Question

a contractor records the areas, in square feet, of several houses in a neighborhood to determine data about the neighborhood. which formula should be used to calculate the standard deviation?

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

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

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

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

why does the formula use \\\(n - 1\\)\ in the denominator?

• the data is a sample and is expected to be more dispersed from the mean.
• the data is a sample and is expected to be less dispersed from the mean.
• the data is a population and is expected to be more dispersed from the mean.
• the data is a population and is expected to be less dispersed from the mean.

Explanation:

Identify the data type

Using the Sample vs Population knowledge point
The contractor records the areas of "several houses in a neighborhood" to determine data about the entire neighborhood. Since they only measure a subset of the houses rather than every single house in the neighborhood, the data represents a sample rather than a population.

Determine the standard deviation formula

Using the Sample Standard Deviation Formula knowledge point
Because the data is a sample, we must use the sample standard deviation formula \(s\), which uses the sample mean \(\bar{x}\) and divides by \(n - 1\) inside the square root:

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

Analyze the denominator \(n - 1\)

Using the Sample Standard Deviation Formula knowledge point
The division by \(n - 1\) (known as Bessel's correction) is used because a sample tends to underestimate the true variability of the entire population. The sample data is expected to be less dispersed from the sample mean than the entire population is from the population mean. Dividing by \(n - 1\) instead of \(n\) increases the calculated standard deviation to correct for this bias.

Answer:

Question 1

• (A) \(s^2 = \frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \dots + (x_n - \bar{x})^2}{n - 1}\)
• (B) \(s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \dots + (x_n - \bar{x})^2}{n - 1}}\) (Correct answer)
• (C) \(\sigma^2 = \frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \dots + (x_n - \mu)^2}{n}\)
• (D) \(\sigma = \sqrt{\frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \dots + (x_n - \mu)^2}{n}}\)

Question 2

• (A) The data is a sample and is expected to be more dispersed from the mean.
• (B) The data is a sample and is expected to be less dispersed from the mean. (Correct answer)
• (C) The data is a population and is expected to be more dispersed from the mean.
• (D) The data is a population and is expected to be less dispersed from the mean.