Question

Variance of 6, 7, 8, 9 is

• A. \( \frac{1}{4} \)
• B. \( \frac{3}{4} \)
• C. \( \frac{2}{3} \)
• D. \( \frac{1}{3} \)
• E. \( \frac{5}{4} \)

Hint:

To calculate the variance of a set of numbers, first find the mean, then calculate the squared differences from the mean for each number, and finally find the average of these squared differences.

Answer 1

Answer: (E)

To calculate the variance of the numbers 6, 7, 8, 9, we use the formula for variance: \[ \sigma^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2, \] where \( \mu \) is the mean of the numbers and \( n \) is the number of observations.
Step 1: Calculate the mean \( \mu \) The numbers are 6, 7, 8, and 9. First, find the mean: \[ \mu = \frac{6 + 7 + 8 + 9}{4} = \frac{30}{4} = 7.5. \] Step 2: Calculate the squared differences from the mean Now, we calculate the squared differences from the mean for each number: 

Step 3: Find the variance The variance is the average of the squared differences: \[ \sigma^2 = \frac{2.25 + 0.25 + 0.25 + 2.25}{4} = \frac{5}{4} = 1.25. \] 
Thus, the variance is \( \frac{5}{4} \). Therefore, the correct answer is option (E).