The modal class and the mode

What you will be able to do

• find the modal class of a grouped table;
• find the mode of grouped data with the formula, also when the modal class is the first or the last class;
• say what to do when two classes tie, and when the class widths are unequal.

The most common value

A shoe shop sells these sizes in one morning: $5, 6, 6, 7, 7, 7, 8, 8, 10$. The owner wants to stock the most popular size. The size that occurs most often is $7$. It is the mode. The mean of the sizes is $7.11$, which is not even a size that the shop can order. The mode is the average that tells the shop what to order. If two sizes tie, the data have two modes and are called bimodal.

The modal class

When the data are grouped, we cannot see which single value occurs most often. We can only see which class holds the most values. That class is the modal class. Here is how fast $40$ students typed in a test.

The largest frequency is $12$, so the modal class is $40$ to $50$ words per minute. Many questions only ask for this class. A better answer is a single number inside the class.

The formula for the mode

Look at the modal class and its two neighbours. The mode is not always the middle of the class. If the class after it has more students than the class before it, the mode leans towards the class after. Here is the formula that does this:

• $l$ is the lower limit of the modal class.
• $h$ is the class width.
• $f_1$ is the frequency of the modal class.
• $f_0$ is the frequency of the class before the modal class.
• $f_2$ is the frequency of the class after the modal class.

For the typing table, $l = 40$, $h = 10$, $f_1 = 12$, $f_0 = 6$ and $f_2 = 9$:

The class mark is $45$, and the mode is above it, because the class after ($9$) is bigger than the class before ($6$).

Why the formula works

Draw the histogram. Join the top right corner of the bar before the modal bar to the top right corner of the modal bar. Then join the top left corner of the modal bar to the top left corner of the bar after it. The two lines cross. The mode is the horizontal position of the crossing.

A little algebra shows why. Let $t$ be the distance of the crossing from $l$, measured in class widths. The first line rises from height $f_0$ to $f_1$, and the second line falls from $f_1$ to $f_2$. At the crossing both lines have the same height, so $f_0 + (f_1 - f_0)t = f_1 + (f_2 - f_1)t$. Solving gives $t = \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2}$. Multiply by $h$ and add $l$.

When the modal class is at an end

If the modal class is the first class, there is no class before it. Take $f_0 = 0$. If it is the last class, take $f_2 = 0$. Here are the days that $30$ students were absent in a term.

The modal class is $0$ to $10$, with $f_1 = 14$, $f_0 = 0$ and $f_2 = 9$. Then

Two modal classes, and classes of different width

A tie. If two classes share the largest frequency, the data are bimodal. Report both modal classes. Do not use the formula to give one mode, because it would hide that the data have two peaks.

Unequal class widths. The formula needs classes of equal width. In an Indian Class 10 histogram the classes are equal, so the height of a bar is the frequency. In A-Level, IGCSE and IB books the area of a bar stands for the frequency, and the height is the frequency density, which is the frequency divided by the class width. The modal class is then the class with the largest frequency density, which may not be the class with the most values.

Inclusive classes. For classes such as $31$ to $40$, use the boundaries $30.5$ and $40.5$. So $l = 30.5$ and $h = 10$.

What other books say. IGCSE, GCSE and IB questions on grouped data ask for the modal class and do not use the formula. The formula is a feature of the Indian syllabus. A-Level books also give the modal class.

Practise with new numbers

Try it in Python

This program finds the modal class of the typing table and then the mode. It also handles a modal class at either end, by using $0$ for a missing neighbour.

Common mistakes

• Swapping $f_0$ and $f_2$. The top line is $f_1 - f_0$. The class before the modal class has $f_0$.
• Using the class mark as the mode. The mode is the class mark only when $f_0 = f_2$.
• Forgetting the end classes. A missing neighbour has frequency $0$, not "nothing to use".
• Using $31$ instead of $30.5$. For inclusive classes, work with the class boundaries.

Remember these

• The modal class has the greatest frequency.
• $\text{mode} = l + \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h$.
• The mode leans towards the neighbouring class with the larger frequency.
• With two equal largest frequencies the data are bimodal. The formula needs equal class widths.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A table has the frequencies $3, 8, 15, 11, 4$ in five classes. Which class is the modal class?

2. For a modal class $20$ to $30$ with $f_1 = 15$, $f_0 = 8$ and $f_2 = 11$, find the mode.

3. When is the mode of a modal class exactly equal to its class mark?

4. What value do you use for $f_0$ when the first class is the modal class?

5. Two classes of a table both have the largest frequency. What should you report?

References

• NCERT, Mathematics for Class 10, the chapter Statistics: the mode of grouped data.
• OpenStax, Introductory Statistics, Chapter 2 Descriptive Statistics: the mode, and skewness.