\setcounter{ExampleCounter}{1}
Suppose you're applying for law school, so you take the LSAT (the Law School Admission Test), and you score 155. If that's all you know, that number is pretty meaningless. What you \textbf{really} want to know is how well you did \emph{relative to everyone else who took the test}. If I told you that you scored better than 63\% of people who took the LSAT, that would be a much better indication of your success.
Therefore, we often want to know where a particular data point (your LSAT score, your baby's weight, etc.) falls in the data set. To do this, we have several \textbf{measures of position}, two of which we'll look at in this section:
\begin{enumerate}
\item Percentiles
\item Quartiles
\end{enumerate}
These two are closely related, as we'll see.
\subsection{Percentiles}
Percentiles are exactly what was described above: if you scored 155 on the LSAT, you did better than 63\% of test-takers, so we would say that you were in the 63rd percentile for the test.\\
\begin{proc}{Percentiles}
\paragraph{Definition:} The percentile of a data point is the percentage of data points that fall below the given one.
Note: being in the 90th percentile on a test does not mean you scored 90\%.
\end{proc}
\begin{example}{Percentiles: Sleep Time}
Fifty college students were asked how much sleep they get per school night (rounded to the nearest hour). The following frequency table records their results.
\begin{center}
\begin{tabular}{c | c}
Hours of Sleep & Frequency\\
\hline
& \\
4 & 2\\
5 & 5\\
6 & 7\\
7 & 12\\
8 & 14\\
9 & 7\\
10 & 3
\end{tabular}
\end{center}
\pagebreak
\begin{enumerate}
\item Find the 28th percentile.
There are 50 data points, and 28\% of 50 is \[(0.28)(50) = 14.\] Therefore, the lowest 28\% consists of the first 14 data values (up through the last 6).
To be above that, a student would have to get between 6 and 7 hours of sleep, so we say that the 28th percentile is 6.5 hours of sleep.
\item Find the 80th percentile (for them).
Again, 80\% of 50 is \[(0.8)(50) = 40,\] so the lowest 80\% consists of the first 40 data points (up through the last 8). Therefore, the 80th percentile is 8.5 hours of sleep.
\item Find the 40th percentile (for them).
Since 40\% of 50 is \[(0.4)(50) = 20,\] we count up through the first 20 data points, which is in the middle of the 7's. Therefore, the 40th percentile is 7 hours of sleep.
\end{enumerate}
\end{example}
\subsection{Quartiles}
Quartiles are nothing more than specific percentiles that split the data into quarters:
\begin{center}
\begin{tabular}{c c c}
25th percentile & 50th percentile & 75th percentile\\
First quartile & Second quartile & Third quartile\\
$Q_1$ & Median or $Q_2$ & $Q_3$
\end{tabular}
\end{center}
\begin{example}{Physics Exam Scores}
A physics class earned the following scores on an exam:
\begin{center}
\begin{tabular}{c c c c c c c c c c}
47 & 48 & 53 & 56 & 57 & 58 & 60 & 61 & 61 & 62\\
63 & 64 & 71 & 72 & 74 & 75 & 76 & 82 & 89 & 95\\
\end{tabular}
\end{center}
(note that the scores are already ordered; if they weren't, we would have to start by ordering them)
\pagebreak
\begin{enumerate}
\item Find the first quartile.
Remember, the first quartile is the same as the 25th percentile. There are 20 test scores, and 25\% of 20 is \[(0.25)(20) = 4,\] so the 25th percentile is between the 4th and 5th scores: \[56.5\]
\item Find the median (for them).
Halfway through the data set is between the 10th and 11th data points, so take the average of 62 and 63: \[62.5\]
\item Find the third quartile (for them).
Again, 75\% of 20 is \[(0.75)(20) = 15,\] so take the average of the 15th and 16th data points: \[74.5\]
\end{enumerate}
\end{example}
\subsection{Five Number Summary}
The five number summary summarizes a data set by giving the following five statistics:
\begin{center}
Min,\ $Q_1$,\ Median,\ $Q_3$,\ Max
\end{center}
We'll use this in the next section to draw \textbf{Box Plots}.
\begin{example}{Physics Exam Scores}
The five number summary for the test score data set in the previous example is \\
\begin{tabular}{l}
Min $=$ 47\\
$Q_1 = $ 56.5\\
Med $=$ 62.5\\
$Q_3 =$ 74.5\\
Max $=$ 95
\end{tabular}
\end{example}
\vfill
\pagebreak
\subsection{Using Your Calculator}
To find the five number summary on your graphing calculator, start by entering the data into \verb|L1| (press \includegraphics[height=0.3in]{CalcStatButton} \includegraphics[height=0.15in]{CalcEdit} to access the data).
Then press \includegraphics[height=0.3in]{CalcStatButton} again and scroll over to the \verb|CALC| menu along the top.
\begin{center}
\includegraphics[width=2.8in]{CalcCalcMenu}
\end{center}
Select the first option: \verb|1-Var-Stats| and you'll see the following:
\begin{center}
\includegraphics[width=2.8in]{Calc1VarStatsMenu}
\end{center}
Leave everything as is (if you entered your data into a different list than \verb|L1|, you could select that here; if your data as a frequency table, you could select which list to use as the frequency list). Select \verb|Calculate| and you'll get something like
\begin{center}
\includegraphics[width=2.8in]{Calc1VarStats}
\end{center}
There's a lot of information here, but if you scroll down, you'll find \verb|minX|, \verb|Q1|, \verb|Med|, \verb|Q3|, and \verb|maxX|, the five number summary.