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%TCIDATA{<META NAME="Title" CONTENT="Measures of Central Tendency">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 16:48:47 1997}
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\section{\label{Statistics, mean}Measures of Central Tendency}

You can compute ordinary measures of central tendency with \textsl{%
Scientific Notebook}. Several of these, such as mean, median, and mode, are
items on the \textsf{Statistics} submenu. Others, such as the geometric mean
and harmonic mean, can be computed directly from the definition using the
computational abilities of \textsl{Scientific Notebook}.

\subsection{\ Arithmetic Mean\label{Mean}}

The \textsl{mean} 
\index{Mean@Mean}(arithmetic mean 
\index{Statistics@Statistics!mean@mean}, average 
\index{Average@Average}\label{Average}) of the numbers $x_{1},x_{2},\ldots
,x_{n}$ is the most commonly used measure of central tendency. It is the sum
of the numbers divided by the number of numbers. 
\[
\dfrac{\sum_{i=1}^{n}x_{i}}{n} 
\]

\begin{quote}
$\blacktriangleright $ To find the mean of the numbers in a list
\end{quote}

\begin{enumerate}
\item[1.]  Place the insertion point in the list.

\item[2.]  From the \textsf{Statistics} submenu choose \textsf{Mean}.
\end{enumerate}

\begin{quote}
$\blacktriangleright $ \textsf{Statistics +} \textsf{Mean}
\end{quote}

\begin{quotation}
$a,b,c$, Mean(s): $.33a+.33b+.33c$

$23,5,-6,18,23,-22,5$, Mean(s): $6.5714$

$16.5,22.1,6.9,14.2,9.0$, Mean(s): $13.74$
\end{quotation}

Applying \textsf{Mean} from the \textsf{Statistics} submenu to a matrix
gives the means of the columns. Applying \textsf{Mean} again, this time to
the list of column means, gives the mean of the matrix entries.

\begin{quote}
$\blacktriangleright $ \textsf{Statistics + Mean, Statistics + Mean}
\end{quote}

\begin{quotation}
$\left[ 
\begin{array}{ccc}
23 & 5 & -6 \\ 
18 & 23 & -22 \\ 
5 & 0 & 0
\end{array}
\right] $ , Mean(s): $15.33,9.333,-9.333$, Mean(s): $5.111\vspace{6pt}$

$\left[ 
\begin{array}{ccc}
\text{x} & \text{y} & \text{z} \\ 
1 & 1 & 4 \\ 
3 & 2 & 5 \\ 
5 & 3 & 6 \\ 
7 & 4 & 7
\end{array}
\right] $, Mean(s): $4.0,\allowbreak 2.5,\allowbreak 5.5$, Mean(s): $4.0$

$\left[ 
\begin{array}{cc}
a & b \\ 
c & d \\ 
f & g
\end{array}
\right] $, Mean(s): $.5c+.5f,\allowbreak .5d+.5g$, Mean(s): $%
.25c+.25f+\allowbreak .25d+.25g$
\end{quotation}

Notice that these final two matrices were interpreted as labeled matrices,
so the first row was ignored.\medskip

\subsection{\ Median\label{Median} 
\index{Statistics@Statistics!median@median}\label{Statistics, median}}

A \textsl{median} 
\index{Median@Median} of a finite list of numbers is a number such that at
least half the numbers in the set are equal to or less than it, and at least
half the numbers in the set are equal to or greater than it. If two
different numbers satisfy this criteria, their midpoint is taken as the
median. You do not have to arrange the numbers in increasing order before
computing the median. Leave the insertion point in a list or set of data,
and from the \textsf{Statistics} submenu, choose \textsf{Median}.

\begin{quote}
$\blacktriangleright $ \textsf{Statistics + Median}
\end{quote}

\begin{quotation}
$1,5,2$, Median(s): $2$

$1,2,3,4$, Median(s): $%
\frac{5}{2}$

$2,3,3,3$, Median(s): $3$

$23,5,-6,18,23,-22,5,7$, Median(s): $6$
\end{quotation}

\strut

For a matrix, you obtain the medians of the columns. The second of the
following matrices is interpreted as a labeled matrix, and the first row is
ignored.

\begin{quote}
$\blacktriangleright $ \textsf{Statistics + Median}
\end{quote}

\begin{quotation}
$\left[ 
\begin{array}{ccc}
23 & 5 & -6 \\ 
18 & 23 & -22 \\ 
5 & 0 & 0
\end{array}
\right] $, Median(s): $\left[ 18,5,-6\right] $\quad \quad $\left[ 
\begin{array}{cc}
a & b \\ 
1 & 2 \\ 
5 & 6 \\ 
3 & 4
\end{array}
\right] $, Median(s): $\left[ 3,4\right] \medskip $
\end{quotation}

