The question asks to find the variance of the given data set: 6, 8, 10, 12, 14, 16, 18, 20, 22, 24.

Calculating the Mean ($\mu$)

First, calculate the mean (average) of the data. The mean is the sum of all observations divided by the number of observations.

• Data points ($x_i$): 6, 8, 10, 12, 14, 16, 18, 20, 22, 24
• Number of observations ($n$): 10
• Sum of observations: $6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 = 150$
• Mean ($\mu$): $ \frac{\sum x_i}{n} = \frac{150}{10} = 15 $

Calculating the Variance ($\sigma^2$)

Variance measures how spread out the data is from its mean. It is calculated as the average of the squared differences from the Mean.

1. Calculate the squared difference for each data point: $(x_i - \mu)^2$.
   • $(6 - 15)^2 = (-9)^2 = 81$
   • $(8 - 15)^2 = (-7)^2 = 49$
   • $(10 - 15)^2 = (-5)^2 = 25$
   • $(12 - 15)^2 = (-3)^2 = 9$
   • $(14 - 15)^2 = (-1)^2 = 1$
   • $(16 - 15)^2 = (1)^2 = 1$
   • $(18 - 15)^2 = (3)^2 = 9$
   • $(20 - 15)^2 = (5)^2 = 25$
   • $(22 - 15)^2 = (7)^2 = 49$
   • $(24 - 15)^2 = (9)^2 = 81$
2. Sum the squared differences: $81 + 49 + 25 + 9 + 1 + 1 + 9 + 25 + 49 + 81 = 330$.
3. Calculate the variance ($\sigma^2$) by dividing the sum of squared differences by the number of observations ($n$). $ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n} $ $ \sigma^2 = \frac{330}{10} = 33 $

Therefore, the variance of the data is 33.