If the arithmetic mean of the following frequency distribution is 21.5, find the value of \( p \).
Show Hint
When solving for unknowns in frequency distribution problems, use the formula for the arithmetic mean and carefully calculate the sums of frequencies and products of frequencies and values.
The arithmetic mean \( \bar{x} \) of a frequency distribution is given by the formula:
\[
\bar{x} = \frac{\sum f_i x_i}{\sum f_i},
\]
where \( f_i \) are the frequencies and \( x_i \) are the corresponding values.
The given arithmetic mean is \( 21.5 \), so:
\[
21.5 = \frac{\sum f_i x_i}{\sum f_i}.
\]
Step 1: Find \( \sum f_i \)
We can find \( \sum f_i \) by adding all the frequencies:
\[
\sum f_i = 6 + 4 + 3 + p + 2 = 15 + p.
\]
Step 2: Find \( \sum f_i x_i \)
Next, calculate \( f_i x_i \) for each value of \( x_i \):
\[
f_1 x_1 = 6 \times 5 = 30, \quad f_2 x_2 = 4 \times 15 = 60, \quad f_3 x_3 = 3 \times 25 = 75, \quad f_4 x_4 = p \times 35 = 35p, \quad f_5 x_5 = 2 \times 45 = 90.
\]
So,
\[
\sum f_i x_i = 30 + 60 + 75 + 35p + 90 = 255 + 35p.
\]
Step 3: Set up the equation for the mean
Using the formula for the mean:
\[
21.5 = \frac{255 + 35p}{15 + p}.
\]
Step 4: Solve for \( p \)
Multiply both sides by \( 15 + p \):
\[
21.5(15 + p) = 255 + 35p.
\]
Expanding both sides:
\[
322.5 + 21.5p = 255 + 35p.
\]
Now, simplify and solve for \( p \):
\[
322.5 - 255 = 35p - 21.5p,
\]
\[
67.5 = 13.5p,
\]
\[
p = \frac{67.5}{13.5} = 5.
\]
Conclusion:
The value of \( p \) is 5.
