Chapter 4, Problem 1.27PE

Considering the data in the Excel file Home Market Value as a population of homeowners on this street, compute the mean, variance, and standard deviation for each of the variables using a spreadsheet and formulas (4.4), (4.7), and (4.9). Verify your calculations using the appropriate Excel function.

Arithmetic Mean

The average is formally called the arithmetic mean (or simply the mean), which is the sum of the observations divided by the number of observations. Mathematically, the mean of a population is denoted by the Greek letter μ, and the mean of a sample is denoted by $\bar{x}$. If a population consists of N observations x₁, x₂, …, x_N, the population mean, μ, is calculated as

$\mu =\frac{{\displaystyle \underset{i=1}{\overset{N}{\sum }}{x}_i}}{N}$    (4.4)

Variance

A more commonly used measure of dispersion is the variance, whose computation depends on all the data. The larger the variance, the more the data are spread out from the mean and the more variability one can expect in the observations. The formula used for calculating the variance is different for populations and samples.

The formula for the variance of a population is

${\sigma }^2=\frac{{\displaystyle \underset{i=1}{\overset{N}{\sum }}\left(x_i-\mu \right)}}{N}$    (4.7)

where x_i is the value of the ith item, N is the number of items in the population, and μ is the population mean. Essentially, the variance is the average of the squared deviations of the observations from the mean.

Standard Deviation

The standard deviation is the square root of the variance. For a population, the standard deviation is computed as

$\sigma =\sqrt{\frac{{{\displaystyle \underset{i=1}{\overset{N}{\sum }}\left(x_i-\mu \right)}}^2}{N}}$    (4.9)