\setcounter{ExampleCounter}{1}
So far in this chapter, we've summarized data visually.  Now we come to summarizing data numerically.  We'll start by summarizing where the data is centered, using three measures:
\begin{enumerate}
\item Mean
\item Median
\item Mode
\end{enumerate}

\begin{proc}{Mean, Median, and Mode}
\paragraph{Mean or Average:} the sum of all the data points divided by the number of data points.\\

For a sample of size $n$, \[\overline{x} = \dfrac{\sum x}{n}\]

For a population of size $N$, \[\mu = \dfrac{\sum x}{N}\]

\paragraph{Median:} The middle of the data set in order; half of the data points are above it and half below.\\

The median is at position \[\dfrac{n+1}{2}\]

\paragraph{Mode:} The most common data value (the highest bar in the histogram).

If there are two data values tied for most common, the data is \textbf{bimodal}.
\end{proc}
\vfill
\pagebreak

The easiest way to find the mean and median is by using your calculator:
\begin{enumerate}
\item Enter the data by pressing \includegraphics[height=0.3in]{CalcStatButton} \includegraphics[height=0.15in]{CalcEdit}
\item Press \includegraphics[height=0.3in]{CalcStatButton} again and scroll over to the \verb|CALC| menu.
\begin{center}
\includegraphics[width=3in]{Calc3Sec1_2}
\end{center}
\item Press \includegraphics[height=0.3in]{CalcEnterButton} to select \verb|1-Var Stats|
\item Scroll down to \verb|Calculate| on the menu and press \includegraphics[height=0.3in]{CalcEnterButton}
\item Scroll through the resulting list to find $\overline{x}$ and \verb|Med|.
\begin{center}
\includegraphics[width=3in]{Calc1VarStats}
\end{center}
\end{enumerate}
\vfill
\pagebreak

\subsection{From a Frequency Table}
\begin{example}{Mean and Median}
Calculate the mean and median of the data set summarized by the following frequency table.
\begin{center}
\begin{tabular}{c c}
Score & Frequency\\
\hline
\\
68 & 3\\
71 & 2\\
75 & 5\\
77 & 3\\
83 & 4\\
89 & 3
\end{tabular}
\end{center}

\paragraph{Mean} Adding up three 68s, two 71s, etc:
\begin{align*}
&68 + 68 + 68 + 71 + 71 + 75 + \ldots\\ 
&= (68)(3)+(71)(2)+(75)(5)+(77)(1)+(83)(4)+(89)(3)\\ 
&= 1551
\end{align*}

Now divide this total by the total number of data points:
\[\overline{x} = \dfrac{1551}{20} = 77.55\]

\paragraph{Median} Remember, the median occurs at position \[\dfrac{n+1}{2} = \dfrac{20+1}{2} = 10.5\] or between the 10th and 11th data points.

In order, the 10th data point is a 75, and the 11th data point is a 77.  Therefore, the median is the average of these two: \[\textrm{Med} = 76\]
\end{example}
\vfill
\pagebreak

This can be done using calculator as well.  Enter the data as a frequency table:
\begin{center}
\includegraphics[width=3in]{CalcMeanFreqTable}
\end{center}

Then \marginnote{On the TI-83, when you hit enter on \texttt{1-Var Stats}, you'll be taken back to the home screen with that entered.  Type in \texttt{L1, L2} by pressing \texttt{2nd} $\longrightarrow$ \texttt{1} $\longrightarrow$ \texttt{,} $\longrightarrow$ \texttt{2nd} $\longrightarrow$ \texttt{2}}when you select \verb|1-Var Stats|, enter \verb|L2| as the \verb|FreqList| (using \verb|2nd| $\longrightarrow$ \verb|2|).
\begin{center}
\includegraphics[width=3in]{CalcMeanFreqTable2}
\end{center}

\begin{example}{Mean and Median}
Find the mean, median, and mode for the data set summarized by the following frequency table, the frequency of airline no-shows.
\begin{center}
\begin{tabular}{c c}
Number of No-Shows & Frequency\\
\hline
\\
0 & 37\\
1 & 31\\
2 & 20\\
3 & 16\\
4 & 12\\
5 & 4
\end{tabular}
\end{center}

\paragraph{Mean:} $\overline{x} = 1.56$

\paragraph{Median:} Med $= 1$

\paragraph{Mode:} Mode $= 0$
\end{example}


