Question

If the variance of the numbers $2, 3, 11$ and $x$ is $\frac{49}{4}$, then the values of $x$ are

• A. $6, \frac{14}{3}$
• B. $4, \frac{13}{5}$
• C. $6, \frac{16}{3}$
• D. $6, \frac{14}{5}$

Hint:

Using the computational formula for variance $\frac{\sum x^2}{n} - \mu^2$ is almost always faster than using the definition formula $\frac{\sum(x-\mu)^2}{n}$ when dealing with unknown variables in the dataset.

Answer 1

The Correct Option is A

Step 1: List the numbers.
The four numbers are $2, 3, 11, x$ and their variance is $\frac{49}{4}$.

Step 2: Write the mean.
\[ \bar{x} = \frac{2 + 3 + 11 + x}{4} = \frac{16 + x}{4} \]
Step 3: Use the variance formula.
\[ \text{Var} = \frac{\sum x_i^2}{4} - \bar{x}^2 \] Here $\sum x_i^2 = 4 + 9 + 121 + x^2 = 134 + x^2$.

Step 4: Put values in.
\[ \frac{134 + x^2}{4} - \left(\frac{16 + x}{4}\right)^2 = \frac{49}{4} \]
Step 5: Simplify to a quadratic.
Multiplying out and clearing denominators leads to
\[ 3x^2 - 32x + 56 = 0 \]
Step 6: Solve the quadratic.
Factoring gives the two roots.
\[ x = 6 \quad\text{or}\quad x = \frac{14}{3} \] \[ \boxed{x = 6,\ \frac{14}{3} \text{ (Option 1)}} \]