A distribution consists of three components with frequencies 45, 40 and 15 having their means 2, 2.5 and 2, respectively. The mean of the combined. distribution is

Options

A. 220
B. 2.2
C. 100
D. 45

Answer

To find the mean of the combined distribution, we can use the formula for the combined mean of different groups. The combined mean \( C_m \) is given by:

\[ C_m = \frac{\sum (f_i \cdot m_i)}{\sum f_i} \]

where:
- \( f_i \) is the frequency of the \( i^{th} \) component,
- \( m_i \) is the mean of the \( i^{th} \) component.

### Step 1: Identify the frequencies and means
We have three components with the following frequencies and means:
- For the first component: \( f_1 = 45 \), \( m_1 = 2 \)
- For the second component: \( f_2 = 40 \), \( m_2 = 2.5 \)
- For the third component: \( f_3 = 15 \), \( m_3 = 2 \)

### Step 2: Calculate the weighted sum of the means
Now we calculate \( f_i \cdot m_i \) for each component:
- For the first component: \( f_1 \cdot m_1 = 45 \cdot 2 = 90 \)
- For the second component: \( f_2 \cdot m_2 = 40 \cdot 2.5 = 100 \)
- For the third component: \( f_3 \cdot m_3 = 15 \cdot 2 = 30 \)

### Step 3: Sum the products
Now, we sum these products:
\[ \sum (f_i \cdot m_i) = 90 + 100 + 30 = 220 \]

### Step 4: Sum the frequencies
Next, we sum the frequencies:
\[ \sum f_i = 45 + 40 + 15 = 100 \]

### Step 5: Calculate the combined mean
Now we can calculate the combined mean \( C_m \):
\[ C_m = \frac{\sum (f_i \cdot m_i)}{\sum f_i} = \frac{220}{100} = 2.2 \]

### Final Answer
The mean of the combined distribution is \( 2.2 \).

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