Mode of Grouped Data and Modal Class Analysis for CBSE Class 10

When a shoe manufacturer decides which shoe sizes to produce in the highest volume, they do not care about the arithmetic mean shoe size (e.g., size $7.34$, which no customer wears!). They want to know the single shoe size purchased by the greatest number of customers. In statistics, the value that appears most frequently in a dataset is called the Mode.

While finding the mode of raw ungrouped data is trivial (simply pick the number with the highest tally), grouped continuous frequency distributions hide individual observations inside class intervals. In CBSE Class 10 Mathematics, Chapter 13 (Statistics) provides the standard formula to identify the Modal Class and interpolate the exact Mode of Grouped Data.

What You Will Learn

• Definition of Mode in statistics
• Why grouped data mode cannot be found by visual inspection alone
• How to locate the Modal Class in a frequency table
• The Mode Formula: $\text{Mode} = l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h$
• Meaning of parameters: $l, h, f_1, f_0, f_2$
• Real-world applications of mode in commerce and healthcare
• Step-by-step solved CBSE board examination problems and common traps

1. What is the Mode?

Definition

The Mode is that measure of central tendency which represents the value of the observation having the maximum frequency in a dataset (i.e., the most common or popular item).

In a grouped frequency distribution, we cannot determine the mode by simply looking at the numbers, because the individual data values are grouped inside class intervals. We can only locate the interval that contains the maximum frequency—called the Modal Class—and then apply a mathematical formula to interpolate the mode within that interval.

2. Locating the Modal Class

The Modal Class is the class interval corresponding to the highest (maximum) frequency in the distribution.

Example: Look at the frequency column. If the frequencies are $6, 11, 21, 23, 14, 5$, the maximum frequency is $23$. The class interval containing $23$ is the Modal Class!

3. The Mode Formula for Grouped Data

Once the modal class is identified, the exact mode is calculated using:

The Mode Formula

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

    Parameters Memory Guide:
    l   = Lower limit of the modal class
    h   = Size of the class interval (assuming equal class sizes)
    f1  = Frequency of the MODAL class (The highest frequency!)
    f0  = Frequency of the class PRECEDING the modal class (f_before)
    f2  = Frequency of the class SUCCEEDING the modal class (f_after)

The $0 - 1 - 2$ Subscript Sequence:

To remember the subscripts without confusion:

• Class before the modal class: frequency $f_0$ (preceding).
• The modal class itself: frequency $f_1$ (maximum).
• Class after the modal class: frequency $f_2$ (succeeding). The numbers line up consecutively in natural order: $0, 1, 2$!

Important: <u>The calculated Mode must ALWAYS lie strictly inside the modal class interval! If your modal class is $35 - 45$ and your calculated answer is $46.8$ or $33.2$, you have made an algebraic error. Use this as an immediate sanity check in board exams!</u>

4. Solved CBSE Board Examination Problems

Solved Example 1: Standard Mode Calculation (NCERT Classic)

Problem: The following table shows the ages of patients admitted in a hospital during a year. Find the mode of the data:

Solution:

1. Identify the Modal Class:
   • Scan the frequency row: the maximum frequency is $23$.
   • Therefore, the Modal Class is $35 - 45$.
2. List All Parameters:
   • Lower limit of modal class: $l = \mathbf{35}$.
   • Class size: $h = 45 - 35 = \mathbf{10}$.
   • Frequency of modal class: $f_1 = \mathbf{23}$.
   • Frequency of preceding class ($25 - 35$): $f_0 = \mathbf{21}$.
   • Frequency of succeeding class ($45 - 55$): $f_2 = \mathbf{14}$.
3. Apply the Mode Formula: $\text{Mode} = l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h$
4. Substitute Numerical Values: $\text{Mode} = 35 + \left( \frac{23 - 21}{2(23) - 21 - 14} \right) \times 10$ $\text{Mode} = 35 + \left( \frac{2}{46 - 35} \right) \times 10 = 35 + \left( \frac{2}{11} \right) \times 10$ $\text{Mode} = 35 + \frac{20}{11} = 35 + 1.818 = \mathbf{36.82\text{ years}}$
5. Sanity Check: $36.82$ lies comfortably inside the modal class interval $35 - 45$.
6. Therefore, <u>the mode of the data is $36.82\text{ years}$</u>.

Solved Example 2: Determining Mode of Family Sizes

Problem: A survey conducted on 20 households in a locality produced the following frequency table for the number of family members in a household:

Find the mode of this data.

Solution:

1. Identify the Modal Class:
   • Maximum frequency is $8$, corresponding to the class $3 - 5$.
   • Modal Class $= 3 - 5$.
2. List Parameters:
   • $l = 3$
   • $h = 5 - 3 = 2$
   • $f_1 = 8$
   • $f_0 = 7$
   • $f_2 = 2$
3. Apply the Mode Formula: $\text{Mode} = 3 + \left( \frac{8 - 7}{2(8) - 7 - 2} \right) \times 2$ $\text{Mode} = 3 + \left( \frac{1}{16 - 9} \right) \times 2 = 3 + \left( \frac{1}{7} \right) \times 2 = 3 + \frac{2}{7} = 3 + 0.286 = \mathbf{3.286}$
4. Therefore, <u>the mode of the family sizes is $3.29$</u>.

5. Summary and Examination Tips

Exam Tip: In questions asking to interpret the mode in real life: State clearly that "Mode represents the most frequent or typical occurrence in the population (e.g., the maximum number of patients admitted were of age $36.82$ years)".

Common Mistake: In the denominator, writing $(f_1 - f_0 - f_2)$ instead of $(2f_1 - f_0 - f_2)$. Remember that $f_1$ is multiplied by $2$ in the denominator!