\strut

\subsection{\ Quantile\label{Quantile}}

The $q$th quantile of a set, where $q$ is a number 
\index{Statistics@Statistics!quantile@quantile}\label{Statistics, quantile}%
between zero and one, is a 
\index{Quantile@Quantile} 
\index{Percentile@Percentile}\label{Percentile}number $Q$ satisfying the
condition that the fraction $q$ of the numbers falls below $Q$ and the
fraction $1-q$ lies above $Q$. The $.5$th quantile is a median or $50$th
percentile, whereas the $.25$th quantile is a first quartile or $25$th
percentile, and so forth. Take the $q$th quantile of a matrix to find the $q$%
th quantiles of the columns.

\begin{quote}
$\blacktriangleright $ \textsf{Statistics + Quantile}
\end{quote}

\begin{quotation}
$1,2,3,4,5,6,7,8,9,10$, $.\,\allowbreak 87$th Quantile(s): $%
\frac{87}{10}\medskip $

$\left\{ 5.6,7,8.3,57,1.4,37,2\right\} $, $.\,\allowbreak 25$th Quantile(s): 
$1.\,\allowbreak 85\medskip $

$\left[ 
\begin{array}{ccc}
23 & 5 & -6 \\ 
9 & -3 & 7 \\ 
18 & 23 & -22
\end{array}
\right] $, $.\,\allowbreak 5$th Quantile(s): $\left[ \frac{27}{2}%
,1,-14\right] \medskip $

$\left[ 
\begin{array}{ccc}
23 & 5 & -6 \\ 
18 & 23 & -22 \\ 
5 & 0 & 0
\end{array}
\right] $, $.\,\allowbreak 33$th Quantile(s): $\left[ 5,0,-22\right]
\medskip $

$\left[ \frac{1137}{100},\frac{49}{20},-\frac{354}{25}\right] $, $%
.\,\allowbreak 75$th Quantile(s): $\frac{117}{25}\medskip $
\end{quotation}

\strut

The value of a quantile may vary according to different conventions. In 
\textsl{Scientific Notebook}, the quantile is interpreted according to the
algorithm implemented by Maple V, Release 4.\medskip

\subsection{\ Mode 
\index{Mode@Mode}\label{Mode}\label{Statistics, mode}}

A \textsl{mode} 
\index{Statistics@Statistics!mode@mode} is a value that occurs with maximum
frequency. To find the modes of a list of numbers, leave the insertion point
in the list, and from the \textsf{Statistics} submenu choose \textsf{Mode}.

\begin{quote}
$\blacktriangleright $ \textsf{Statistics + Mode}
\end{quote}

\begin{quotation}
$23,5,-6,18,23,-22,5$ , Mode(s): $5,23$

$1,1,5,5,5,7,7,8,9,9,9$ , Mode(s): $5,9$
\end{quotation}

With a matrix, \textsl{Scientific Notebook} gives the mode or modes for each
column.

\begin{quote}
$\blacktriangleright $ \textsf{Statistics + Mode}
\end{quote}

\begin{quotation}
$\left[ 
\begin{array}{ccc}
23 & 5 & -6 \\ 
18 & 23 & -22 \\ 
5 & 23 & 0
\end{array}
\right] $ , Mode(s): $\left[ 5,18,23\right] ,\left[ 23\right] ,\left[
0,-6,-22\right] 
\vspace{6pt}$

$\left[ 
\begin{array}{cc}
a & b \\ 
3 & 4 \\ 
1 & 2
\end{array}
\right] $, Mode(s): $\left[ 1,3\right] ,\left[ 2,4\right] $
\end{quotation}

The previous matrix is interpreted as a labeled matrix, and the modes
returned are the modes of the matrix $\left[ 
\begin{array}{cc}
3 & 4 \\ 
1 & 2
\end{array}
\right] .\medskip $

\subsection{\ Geometric Mean 
\index{Geometric mean@Geometric mean}\label{Geometric mean}}

The \textsl{geometric mean }of $n$ nonnegative numbers $x_{1},x_{2},\ldots
,x_{n}$ is the $n$th root of the product of the numbers 
\[
\sqrt[n]{x_{1}x_{2}\cdots x_{n}} 
\]

\begin{quote}
$\blacktriangleright $ To find the geometric mean of $\left\{
3,56,14,2\right\} $
\end{quote}

\begin{enumerate}
\item[1.]  Enter the product $3\times 56\times 14\times 2$.

