Histograms and frequency polygons

Large data sets are grouped into class intervals, usually of equal width, such as $10\le h<20$. The frequency is the number of values in each class.

Example. Heights (cm) of 40 plants:

• $10\le h<20$: 6
• $20\le h<30$: 10
• $30\le h<40$: 12
• $40\le h<50$: 8
• $50\le h<60$: 4

The modal class is the class with the highest frequency: $30\le h<40$.

Histogram

• Class boundaries on the horizontal axis, frequency on the vertical axis. This works when the classes have equal widths, which is the case in school questions.
• Bars touch, because the data is continuous.
• Label both axes and give the graph a heading.

Frequency polygon

• Plot the frequency at the midpoint of each class.
• Join the points with straight lines.
• Close the polygon by adding a point with frequency 0 at the midpoint of an empty class at each end.

For the plant data, the midpoints are 15, 25, 35, 45 and 55, and the polygon starts at $(5;0)$ and ends at $(65;0)$.

Use each class midpoint as the value for every item in that class:

$\bar{x}\approx \frac{\sum f\cdot x_{\text{mid}}}{\sum f}$

Example.

$\bar{x}\approx \frac{6(15)+10(25)+12(35)+8(45)+4(55)}{40}=\frac{1340}{40}=33,5\text{ cm}$

This is an estimate, because the actual values inside each class are unknown.

The median class contains the middle value. With 40 values, the 20th and 21st values lie in $30\le h<40$ (cumulative frequencies 6, 16, 28).