Mode and the Modal Class
A measure of central tendency is one value used to describe the centre of a data set. The three measures in this chapter answer different questions: mode means most common, mean means equal share, and median means middle after sorting. Start with the mode.
Exact values and grouped data answer different questions
raw list
2, 4, 4, 4, 7 → exact mode = 4
grouped intervals
densities = 6, 9 and 7
For classes 0–4, 4–6 and 6–10; individual values are hidden.
conclusion
greatest density 9 → modal class 4 ≤ x < 6
Grouped data cannot reveal an exact mode.
Raw data can reveal an exact mode
- For raw data, the exact observations are visible. For discrete data, the possible numerical values are separate and countable. In either form, the mode is the value with the greatest frequency, where frequency means number of occurrences.
- A data set can have one mode, more than one mode, or no mode.
- Mode is the only one of the three that can describe categorical data: labels such as eye colour rather than numerical measurements.
Worked example
One mode, two modes, no mode
- 3, 5, 5, 5, 8 has mode 5.
- 2, 2, 7, 7, 9 has modes 2 and 7.
- 1, 4, 6, 8 has no mode.
Grouped data reveals only a modal class
Once values are grouped into intervals, their exact values are hidden. You can name the modal class, the interval with the greatest frequency density, but not an exact mode.
Here is the class frequency and is the class width.
- Find each class width. Subtract the lower class boundary from the upper boundary. For values written as 1–4 and rounded to the nearest whole number, the boundaries are 0.5 and 4.5, so the width is 4.
- Calculate each density. Divide frequency by class width.
- Choose the greatest density. That interval is the modal class. On a histogram it is the tallest bar.
Worked example
Largest frequency is not always the modal class
In the figure, the three frequencies are 24, 18 and 28, but their densities are 6, 9 and 7. The class with frequency 18 is tallest because it is much narrower.
Common mistake
Now you try
Three classes have frequencies 40, 22 and 45 for the intervals , and . State the class with the greatest frequency, calculate all three frequency densities, and identify the modal class.
Show worked answer
The greatest frequency is 45 in . The densities are , and .