Mode of Grouped Data: Formula, Modal Class and Bimodal Data | Class 10 Maths

Quick answer: To find the mode of grouped data, first find the modal class (the class with the highest frequency), then use $\text{Mode} = l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$. Here $l$ is the lower limit of the modal class, $f_1$ its frequency, $f_0$ and $f_2$ the frequencies of the classes before and after it, and $h$ the class size.

The mean is the average of a data set and the median is its middle value. The mode is the value that occurs most often. For example, a shoe company plans production around the most commonly bought size (the mode), not the average foot length.

For ungrouped data, the mode is simply the most repeated value. For grouped data (data in class intervals), we first find the modal class and then use a formula to estimate the mode within that class.

In CBSE Class 10 Mathematics, Chapter 13 (Statistics), finding the mode of grouped data is a common board exam question, often together with the mean and median.

What You Will Learn

• What the mode means and why it is useful
• How to find the modal class in a frequency table
• The formula for the mode of grouped data and what $l$, $f_1$, $f_0$, $f_2$ and $h$ stand for
• An easy way to remember $f_0$, $f_1$ and $f_2$
• A fully solved NCERT problem
• What to do when two classes have the same highest frequency (bimodal data)
• The empirical relationship between mean, median and mode

1. Formula for the Mode of Grouped Data

$\text{Mode} = l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$

Easy Way to Remember $f_0$, $f_1$ and $f_2$

The subscripts go in order, $0 \rightarrow 1 \rightarrow 2$, just like the classes in the table:

• $f_0$: the class before the modal class
• $f_1$: the modal class itself
• $f_2$: the class after the modal class

2. Solved NCERT Problem: Modal Monthly Expenditure

Question

The following data gives the distribution of total monthly household expenditure of $200$ families of a village. Find the modal monthly expenditure of the families.

Solution

Step 1: Find the modal class. The highest frequency is $40$, which belongs to the class $1500 – 2000$. So the modal class is $1500 – 2000$.

Step 2: Write the values.

• Lower limit of the modal class: $l = 1500$
• Frequency of the modal class: $f_1 = 40$
• Frequency of the class before it ($1000 – 1500$): $f_0 = 24$
• Frequency of the class after it ($2000 – 2500$): $f_2 = 33$
• Class size: $h = 2000 - 1500 = 500$

Step 3: Substitute into the formula.

$\text{Mode} = 1500 + \left(\frac{40 - 24}{2(40) - 24 - 33}\right) \times 500$

Step 4: Simplify.

• Numerator: $40 - 24 = 16$
• Denominator: $80 - 24 - 33 = 23$

$\text{Mode} = 1500 + \frac{16}{23} \times 500 = 1500 + \frac{8000}{23} \approx 1500 + 347.83 = 1847.83$

Step 5: Check your answer. The mode should lie inside the modal class. Since $1500 < 1847.83 < 2000$, the answer is reasonable.

Answer: The modal monthly expenditure of the families is approximately ₹1847.83.

3. What If Two Classes Have the Same Highest Frequency?

If two classes share the highest frequency, the data has two modes and is called bimodal. In that case:

• Both classes are modal classes, and you can apply the formula to each one separately.
• If the two classes are next to each other, the formula may not give a meaningful answer, because $f_0$ or $f_2$ equals $f_1$.

Board questions are usually set so that there is one clear modal class, so always look carefully for a single highest frequency first.

4. Empirical Relationship Between Mean, Median and Mode

For distributions that are only slightly skewed, the three measures of central tendency are approximately related by:

$\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}$

The same relationship can be rearranged as:

$3\,\text{Median} = \text{Mode} + 2\,\text{Mean} \qquad \text{Mean} = \frac{3\,\text{Median} - \text{Mode}}{2}$

Example

Question: In a distribution, the mean is $28$ and the median is $30$. Find the mode.

$\text{Mode} = 3(30) - 2(28) = 90 - 56 = 34$

Answer: The mode is $34$.

5. Summary Table

6. Exam Tips and Common Mistakes

Exam Tip: If your answer falls outside the modal class (for example, $2050$ when the modal class is $1500 – 2000$), recheck your arithmetic, especially the denominator.

Exam Tip: Unlike the median, the mode does not need a cumulative frequency column, so don't spend time making one.

Common Mistake: Writing $2f_0$ instead of $2f_1$ in the denominator. It is always twice the frequency of the modal class, $2f_1$.

Frequently Asked Questions

What is the formula for the mode of grouped data?

$\text{Mode} = l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$, where $l$ is the lower limit of the modal class, $f_1$ is its frequency, $f_0$ and $f_2$ are the frequencies of the classes before and after it, and $h$ is the class size.

What is a modal class?

The modal class is the class interval with the highest frequency in a grouped frequency distribution.

What is the empirical formula for the mode?

$\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}$. It is an approximate relationship for distributions that are only slightly skewed.

What is bimodal data?

Data that has two modes, because two values or two classes share the highest frequency, is called bimodal.

Do we need cumulative frequency to find the mode?

No. Cumulative frequency is needed for the median, not for the mode.