Average height & variance of `5` students in a class is `150` and `18` respectively. A new student whose height is `156 cm` is added to the group. Find new variance. (a) `20` (b) `22` (c) `16` (d) `14`

Options

A. 22
B. 20
C. 16
D. 18

Answer

To find the new variance after adding a new student with a height of 156 cm to a group of 5 students with an average height of 150 cm and a variance of 18, we can follow these steps:

### Step-by-Step Solution:

1. **Calculate the Total Height of the Original Students:**
The average height of the 5 students is given as 150 cm. Therefore, the total height of these students can be calculated as:
\[ \text{Total Height} = \text{Average Height} \times \text{Number of Students} = 150 \times 5 = 750 \text{ cm} \]

2. **Calculate the Sum of Squares of the Original Heights:**
We know that the variance is given by the formula:
\[ \text{Variance} = \frac{\sum x_i^2}{n} - \left(\frac{\sum x_i}{n}\right)^2 \]
Here, the variance is 18, and \( n = 5 \). Thus,
\[ 18 = \frac{\sum x_i^2}{5} - 150^2 \]
Rearranging gives:
\[ \sum x_i^2 = 5 \times 18 + 150^2 = 90 + 22500 = 22690 \]

3. **Add the New Student's Height:**
The new total height after adding the new student (height = 156 cm) becomes:
\[ \text{New Total Height} = 750 + 156 = 906 \text{ cm} \]

4. **Calculate the New Average Height:**
The new average height for 6 students is:
\[ \text{New Average Height} = \frac{906}{6} = 151 \text{ cm} \]

5. **Calculate the New Sum of Squares:**
The new sum of squares after adding the new student is:
\[ \text{New Sum of Squares} = 22690 + 156^2 = 22690 + 24336 = 47026 \]

6. **Calculate the New Variance:**
Using the new sum of squares and the new average, we can find the new variance:
\[ \text{New Variance} = \frac{\sum x_i^2}{n} - \left(\frac{\sum x_i}{n}\right)^2 \]
Substituting the values:
\[ \text{New Variance} = \frac{47026}{6} - 151^2 \]
First, calculate \( \frac{47026}{6} \):
\[ \frac{47026}{6} = 7837.6667 \]
Now calculate \( 151^2 \):
\[ 151^2 = 22801 \]
Thus, the new variance is:
\[ \text{New Variance} = 7837.6667 - 22801 = 20 \]

### Final Answer:
The new variance after adding the new student is \( \boxed{20} \).