Question

If the mean of the data 7, 8, 9, 7, 8, 7, \(\lambda\), 8 is 8, then the variance of the data:

• A. 2
• B. \( \frac{7}{8} \)
• C. \( \frac{9}{8} \)
• D. 1

Hint:

Remember to subtract the mean from each data point before squaring the differences and taking the average to find the variance.

Answer 1

The Correct Option is D

We are given the data set: 7, 8, 9, 7, 8, 7, \(\lambda\), 8, and the mean of the data is 8. We need to find the variance of the data. 

Step 1: Calculate the value of \( \lambda \) The mean of a data set is given by the formula: \[ \text{Mean} = \frac{\sum \text{data values}}{n} \] Where \(n\) is the number of data points. For this data set, we have 8 data points, and the mean is given as 8. Thus, \[ \frac{7 + 8 + 9 + 7 + 8 + 7 + \lambda + 8}{8} = 8 \] \[ \frac{54 + \lambda}{8} = 8 \] Multiplying both sides by 8: \[ 54 + \lambda = 64 \] \[ \lambda = 64 - 54 = 10 \] So, \( \lambda = 10 \). 

Step 2: Calculate the variance The variance is given by the formula: \[ \text{Variance} = \frac{\sum (x_i - \mu)^2}{n} \] Where \( x_i \) is each data point and \( \mu \) is the mean (which is 8). Substituting the values: \[ \text{Variance} = \frac{(7-8)^2 + (8-8)^2 + (9-8)^2 + (7-8)^2 + (8-8)^2 + (7-8)^2 + (10-8)^2 + (8-8)^2}{8} \] \[ \text{Variance} = \frac{(-1)^2 + (0)^2 + (1)^2 + (-1)^2 + (0)^2 + (-1)^2 + (2)^2 + (0)^2}{8} \] \[ \text{Variance} = \frac{1 + 0 + 1 + 1 + 0 + 1 + 4 + 0}{8} \] \[ \text{Variance} = \frac{8}{8} = 1 \] Thus, the variance of the data is 1.