For two data sets each of size 5 the variances are given by 4 and 5 and the corresponding means are given to be 2 and 4 respectively. Then find the variance of the combined data.

Answer

To find the variance of the combined data sets, we can follow these steps:

### Step 1: Identify the given data
- Size of each dataset (n1 = n2) = 5
- Variance of the first dataset (σ1²) = 4
- Variance of the second dataset (σ2²) = 5
- Mean of the first dataset (x̄1) = 2
- Mean of the second dataset (x̄2) = 4

### Step 2: Calculate the sum of squares for each dataset
The formula for variance is given by:
\[ \sigma^2 = \frac{\sum x_i^2}{n} - \bar{x}^2 \]

For the first dataset:
\[ 4 = \frac{\sum x_i^2}{5} - 2^2 \]
\[ 4 = \frac{\sum x_i^2}{5} - 4 \]
\[ \sum x_i^2 = 5 \times 8 = 40 \]

For the second dataset:
\[ 5 = \frac{\sum y_i^2}{5} - 4^2 \]
\[ 5 = \frac{\sum y_i^2}{5} - 16 \]
\[ \sum y_i^2 = 5 \times 21 = 105 \]

### Step 3: Calculate the combined variance
The formula for the variance of the combined dataset is:
\[ \sigma^2 = \frac{n_1 \sigma_1^2 + n_2 \sigma_2^2 + n_1(\bar{x}_1^2) + n_2(\bar{x}_2^2)}{n_1 + n_2} - \left(\frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}\right)^2 \]

Substituting the values:
- \(n_1 = n_2 = 5\)
- \(\sigma_1^2 = 4\)
- \(\sigma_2^2 = 5\)
- \(\bar{x}_1 = 2\)
- \(\bar{x}_2 = 4\)

Calculating the combined variance:
\[ \sigma^2 = \frac{5 \cdot 4 + 5 \cdot 5 + 5 \cdot 2^2 + 5 \cdot 4^2}{5 + 5} - \left(\frac{5 \cdot 2 + 5 \cdot 4}{5 + 5}\right)^2 \]
\[ = \frac{20 + 25 + 20 + 80}{10} - \left(\frac{10 + 20}{10}\right)^2 \]
\[ = \frac{145}{10} - (3)^2 \]
\[ = 14.5 - 9 \]
\[ = 5.5 \]

### Step 4: Final result
The variance of the combined data is:
\[ \sigma^2 = 5.5 \]