Question

the data set on the right represents the population.

which formula should be used to calculate the variance? c

what is the variance?

a teacher recorded all of his students grades on the final exam as:
62, 77, 78, 80, 82, 82, 83, 84, 85, 87, 89, 95

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}}\\)

Explanation:

Identify the given values and target

Using the Sample vs Population knowledge point, we identify that the data set represents all of the teacher's students, which constitutes the population.
The given population data set is:

$$ \{62, 77, 78, 80, 82, 82, 83, 84, 85, 87, 89, 95\} $$

The number of data points is \(N = 12\).
Our target is to calculate the population variance, denoted by \(\sigma^2\).

Calculate the population mean

Using the Population Mean Symbol knowledge point

$$ \mu = \frac{\sum x_i}{N} = \frac{62 + 77 + 78 + 80 + 82 + 82 + 83 + 84 + 85 + 87 + 89 + 95}{12} = \frac{984}{12} = 82 $$

Calculate the squared deviations

Using the Population Variance Calculation knowledge point

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Sum the squared deviations

Using the Population Variance Calculation knowledge point

$$ \sum (x_i - \mu)^2 = 400 + 25 + 16 + 4 + 0 + 0 + 1 + 4 + 9 + 25 + 49 + 169 = 702 $$

Calculate the population variance

Using the Population Variance Calculation knowledge point

$$ \sigma^2 = \frac{\sum (x_i - \mu)^2}{N} = \frac{702}{12} = 58.5 $$

Answer:

What is the variance? <blank>58.5</blank>