Definition 3.2.1.
The range is the difference between the highest and lowest data values in a data set.
\begin{equation*}
\text{Range} = \text{highest data value} - \text{lowest data value}
\end{equation*}
Section 3.2 Measures of Variability
Measures of variability (or dispersion) describe the spread of a data set. We will define the most common ones.
Exercise 3.2.2.
The Excel file below shows the effective property tax rate for states in the United States in 2023. (This data is from the Tax Foundation)
Find the range of property tax rates in the United States in 2023.
Definition 3.2.3.
Standard deviation is another measure of variability, and standard deviation is found by determining how much each data item differs from the mean.
Computing the Standard Deviation for a Data Set:
-
Find the mean of the data items
-
Find the deviation of each data item from the mean:\begin{equation*} \text{data item} - \text{mean} \end{equation*}
-
Sum the squared deviations:\begin{equation*} \sum(\text{data item} - \text{mean})^2 \end{equation*}
-
Divide the sum in step 4 by \(n-1\text{,}\) where \(n\) represents the number of data items:\begin{equation*} \frac{\sum(\text{data item} - \text{mean})^2}{n-1} \end{equation*}
-
Take the square root of the quotient in step 5. This value is the standard deviation for the data set.\begin{equation*} \text{Standard deviation}=\sqrt{\frac{\sum(\text{data item} - \text{mean})^2}{n-1}} \end{equation*}or\begin{equation*} s=\sqrt{\frac{\sum(\text{data item} - \text{mean})^2}{n-1}} \end{equation*}
We can also find the standard deviation more easily in Excel:
\begin{equation*}
\boxed{ \text{Excel formula:} \text{ STDEV.S} }
\end{equation*}
-
Low standard deviation means values tend to be close to the mean
-
High standard deviation means values are more spread out.
The formulas above are for finding the sample standard deviation, which we denote by \(s\text{.}\)
The population standard deviation requires us to have information about the entire population, and we would use a different formula to find it. We denote the population standard deviation by \(\sigma\text{.}\)
Exercise 3.2.4.
Find the standard deviation for the effective tax rates in the file from Exercise 3.2.2
Exercise 3.2.5.
Shown below are the means and standard deviations of the yearly returns on two investments from 1926 through 2004.
| Investment | Mean Yearly Return | Standard Deviation |
|---|---|---|
| Small-Company Stocks | \(17.5\%\) | \(33.3\%\) |
| Large-Company Stocks | \(12.4\%\) | \(20.4\%\) |
(a)
Use the means to determine which investment provided the greater yearly return.
(b)
Use the standard deviations to determine which investment had the greater risk. (Explain your answer.)
Definition 3.2.6.
The variance (\(\sigma^2\) or \(s^2\)) measures the variability, or spread, of the data points in a set around the set’s mean.
\begin{equation*}
s^2=\frac{\sum (x_i-\overline{x})^2}{n-1}
\end{equation*}
\begin{equation*}
\boxed{\text{In Excel:} \text{ VAR.S}}
\end{equation*}
(The formulas above are for the sample variance, not the population variance.)
Exercise 3.2.7.
(Donnelly, Your Turn 4)
The data in the file below lists the number of books that seven adults have read during the last 12 months.
external/sheets/BooksRead.xlsx
Calculate the variance and standard deviation.
