Kerala KEAM 5 June 2024 Solved Paper

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 meanNow, we calculate the squared differences from the mean for each number: $$(6 - 7.5)^2 = (-1.5)^2 = 2.25, (7 - 7.5)^2 = (-0.5)^2 = 0.25, (8 - 7.5)^2 = (0.5)^2 = 0.25, (9 - 7.5)^2 = (1.5)^2 = 2.25.$$Step 3: Find the varianceThe 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). Quick Tip: 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.