Question

If the mean of the following frequency distribution is 15, then the value of y is

x	5	10	15	20	25
f	6	8	6	y	5

• A. 8
• B. 7
• C. 10
• D. 9

Answer 1

The Correct Option is C

Given: The mean of the frequency distribution is 15.

Step 1: Understanding the Mean Formula 

The mean is given by:

\[ \text{Mean} = \frac{\sum fx}{\sum f} \]

Where:

• \( f \) = Frequency
• \( x \) = Midpoint of class
• \( fx \) = Product of \( x \) and \( f \)
• \( \sum fx \) = Sum of all \( fx \) values
• \( \sum f \) = Sum of all frequencies

Step 2: Calculating \( \sum fx \) and \( \sum f \)

Calculating \( fx \) for given values:

\[ (5 \times 6) + (10 \times 8) + (15 \times 6) + (20 \times y) + (25 \times 5) \]

\[ = 30 + 80 + 90 + 20y + 125 \]

\[ \sum fx = 325 + 20y \]

Calculating \( \sum f \):

\[ \sum f = 6 + 8 + 6 + y + 5 = 25 + y \]

Step 3: Substituting into Mean Formula

\[ 15 = \frac{325 + 20y}{25 + y} \]

Multiplying both sides by \( (25 + y) \):

\[ 15(25 + y) = 325 + 20y \]

\[ 375 + 15y = 325 + 20y \]

Rearranging:

\[ 375 - 325 = 20y - 15y \]

\[ 50 = 5y \]

\[ y = 10 \]

Final Answer: \( \mathbf{10} \)

Answer 2

To find the value of y in the frequency distribution table, given that the mean is 15, we start by applying the formula for the mean of a grouped frequency distribution, which is:

Mean = Σ(f_i×x_i)/Σf_i

 

Given:

x	5	10	15	20	25
f	6	8	6	y	5

Calculate Σ(f_i×x_i):

• 6×5 = 30
• 8×10 = 80
• 6×15 = 90
• y×20 = 20y
• 5×25 = 125

So, Σ(f_i×x_i) = 30 + 80 + 90 + 20y + 125 = 325 + 20y

Next, calculate Σf_i:

6 + 8 + 6 + y + 5 = 25 + y

Using the mean formula, substitute the values:

15 = (325 + 20y)/(25 + y)

\(\Rightarrow\) Cross multiply to solve for y:

\(\Rightarrow\) 15(25 + y) = 325 + 20y

\(\Rightarrow\) 375 + 15y = 325 + 20y

Rearrange to solve for y:

\(\Rightarrow\) 375 - 325 = 20y - 15y

\(\Rightarrow\) 50 = 5y

Divide both sides by 5:

\(\Rightarrow\) y = 10

The value of y is 10.