Frequency Polygons GCSE: Drawing and Comparing Distributions

What is a frequency polygon used for?

A frequency polygon gives a quick visual impression of a distribution's shape — whether it is symmetrical, skewed left or right, or peaked. It is particularly powerful when you overlay two polygons on the same diagram to compare two groups side by side (e.g. boys' and girls' test scores).

How do you draw a frequency polygon from a grouped frequency table?

Method at a glance:

1. Find each midpoint. For each class interval, add the lower and upper bounds and divide by 2.
2. Plot the midpoints against the frequencies. Use the midpoint as the x-value and the frequency as the y-value.
3. Join the plotted points with straight lines. Work from left to right.
4. Close the polygon. Join the leftmost point to zero on the left (at the midpoint of the class before the first class) and the rightmost point to zero on the right. At GCSE, many mark schemes accept leaving the polygon open — check the question wording.
5. Label axes and give the diagram a title.

Worked example: drawing a frequency polygon

The table shows the heights of 40 students.

Plot the five points: (155, 5), (165, 14), (175, 12), (185, 7), (195, 2). Join them left to right with straight lines. The distribution peaks at 160–170 cm and tails off to the right — a slight positive skew.

Midpoint calculation check: for 150 ≤ h < 160, midpoint = (150 + 160) ÷ 2 = 155. Apply this to every row before plotting.

How do you compare two distributions using frequency polygons?

Draw both polygons on the same axes using different colours or line styles. Then compare:

• Mode class: the interval with the highest point on each polygon.
• Shape: is one distribution more spread out than the other?
• Centre: which polygon peaks higher up the x-axis (suggesting a higher average)?

Example comparison statement: "The girls' distribution peaks at 165–170 cm, while the boys' peaks at 175–180 cm, suggesting boys in this sample are typically taller. The boys' polygon is also more spread out, indicating greater variation in height."

Always reference both the graph and the context in your written comparison.

What scale should you use on the axes?

• The x-axis starts from the lower boundary of the first class interval (or the midpoint of a "zero frequency" class before it). Use an even scale matching the class width.
• The y-axis starts at zero and extends comfortably above the highest frequency. Use an evenly spaced scale.

Label the x-axis with the quantity and units (e.g. "Height (cm)"), and the y-axis with "Frequency". Unlabelled axes lose marks.

What is the difference between a frequency polygon and a frequency diagram?

A frequency diagram (also called a frequency bar chart) uses bars drawn at the midpoints of each class, with the height of each bar representing the frequency. A frequency polygon connects those midpoints with lines. The polygon is more useful for comparison; the bar chart makes individual frequencies easier to read.

Note: do not connect the midpoints of a frequency polygon to the corners of the class intervals — that is a common error. The line joins the midpoints only.

Frequently asked questions

Can I draw a frequency polygon from an ungrouped frequency table?

Yes — if data is not grouped, the x-values are the actual data values and the y-values are their frequencies. The method is identical: plot (value, frequency) pairs and join them. However, frequency polygons are most common with grouped data at GCSE.

Do I have to close the polygon all the way to the x-axis?

Closing the polygon to zero (x-axis) at both ends turns it into an actual polygon shape, which is why the diagram is called a frequency polygon. Some GCSE mark schemes award full marks without the closing lines, but it is good practice to add them. If the question explicitly says "draw a frequency polygon," close it.

Why do we use midpoints rather than the class boundaries?

The midpoint represents the best single value for the whole class interval. Since we do not know where within each class the data values actually fall, the midpoint is the most reasonable estimate of the "typical" value for that group.

How do I estimate the mean from a frequency polygon?

Read off each midpoint and its frequency from the polygon (or the original table). Multiply each midpoint by its frequency, sum all the products, then divide by the total frequency. This is identical to estimating the mean from a grouped frequency table — the polygon simply gives a visual version of the same information.

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