Question

find the mean, variance, and standard deviation for the data set 6, 15, 9, 2, 12, 9, 14, 4, 17, 2 the mean is \\( \bar{x} = \square \\) (round to two decimal places as needed.) the variance is \\( \sigma^2 = \square \\) (round to two decimal places as needed.) the standard deviation is \\( \sigma \approx \square \\) (round to two decimal places as needed.)

Explanation:

Step1: Calculate the sum of data

$\sum x = 6 + 15 + 9 + 2 + 12 + 9 + 14 + 4 + 17 + 2 = 90$

Step2: Compute the sample mean

$\bar{x} = \frac{\sum x}{n} = \frac{90}{10} = 9.00$

Step3: Find squared deviations

$\sum (x-\bar{x})^2 = (6-9)^2 + (15-9)^2 + (9-9)^2 + (2-9)^2 + (12-9)^2 + (9-9)^2 + (14-9)^2 + (4-9)^2 + (17-9)^2 + (2-9)^2$
$= 9 + 36 + 0 + 49 + 9 + 0 + 25 + 25 + 64 + 49 = 266$

Step4: Calculate population variance

$\sigma^2 = \frac{\sum (x-\bar{x})^2}{n} = \frac{266}{10} = 26.60$

Step5: Compute standard deviation

$\sigma = \sqrt{\sigma^2} = \sqrt{26.60} \approx 5.16$

Answer:

The mean is $\bar{x} = 9.00$
The variance is $\sigma^2 = 26.60$
The standard deviation is $\sigma \approx 5.16$