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

Options

A. 7.1
B. 7.3
C. 7.5
D. 7.7

Explanation

To find the combined standard deviation of the two groups of items, we can follow these steps:

### Step 1: Calculate the combined mean
The mean of the first group (n1 = 100) is 50, and the mean of the second group (n2 = 150) is 40.

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

Substituting the values:

\[ \text{Combined Mean} = \frac{(100 \times 50) + (150 \times 40)}{100 + 150} \]

Calculating the numerator:

\[ = \frac{5000 + 6000}{250} = \frac{11000}{250} = 44 \]

### Step 2: Calculate the deviations from the combined mean
For the first group:
- Mean = 50, Combined Mean = 44
- Deviation (d1) = 50 - 44 = 6

For the second group:
- Mean = 40, Combined Mean = 44
- Deviation (d2) = 40 - 44 = -4

### Step 3: Calculate the squares of the deviations
Now we calculate the squares of the deviations:

\[ d_1^2 = 6^2 = 36 \]
\[ d_2^2 = (-4)^2 = 16 \]

### Step 4: Use the formula for combined standard deviation
The formula for combined standard deviation (SD) is:

\[ SD = \sqrt{\frac{n_1 \times SD_1^2 + n_2 \times SD_2^2 + n_1 \times d_1^2 + n_2 \times d_2^2}{n_1 + n_2}} \]

Where:
- \(SD_1 = 5\) (Standard deviation of the first group)
- \(SD_2 = 6\) (Standard deviation of the second group)

Substituting the values:

\[ SD = \sqrt{\frac{100 \times 5^2 + 150 \times 6^2 + 100 \times 36 + 150 \times 16}{250}} \]

Calculating each term:

\[ = \sqrt{\frac{100 \times 25 + 150 \times 36 + 3600 + 2400}{250}} \]
\[ = \sqrt{\frac{2500 + 5400 + 3600 + 2400}{250}} \]
\[ = \sqrt{\frac{13900}{250}} = \sqrt{55.6} \]

### Step 5: Calculate the final standard deviation
Calculating the square root:

\[ SD \approx 7.45 \]

### Final Answer
Thus, the combined standard deviation of all 250 items is approximately **7.45**.

---