Question

Find the mode of the following frequency table : 

Hint:

The Mode should always lie within the modal class (30-40). If your answer is outside this range, re-check your calculations.

Answer 1

Step 1: Identify the modal class.
First, we find the class interval with the highest frequency, which is called the modal class.
From the table:
10–20 → 12
20–30 → 10
30–40 → 15
40–50 → 11
50–60 → 7
60–70 → 5

The highest frequency is 15, corresponding to the class interval 30–40. Therefore, the modal class is 30–40.

Step 2: Write the formula for mode of grouped data.
The formula used is:
Mode \(=\; l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h\)

Where:
l = lower limit of modal class
h = class width
f₁ = frequency of modal class
f₀ = frequency of class preceding modal class
f₂ = frequency of class succeeding modal class

Step 3: Substitute the values.
Modal class = 30–40
\(l = 30\)
\(h = 10\)
\(f_1 = 15\)
\(f_0 = 10\) (frequency of 20–30)
\(f_2 = 11\) (frequency of 40–50)

Substituting into the formula:
Mode \(=\; 30 + \frac{15 - 10}{2(15) - 10 - 11} \times 10\)

Step 4: Simplify the expression.
Mode \(=\; 30 + \frac{5}{30 - 21} \times 10\)
Mode \(=\; 30 + \frac{5}{9} \times 10\)
Mode \(=\; 30 + \frac{50}{9}\)

Step 5: Final calculation.
\(\frac{50}{9} \approx 5.56\)
Mode \(=\; 30 + 5.56\)
Mode \( \approx 35.56\)

Final Answer:
The mode of the given frequency distribution is approximately 35.56.