph may look unfinished and may not show the distribution returning to the x-axis.
- Using unequal class widths without noting them is wrong because the visual shape can become misleading when intervals do not represent the same span of values.
- Connecting points out of order is wrong because the line must follow the classes from lowest midpoint to highest midpoint to show the distribution correctly.
Practice Questions
- 1 A grouped data table has class intervals 0 to 9, 10 to 19, 20 to 29, and 30 to 39 with frequencies 3, 8, 12, and 7. Find the class midpoint for each interval and list the points to plot for the frequency polygon.
- 2 For the classes 40 to 49, 50 to 59, 60 to 69, 70 to 79, and 80 to 89, the frequencies are 4, 10, 15, 9, and 2. Draw the frequency polygon, including zero-frequency endpoints.
- 3 Two frequency polygons are drawn on the same axes. Distribution A has one high peak near the middle and low frequencies at both ends, while Distribution B is flatter and spread across many midpoints. Explain which distribution has greater variability and why.