\item[2.]  Select the product and click 
\FRAME{itbpF}{0.3009in}{0.3009in}{0.0701in}{}{}{radical.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";display "PICT";valid-file
"F";width 0.3009in;height 0.3009in;depth 0.0701in;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'RADICAL.WMF';file-properties "XNPEU";}}.

\item[3.]  With the insertion point immediately inside the radical sign,
press the \textsc{tab} key (or select the radical and click \FRAME{itbpF}{%
0.3009in}{0.3009in}{0.0701in}{}{}{property.wmf}{\special{language
"Scientific Word";type "GRAPHIC";display "PICT";valid-file "F";width
0.3009in;height 0.3009in;depth 0.0701in;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'Property.wmf';file-properties "XNPEU";}}.

\item[4.]  Fill in the small box with $4$ for the fourth root.

\item[5.]  Leave the insertion point in the expression.

\item[6.]  Choose \textsf{Evaluate Numerically}
\end{enumerate}

\begin{quotation}
$\sqrt[4]{3\times 56\times 14\times 2}=\,8.2816\medskip $
\end{quotation}

\subsection{\ Harmonic Mean\label{Harmonic mean}}

The \textsl{harmonic mean } 
\index{Harmonic mean@Harmonic mean}of $n$ positive numbers $%
x_{1},x_{2},\ldots ,x_{n}$ is the reciprocal of the mean of the reciprocals. 
\[
\frac{n}{\sum_{i=1}^{n}\frac{1}{x_{i}}} 
\]
The harmonic mean can be used in averaging speeds, where the distances
applying to each speed are the same.

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate Numerically}
\end{quote}

\begin{quotation}
Harmonic mean of $2,4,6,8$:

$\qquad \qquad \frac{4}{\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}}%
=\,3.84\bigskip $

Harmonic mean of $0.67,1.9,6.2,5.8,4.7$:

$\qquad \qquad 5\left( \left( 0.67\right) ^{-1}+\left( 1.9\right)
^{-1}+\left( 6.2\right) ^{-1}+\left( 5.8\right) ^{-1}+\left( 4.7\right)
^{-1}\right) ^{-1}=\,1.9491$
\end{quotation}

\strut

\begin{example}
If you average $20$ mph driving from your home to a friend's home and $30$
mph driving back home over the same route, then your ``average'' speed for
the round trip is the harmonic mean 
\[
\frac{2}{\frac{1}{20}+\frac{1}{30}}=\,24\text{ mph} 
\]
This computation gives the speed that you would have to travel if you did
the round trip at a constant speed, taking the same total amount of
time.\medskip
\end{example}

\subsection{\ Related topics}

\begin{itemize}
\item  \hyperref{Menu Items, Lists, and Matrices}{}{}{DM11-1.tex#Statistics}

\item  \hyperref{Measures of Dispersion}{}{}{DM11-3.tex#Mean deviation}

\item  \hyperref{Distributions and Densities}{}{}{DM11-4.tex#Distribution}

\item  \hyperref{Families of Continuous Distributions}{}{}{%
DM11-5.tex#Continuous distributions}

\item  \hyperref{Families of Discrete Distributions}{}{}{DM11-6.tex#Discrete
distributions}

\item  \hyperref{Random Numbers}{}{}{DM11-7.tex#Statistics, random numbers}

\item  \hyperref{Curve Fitting}{}{}{DM11-8.tex#Curve fitting}

\item  \hyperref{Exercises and Solutions}{}{}{DM11-9.tex#Exercises}
\end{itemize}

\rule{0.67in}{0.01in}

\hyperref{\FRAME{itbpF}{0.5275in}{0.3399in}{0.0519in}{}{}{index.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";display "PICT";valid-file
"F";width 0.5275in;height 0.3399in;depth 0.0519in;original-width
36.875pt;original-height 22.5625pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'index.wmf';file-properties "XNPEU";}}}{}{}{%
DMIndex.tex#Math Index}\quad {\small Index entries: \hyperref{Data}{}{}{%
DMIndex.tex#Data}, \hyperref{Matrices}{}{}{DMIndex.tex#Matrices}, \hyperref{%
Reshape}{}{}{DM8-1.tex#Reshaping}, \hyperref{Statistics}{}{}{%
DMIndex.tex#Statistics}}

\end{document}
