3.9: Additional Measures
- Page ID
- 2303
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Learning Objectives
- Compute the trimean
- Compute the geometric mean directly
- Compute the geometric mean using logs
- Use the geometric to compute annual portfolio returns
- Compute a trimmed mean
Although the mean, median, and mode are by far the most commonly used measures of central tendency, they are by no means the only measures. This section defines three additional measures of central tendency: the trimean, the geometric mean, and the trimmed mean. These measures will be discussed again in the section "Comparing Measures of Central Tendency."
Trimean
The trimean is a weighted average of the \(25^{th}\) percentile, the \(50^{th}\) percentile, and the \(75^{th}\) percentile. Letting \(P25\) be the \(25^{th}\) percentile, \(P50\) be the \(50^{th}\)and \(P75\) be the \(75^{th}\) percentile, the formula for the trimean is:
\[Trimean = \dfrac{P25 + 2P50 + P75}{4}\]
As you can see from the formula, the median is weighted twice as much as the \(25^{th}\) and \(75^{th}\) percentiles. Table \(\PageIndex{1}\) shows the number of touchdown (TD) passes thrown by each of the \(31\) teams in the National Football League in the \(2000\) season. The relevant percentiles are shown in Table \(\PageIndex{2}\).
Table \(\PageIndex{1}\): Number of touchdown passes
\[\begin{matrix} 37 & 33 & 33 & 32 & 29 & 28 & 28 & 23\\ 22 & 22 & 22 & 21 & 21 & 21 & 20 & 20\\ 19 & 19 & 18 & 18 & 18 & 18 & 16 & 15\\ 14 & 14 & 14 & 12 & 12 & 9 & 6 & \end{matrix}\]
| Percentile | Value |
|---|---|
| 25 | 15 |
| 50 | 20 |
| 75 | 23 |
The trimean is therefore
\[\dfrac{15 + 2 \times 20 + 23}{4} = \dfrac{78}{4} = 19.5.\]
Geometric Mean
The geometric mean is computed by multiplying all the numbers together and then taking the \(n^{th}\) root of the product. For example, for the numbers \(1, 10\) and \(100\), the product of all the numbers is:
\[1 \times 10 \times 100 = 1,000.\]
Since there are three numbers, we take the cubed root of the product (\(1,000\)) which is equal to \(10\). The formula for the geometric mean is therefore
\[ \large\left(\Pi \, X \right)^{1/N}\]
where the symbol \(Π\) means to multiply. Therefore, the equation says to multiply all the values of \(X\) and then raise the result to the \(1/N\)th power. Raising a value to the \(\dfrac{1}{N}^{th}\) power is, of course, the same as taking the \(N^{th}\) root of the value. In this case, \(1000^{1/3}\) is the cube root of \(1,000\).
The geometric mean has a close relationship with logarithms. Table \(\PageIndex{3}\) shows the logs (base \(10\)) of these three numbers. The arithmetic mean of the three logs is \(1\). The anti-log of this arithmetic mean of \(1\) is the geometric mean. The anti-log of \(1\) is \(10^1 = 10\). Note that the geometric mean only makes sense if all the numbers are positive.
| X | \(\log _{10}(X)\) |
|---|---|
| 1 | 0 |
| 10 | 1 |
| 100 | 2 |
The geometric mean is an appropriate measure to use for averaging rates. For example, consider a stock portfolio that began with a value of \(\$1,000\) and had annual returns of \({13\%, 22\%, 12\%, -5\%, and -13\%}\). Table \(\PageIndex{4}\) shows the value after each of the five years.
| Year | Return | Value |
|---|---|---|
| 1 | 13% | 1,130 |
| 2 | 22% | 1,379 |
| 3 | 12% | 1,544 |
| 4 | -5% | 1,467 |
| 5 | -13% | 1,276 |
The question is how to compute average annual rate of return. The answer is to compute the geometric mean of the returns. Instead of using the percents, each return is represented as a multiplier indicating how much higher the value is after the year. This multiplier is \(1.13\) for a \(13\%\) return and \(0.95\) for a \(5\%\) loss. The multipliers for this example are \({1.13, 1.22, 1.12, 0.95,\: and\; 0.87}\). The geometric mean of these multipliers is \(1.05\). Therefore, the average annual rate of return is \(5\%\). Table \(\PageIndex{5}\) shows how a portfolio gaining \(5\%\) a year would end up with the same value (\(\$1,276\)) as shown in Table \(\PageIndex{4}\).
| Year | Return | Value |
|---|---|---|
| 1 | 5% | 1,050 |
| 2 | 5% | 1,103 |
| 3 | 5% | 1,158 |
| 4 | 5% | 1,216 |
| 5 | 5% | 1,276 |
Trimmed Mean
To compute a trimmed mean, you remove some of the higher and lower scores and compute the mean of the remaining scores. A mean trimmed \(10\%\) is a mean computed with \(10\%\) of the scores trimmed off: \(5\%\) from the bottom and \(5\%\) from the top. A mean trimmed \(50\%\) is computed by trimming the upper \(25\%\) of the scores and the lower \(25\%\) of the scores and computing the mean of the remaining scores. The trimmed mean is similar to the median which, in essence, trims the upper \(49\%\) and the lower \(49\%\) of the scores. Therefore the trimmed mean is a hybrid of the mean and the median. To compute the mean trimmed \(20\%\) for the touchdown pass data shown in Table \(\PageIndex{1}\), you remove the lower \(10\%\) of the scores (\({6, 9,\: and\; 12}\)) as well as the upper \(10\%\) of the scores (\({33, 33,\: and\; 37}\)) and compute the mean of the remaining \(25\) scores. This mean is \(20.16\).