Question

find the sample variance and standard deviation.

18, 14, 5, 8, 11

(s^2 = \\) (type an exact answer. type an integer or a decimal.)

(s = \\) (round to the nearest tenth as needed.)

Explanation:

🆕 New Concept Discovered: Sample Variance and Standard Deviation
How spread out data is from the average.

Step 1: Calculate the sample mean

Find the mean (\(\bar{x}\)) of the data set: \(18, 14, 5, 8, 11\).

$$ n = 5 $$

$$ \bar{x} = \frac{18 + 14 + 5 + 8 + 11}{5} = \frac{56}{5} = 11.2 $$

Step 2: Calculate the squared deviations

Subtract the mean from each data value and square the result:

$$ (18 - 11.2)^2 = (6.8)^2 = 46.24 $$

$$ (14 - 11.2)^2 = (2.8)^2 = 7.84 $$

$$ (5 - 11.2)^2 = (-6.2)^2 = 38.44 $$

$$ (8 - 11.2)^2 = (-3.2)^2 = 10.24 $$

$$ (11 - 11.2)^2 = (-0.2)^2 = 0.04 $$

Step 3: Sum the squared deviations

$$ \sum (x - \bar{x})^2 = 46.24 + 7.84 + 38.44 + 10.24 + 0.04 = 102.8 $$

Step 4: Calculate the sample variance

Divide the sum of squared deviations by \(n - 1\):

$$ s^2 = \frac{\sum (x - \bar{x})^2}{n - 1} = \frac{102.8}{5 - 1} = \frac{102.8}{4} = 25.7 $$

Step 5: Calculate the sample standard deviation

Take the square root of the sample variance and round to the nearest tenth:

$$ s = \sqrt{25.7} \approx 5.0695 \approx 5.1 $$

Answer:

$$ s^2 = 25.7 $$

$$ s = 5.1 $$