For two data sets each of size 5, then variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4 respectively the variance of the combined data set is

Options

A. `5/2`
B. `(11)/(2)`
C. `6`
D. `(13)/(2)`

Answer

To find the variance of the combined data set from two separate data sets, we can follow these steps:

### Step 1: Identify the given data
We have two data sets, each of size 5. The variances and means are given as follows:
- Variance of the first data set (Var1) = 4
- Variance of the second data set (Var2) = 5
- Mean of the first data set (Mean1) = 2
- Mean of the second data set (Mean2) = 4

### Step 2: Calculate the total number of observations
Since each data set has 5 observations, the total number of observations (N) in the combined data set is:
\[ N = 5 + 5 = 10 \]

### Step 3: Calculate the sum of the observations
Using the means, we can find the sum of the observations for both data sets:
- Sum of the first data set (Sum1) = Mean1 × Size = 2 × 5 = 10
- Sum of the second data set (Sum2) = Mean2 × Size = 4 × 5 = 20

### Step 4: Calculate the combined mean
The combined mean (Mean_combined) can be calculated as:
\[ \text{Mean_combined} = \frac{\text{Sum1} + \text{Sum2}}{N} = \frac{10 + 20}{10} = \frac{30}{10} = 3 \]

### Step 5: Calculate the sum of squares for both data sets
The formula for variance is:
\[ \text{Variance} = \frac{\text{Sum of squares}}{N} - \text{Mean}^2 \]
We can rearrange this to find the sum of squares:
\[ \text{Sum of squares} = \text{Variance} \times N + \text{Mean}^2 \times N \]

For the first data set:
\[ \text{Sum of squares}_1 = Var1 \times 5 + Mean1^2 \times 5 = 4 \times 5 + 2^2 \times 5 = 20 + 20 = 40 \]

For the second data set:
\[ \text{Sum of squares}_2 = Var2 \times 5 + Mean2^2 \times 5 = 5 \times 5 + 4^2 \times 5 = 25 + 80 = 105 \]

### Step 6: Calculate the total sum of squares for the combined data set
The total sum of squares for the combined data set is:
\[ \text{Total Sum of squares} = \text{Sum of squares}_1 + \text{Sum of squares}_2 = 40 + 105 = 145 \]

### Step 7: Calculate the variance of the combined data set
Now, we can calculate the variance of the combined data set using the total sum of squares:
\[ \text{Variance_combined} = \frac{\text{Total Sum of squares}}{N} - \text{Mean_combined}^2 \]
\[ \text{Variance_combined} = \frac{145}{10} - 3^2 \]
\[ \text{Variance_combined} = 14.5 - 9 = 5.5 \]

### Step 8: Express the variance in fractional form
The variance can be expressed as:
\[ \text{Variance_combined} = 5.5 = \frac{11}{2} \]

### Final Answer
The variance of the combined data set is \( \frac{11}{2} \).

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