Question

sabine records the daily heights of a random sample of bamboo stalks, in inches. they are:

20, 19, 17, 16, 18, 15, 20, 21

consider the formulas:

a. \\(s^2 = \frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \dots + (x_n - \bar{x})^2}{n - 1}\\)

b. \\(s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \dots + (x_n - \bar{x})^2}{n - 1}}\\)

c. \\(\sigma^2 = \frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \dots + (x_n - \mu)^2}{n}\\)

d. \\(\sigma = \sqrt{\frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \dots + (x_n - \mu)^2}{n}}\\)

which formula should you use for variance?
which formula should you use for standard deviation?

the mean of the data is 18.25.
find the variance.
find the standard deviation.

Explanation:

Identify the sample data and mean

Using the Sample Variance Formula and Sample Standard Deviation Formula knowledge points
The data represents a random sample of \(n = 8\) bamboo stalks:

$$ \{20, 19, 17, 16, 18, 15, 20, 21\} $$

The given sample mean is:

$$ \bar{x} = 18.25 $$

Calculate squared deviations

Using the Sample Variance Formula knowledge point

$$
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$$

Sum the squared deviations

Using the Sample Variance Formula knowledge point

$$ \sum (x_i - \bar{x})^2 = 3.0625 + 0.5625 + 1.5625 + 5.0625 + 0.0625 + 10.5625 + 3.0625 + 7.5625 = 31.5 $$

Calculate the sample variance

Using the Sample Variance Formula knowledge point

$$ s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1} = \frac{31.5}{8 - 1} = \frac{31.5}{7} = 4.5 $$

Calculate the sample standard deviation

Using the Sample Standard Deviation Formula knowledge point

$$ s = \sqrt{s^2} = \sqrt{4.5} \approx 2.12 $$

Answer:

Question 1

The variance is <blank>4.5</blank>.

Question 2

The standard deviation is <blank>2.12</blank>.