Chapter 4, Problem 1.28PE

Considering the data in the Excel file Home Market Value as a sample of homeowners on this street, compute the mean, variance, and standard deviation for each of the variables using formulas (4.5), (4.8), and (4.10). Verify your calculations using the appropriate Excel function.

The mean of a sample of n observations, x₁, x₂, …, x_n, denoted by $\bar{x}$, is calculated as

$\bar{x}=\frac{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{x}_i}}{n}$    (4.5)

A significant difference exists between the formulas for computing the variance of a population and that of a sample. The variance of a sample is calculated using the formula

$s^2=\frac{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(x_i-\bar{x}\right)}^2}}{n-1}$    (4.8)

where n is the number of items in the sample and $\bar{x}$ is the sample mean. It may seem peculiar to use a different denominator to “average” the squared deviations from the mean for populations and samples, but statisticians have shown that the formula for the sample variance provides a more accurate representation of the true population variance. We discuss this more formally in Chapter 6. For now, simply understand that the proper calculations of the population and sample variance use different denominators based on the number of observations in the data.

and for samples, it is

$s=\sqrt{\frac{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(x_i-\bar{x}\right)}^2}}{n-1}}$    (4.10)