The mean and standard diviation of 100 items are 50,5 and of 150 items are 40,6 respectively
What is the combined mean of all 250 items?

Options

A. 43
B. 44
C. 45
D. 46

Answer

To find the combined mean of all 250 items, we can use the formula for combined mean:

\[ \text{Combined Mean} (\bar{x}_c) = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2} \]

Where:
- \( n_1 \) = number of items in the first group
- \( \bar{x}_1 \) = mean of the first group
- \( n_2 \) = number of items in the second group
- \( \bar{x}_2 \) = mean of the second group

### Step 1: Identify the values
From the question:
- \( n_1 = 100 \)
- \( \bar{x}_1 = 50 \)
- \( n_2 = 150 \)
- \( \bar{x}_2 = 40 \)

### Step 2: Substitute the values into the formula
Now we can substitute these values into the combined mean formula:

\[ \bar{x}_c = \frac{100 \cdot 50 + 150 \cdot 40}{100 + 150} \]

### Step 3: Calculate the numerator
Calculate \( 100 \cdot 50 \) and \( 150 \cdot 40 \):

\[ 100 \cdot 50 = 5000 \]
\[ 150 \cdot 40 = 6000 \]

Now add these two results:

\[ 5000 + 6000 = 11000 \]

### Step 4: Calculate the denominator
Now calculate the total number of items:

\[ 100 + 150 = 250 \]

### Step 5: Calculate the combined mean
Now substitute the values back into the formula:

\[ \bar{x}_c = \frac{11000}{250} \]

### Step 6: Perform the division
Now perform the division:

\[ \bar{x}_c = 44 \]

### Final Answer
The combined mean of all 250 items is **44**.

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