Question

The mean of the following distribution is 53. Find the missing frequency p.

Hence, find mode of the distribution.}

Hint:

Double-check the modal class after finding missing frequencies, as it might change depending on the calculated value.

Answer 1

Step 1: Understanding the Concept:
Mean is the weighted average of class marks. Mode is the value with the highest frequency.
Step 2: Key Formula or Approach:
Mean \(\bar{x} = \frac{\sum f_i x_i}{\sum f_i}\)
Mode \(= l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h\)
Step 3: Detailed Explanation:
Calculating Mean and Finding p:

Given Mean \(= 53\).
\[ 53 = \frac{3700 + 50p}{68 + p} \]
\[ 53(68 + p) = 3700 + 50p \implies 3604 + 53p = 3700 + 50p \]
\[ 3p = 3700 - 3604 = 96 \implies p = 32 \]
Calculating Mode:
Since \(p = 32\), the maximum frequency is 32.
Modal Class: 40-60.
\(l = 40, f_1 = 32, f_0 = 15, f_2 = 28, h = 20\).
\[ \text{Mode} = 40 + \left( \frac{32 - 15}{2(32) - 15 - 28} \right) \times 20 \]
\[ \text{Mode} = 40 + \left( \frac{17}{64 - 43} \right) \times 20 = 40 + \frac{17}{21} \times 20 \]
\[ \text{Mode} = 40 + \frac{340}{21} \approx 40 + 16.19 = 56.19 \]
Step 4: Final Answer:
The missing frequency \(p\) is 32 and the mode of the distribution is approximately 56.19.