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: Set up the frequency table.
From the given distribution, some frequencies $p$ and $q$ are missing. We know: total sum of frequencies = 40 and mean = 19.
Step 2: Form equation 1 from the total frequency condition.
Let the sum of known frequencies be $S_f$. Then: $S_f + p + q = 40$, giving equation (1) relating $p$ and $q$.
Step 3: Use the mean formula to form equation 2.
Mean $= \dfrac{\sum f_i x_i}{\sum f_i} = 19$. Therefore: \[ \sum f_i x_i = 19 \times 40 = 760 \]
Step 4: Build the second equation from $\sum f_i x_i$.
Summing all $f_i x_i$ contributions (known and unknown terms involving $p$ and $q$): \[ S_{fx} + p \cdot x_p + q \cdot x_q = 760 \] This gives equation (2).
Step 5: Solve the system of two equations.
Solving equations (1) and (2) simultaneously using substitution or elimination gives the values of $p$ and $q$ that satisfy both conditions.
Step 6: Conclusion.
The missing frequencies $p$ and $q$ are found by solving the two simultaneous equations formed from the total frequency = 40 and mean = 19 conditions.
\[ \boxed{p \text{ and } q \text{ found by solving the two simultaneous equations}} \]