If mean of 5 terms= `24/5` & variance of terms = `194/25` & mean of 4 terms = `7/2`, then find the variance of 4 terms?

Options

A. `5/2`
B. `5/3`
C. `5/4`
D. 1

Answer

To solve the problem, we need to find the variance of 4 terms given the mean and variance of 5 terms and the mean of 4 terms. Let's break it down step by step.

### Step 1: Find the sum of the 5 terms
Given that the mean of 5 terms is \( \frac{24}{5} \):
\[ \text{Sum of 5 terms} = \text{Mean} \times \text{Number of terms} = \frac{24}{5} \times 5 = 24 \]

### Step 2: Find the sum of the first 4 terms
Given that the mean of 4 terms is \( \frac{7}{2} \):
\[ \text{Sum of 4 terms} = \text{Mean} \times \text{Number of terms} = \frac{7}{2} \times 4 = 14 \]

### Step 3: Find the 5th term
Using the sums calculated in Steps 1 and 2:
\[ \text{5th term} = \text{Sum of 5 terms} - \text{Sum of 4 terms} = 24 - 14 = 10 \]

### Step 4: Use the variance formula for 5 terms
The variance of the 5 terms is given as \( \frac{194}{25} \). The formula for variance is:
\[ \text{Variance} = \frac{x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2}{5} - \left(\text{Mean}\right)^2 \]
Substituting the known values:
\[ \frac{x_1^2 + x_2^2 + x_3^2 + x_4^2 + 10^2}{5} - \left(\frac{24}{5}\right)^2 = \frac{194}{25} \]

### Step 5: Calculate \( \left(\frac{24}{5}\right)^2 \)
\[ \left(\frac{24}{5}\right)^2 = \frac{576}{25} \]

### Step 6: Substitute and solve for the sum of squares
Substituting back into the variance equation:
\[ \frac{x_1^2 + x_2^2 + x_3^2 + x_4^2 + 100}{5} - \frac{576}{25} = \frac{194}{25} \]
Multiplying through by 5 to eliminate the fraction:
\[ x_1^2 + x_2^2 + x_3^2 + x_4^2 + 100 - \frac{576}{5} = \frac{970}{5} \]
\[ x_1^2 + x_2^2 + x_3^2 + x_4^2 + 100 = \frac{970 + 576}{5} = \frac{1546}{5} \]
\[ x_1^2 + x_2^2 + x_3^2 + x_4^2 = \frac{1546}{5} - 100 = \frac{1546 - 500}{5} = \frac{1046}{5} \]

### Step 7: Find the variance of the 4 terms
The variance of the 4 terms can be calculated as:
\[ \text{Variance} = \frac{x_1^2 + x_2^2 + x_3^2 + x_4^2}{4} - \left(\text{Mean of 4 terms}\right)^2 \]
Substituting the values:
\[ \text{Variance} = \frac{\frac{1046}{5}}{4} - \left(\frac{7}{2}\right)^2 \]
Calculating \( \left(\frac{7}{2}\right)^2 \):
\[ \left(\frac{7}{2}\right)^2 = \frac{49}{4} \]
Now substituting:
\[ \text{Variance} = \frac{1046}{20} - \frac{49}{4} \]
Finding a common denominator (20):
\[ \frac{49}{4} = \frac{245}{20} \]
Thus:
\[ \text{Variance} = \frac{1046 - 245}{20} = \frac{801}{20} \]

### Final Answer
The variance of the 4 terms is:
\[ \frac{801}{20} = 40.05 \]