The 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 follow these steps:

### Step 1: Identify the given data
We have three components with the following frequencies and means:
- Frequency \( f_1 = 45 \), Mean \( x_1 = 2 \)
- Frequency \( f_2 = 40 \), Mean \( x_2 = 2.5 \)
- Frequency \( f_3 = 15 \), Mean \( x_3 = 2 \)

### Step 2: Calculate the total frequency
The total frequency \( N \) is the sum of all individual frequencies:
\[ N = f_1 + f_2 + f_3 = 45 + 40 + 15 = 100 \]

### Step 3: Calculate the weighted sum of the means
To find the combined mean, we need to calculate the weighted sum of the means, which is given by:
\[ \text{Weighted Sum} = (f_1 \cdot x_1) + (f_2 \cdot x_2) + (f_3 \cdot x_3) \]
Calculating each term:
- \( f_1 \cdot x_1 = 45 \cdot 2 = 90 \)
- \( f_2 \cdot x_2 = 40 \cdot 2.5 = 100 \)
- \( f_3 \cdot x_3 = 15 \cdot 2 = 30 \)

Now, summing these values:
\[ \text{Weighted Sum} = 90 + 100 + 30 = 220 \]

### Step 4: Calculate the combined mean
The combined mean \( \bar{x} \) is calculated using the formula:
\[ \bar{x} = \frac{\text{Weighted Sum}}{N} \]
Substituting the values we found:
\[ \bar{x} = \frac{220}{100} = 2.2 \]

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

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