Question

Find the missing frequencies p and q in the following frequency distribution, when sum of frequencies is 40 and mean is 19 :

Hint:

Double check your final values by verifying the sum: \(7 + 6 = 13\).
Since the sum of missing frequencies matches perfectly with the total count, your algebraic solution is confirmed correct.

Answer 1

Correct Answer: 6

Step 1: Understanding the Question:
We are given a grouped frequency distribution with two missing frequencies, \(p\) and \(q\).
We are given:
- The total sum of all frequencies is 40.
- The arithmetic mean of the distribution is 19.
We need to set up a system of two linear equations and solve for the unknown frequencies \(p\) and \(q\).

Step 2: Key Formula or Approach: 1. The sum of all frequencies is:
\[ \sum f_i = 40 \]
2. The formula for calculating the mean of a grouped frequency distribution is:
\[ \text{Mean } (\bar{x}) = \frac{\sum f_i x_i}{\sum f_i} \]
Where \(x_i\) is the class mark (midpoint) of each class, and \(f_i\) is the frequency.

Step 3: Detailed Explanation:

• Let us first use the total frequency condition:
- Sum of the given frequencies:
\[ \sum f_i = 2 + 5 + 6 + p + 10 + q + 4 = 40 \] \[ 27 + p + q = 40 \] \[ p + q = 40 - 27 \] \[ p + q = 13 \] --- (Equation 1)

• Let us construct a table to calculate \(\sum f_i x_i\):

• Use the given mean (\(\bar{x} = 19\)) to set up the second equation:
\[ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \] \[ 19 = \frac{472.5 + 17.5p + 27.5q}{40} \] \[ 19 \times 40 = 472.5 + 17.5p + 27.5q \] \[ 760 = 472.5 + 17.5p + 27.5q \] \[ 17.5p + 27.5q = 760 - 472.5 \] \[ 17.5p + 27.5q = 287.5 \]

• Multiply the entire equation by 2 to eliminate decimals:
\[ 35p + 55q = 575 \] - Divide by 5 to simplify:
\[ 7p + 11q = 115 \] --- (Equation 2)

• Solve the system of linear equations (Equation 1 and Equation 2):
- From Equation 1, express \(p\) in terms of \(q\):
\[ p = 13 - q \] - Substitute this expression into Equation 2:
\[ 7(13 - q) + 11q = 115 \] \[ 91 - 7q + 11q = 115 \] \[ 91 + 4q = 115 \] \[ 4q = 115 - 91 = 24 \] \[ q = 6 \]

• Find the value of \(p\):
\[ p = 13 - 6 = 7 \]


Step 4: Final Answer:
The missing frequencies are \(p = 7\) and \(q = 6\).