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

Options

A. 50.6
B. 53.3
C. 55.6
D. 59.6

Explanation

To find the variance of all 250 items, we will follow these steps:

### Step 1: Calculate the combined mean of the two groups of items.

The mean of the first group (100 items) is 50, and the mean of the second group (150 items) is 40. The combined mean can be calculated using the formula:

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

Where:
- \( n_1 = 100 \)
- \( \text{mean}_1 = 50 \)
- \( n_2 = 150 \)
- \( \text{mean}_2 = 40 \)

Substituting the values:

\[ \text{Combined Mean} = \frac{(100 \cdot 50) + (150 \cdot 40)}{100 + 150} = \frac{5000 + 6000}{250} = \frac{11000}{250} = 44 \]

### Step 2: Calculate the deviations from the combined mean for both groups.

For the first group:
\[ d_1 = \text{mean}_1 - \text{Combined Mean} = 50 - 44 = 6 \]

For the second group:
\[ d_2 = \text{mean}_2 - \text{Combined Mean} = 40 - 44 = -4 \]

### Step 3: Calculate the variance of each group.

The variance is the square of the standard deviation. The standard deviations are given as follows:
- Standard deviation of the first group (\( \sigma_1 \)) = 5
- Standard deviation of the second group (\( \sigma_2 \)) = 6

Thus, the variances are:
\[ \text{Variance}_1 = \sigma_1^2 = 5^2 = 25 \]
\[ \text{Variance}_2 = \sigma_2^2 = 6^2 = 36 \]

### Step 4: Calculate the combined variance using the formula for combined variance.

The formula for combined variance is given by:

\[ \text{Combined Variance} = \frac{n_1(\text{Variance}_1 + d_1^2) + n_2(\text{Variance}_2 + d_2^2)}{n_1 + n_2} \]

Substituting the values:

\[ \text{Combined Variance} = \frac{100(25 + 6^2) + 150(36 + (-4)^2)}{250} \]

Calculating the terms:

\[ = \frac{100(25 + 36) + 150(36 + 16)}{250} \]
\[ = \frac{100 \cdot 61 + 150 \cdot 52}{250} \]
\[ = \frac{6100 + 7800}{250} \]
\[ = \frac{13900}{250} = 55.6 \]

### Final Step: The variance of all 250 items is:

\[ \text{Variance} = 55.6 \]