Frequency Histogram and Polygons
A frequency histogram is a column graph for numerical data in which the columns touch. In NSW Year 8 maths (NESA outcome MA4‑DAT‑C‑01) a frequency polygon is drawn over it by joining the middle of each column top and closing the line down to the axis at both ends.
You will read histograms of single values and of grouped data, find class centres, the modal class, the total and a percentage, and calculate a mean from a histogram. You will also spot a mistake in a histogram’s scale, compare two polygons on the same axes, combine two classes’ histograms and rebuild a histogram from clues. Data includes flathead lengths, parkrun times and screen time. It is part of Year 8 Investigating Data.
Theory
A frequency histogram shows numerical data as columns that touch; each column’s height is the frequency. A frequency polygon joins the middles of the column tops. For \(15\) test scores with a tallest column at \(6\), the mode is \(6\). This Year 8 maths page shows how.
A histogram has the values on the horizontal axis and the frequency on the vertical axis. The columns touch, because the values or classes join with no gaps. (A column graph of categories has gaps.)
Single values: centre each column on its value. Grouped data: each column covers one class of equal width; the class \(20\)–\(30\) holds values from \(20\) up to (but not including) \(30\).
The class centre is halfway across the class: \(\dfrac{\text{lower end} + \text{upper end}}{2}\).
A frequency polygon: put a point at the middle of the top of each column (height \(=\) frequency) and join the points with straight lines. Close it down to the axis at the centre of the empty class on each side; those two end points are not data.
Reading the graph. The total is the sum of the frequencies. The mode (or modal class) has the tallest column. The median comes from a running total of the frequencies, at position \(\dfrac{n + 1}{2}\). For single values, mean \(= \dfrac{\text{sum of (value} \times \text{frequency)}}{\text{total}}\).
Grouped data hides the exact values, so its range can only be estimated. Two polygons on the same axes are compared by where they sit: the one further right has the higher values.
Label it. A title, axis labels with units, and a frequency scale in equal steps.
Class centres:
| Class | Class centre |
|---|---|
| \(10\)–\(20\) | \(15\) |
| \(\$5\)–\(\$10\) | \(\$7.50\) |
| \(2\)–\(4\) km | \(3\) km |
| \(140\)–\(150\) cm | \(145\) cm |
Drawing a histogram and polygon
- Draw the axes: values (or classes) across, frequency up, in equal steps.
- Draw touching columns, each as high as its frequency.
- Mark the middle of each column top and join the points.
- Close the polygon to the axis at the centres of the empty classes at each end; add a title and labels.
Reading statistics
- Total: add the column heights. Mode: the tallest column.
- Median: keep a running total to position \(\dfrac{n + 1}{2}\) (or the two middle positions).
- Mean (single values): sum of value \(\times\) frequency, divided by the total.
(i) the number of students
(ii) the mode
(iii) the range
(iv) the median
(v) the mean.
The column heights are \(1, 2, 3, 4, 2, 1, 2\) for the scores \(3\) to \(9\).
| \(1 + 2 + 3 + 4\) | \(=\) | \(10\) |
| \(10 + 2 + 1 + 2\) | \(=\) | \(15\) |
| \(\text{Range}\) | \(=\) | \(9 - 3 = 6\) |
| \(\text{Median}\) | \(=\) | \(\text{8th score} = 6\) |
| \(\text{Mean}\) | \(=\) | \(\dfrac{90}{15} = 6\) |
(i) \(15\)
(ii) \(6\) (the tallest column)
(iii) \(6\)
(iv) \(6\): with \(15\) scores the median is in position \(\dfrac{15 + 1}{2} = 8\); running totals \(1, 3, 6, 10\) put the 8th score at \(6\).
(v) \(6\)
(i) more than \(6\)
(ii) less than \(6\)?
| \(2 + 1 + 2\) | \(=\) | \(5\) |
| \(\dfrac{5}{15} \times 100\%\) | \(=\) | \(33\dfrac{1}{3}\%\) |
| \(1 + 2 + 3\) | \(=\) | \(6\) |
| \(\dfrac{6}{15} \times 100\%\) | \(=\) | \(40\%\) |
(i) \(33\dfrac{1}{3}\%\)
(ii) \(40\%\). On a frequency polygon of this data the points sit at the same heights, and it closes to the axis at \(2\) and \(10\).
(i) the number of students
(ii) the mode
(iii) the percentage who do \(3\) or more.
| \(3 + 7 + 9 + 4 + 2\) | \(=\) | \(25\) |
| \(\dfrac{4 + 2}{25} \times 100\%\) | \(=\) | \(24\%\) |
(i) \(25\)
(ii) \(2\) activities (tallest column, \(9\))
(iii) \(24\%\)
(i) List the class centres.
(ii) Give the highest point of the polygon.
(iii) Name the modal class.
| \(\dfrac{5 + 10}{2}\) | \(=\) | \(7.5\) |
(i) \(\$7.50\), \(\$12.50\), \(\$17.50\), \(\$22.50\) (steps of \(\$5\))
(ii) \((\$12.50,\ 13)\)
(iii) \(\$10\)–\(\$15\)
(i) the mean
(ii) the median.
| \(0 + 3 + 8\) | \(=\) | \(11\) |
| \(15 + 16 + 10\) | \(=\) | \(41\) |
| \(\text{Mean}\) | \(=\) | \(\dfrac{11 + 41}{20} = 2.6\) |
(i) \(2.6\) days
(ii) \(3\) days: the 10th and 11th values both sit in the column at \(3\) (running totals \(2, 5, 9, 14\)).
(i) the modal class
(ii) the class holding the median
(iii) the percentage who did \(60\) minutes or more.
| \(\dfrac{6 + 3}{40} \times 100\%\) | \(=\) | \(22.5\%\) |
(i) \(20\)–\(40\) min
(ii) \(40\)–\(60\) min (running totals \(5, 19, 31\) reach the 20th and 21st values in the third class)
(iii) \(22.5\%\)
Common pitfalls
Frequently asked questions
What is a frequency histogram?
A graph of numerical data with touching columns, where each column’s height is the frequency.
How do you draw a frequency polygon?
Put a point at the middle of the top of each column, join them with straight lines, and close the shape to the axis half a class beyond each end.
What is the class centre?
The middle of a class: add the two ends and divide by \(2\).
What is the modal class?
The class with the tallest column.
Why do histogram columns touch?
Numerical classes follow on from each other with no gaps between them.