\chapter{Statistics} \label{chap:statistics}   
   
\Citation{There are three kinds of lies: lies, damned lies and statistics.\\
  Attributed to Benjamin \textsc{Disraeli}\footnotemark\\
  \auteur{Mark}{Twain}{1835}{1910}{American writer}{Autobiography}{1924}\footnotemark} \setcounter{footnote}{1}
\footnotetext{Benjamin \textsc{Disraeli} \pron{d£iz£hr£ql£i} \Detantatant{1804}{1881} 1st Earl of Beaconsfield
  \pron{£hb£Ik£enzf£Ild}, British Conservative politician and novelist; Prime Minister, \oldstylenums{1868},
  \detantatant{1874}{1880}.}\stepcounter{footnote}\footnotetext{`Mark Twain' is the pseudonym \pron{£hsj£Ud£en£im} of
  Samuel Langhorne \textsc{Clemens}.}
   
\dicodef{Statistics} \pron{st£e£ht£ist£iks} \textbf{1} (\textsl{usually treated as singular}) the science of collecting 
and analysing numerical \pron{nj£U£hmer£ikl} data, especially in or for large quantities, and usually inferring 
proportions in a whole from proportions in a representative sample. \textbf{2} any systematic collection or presentation 
of such facts. 
   
\section{Introduction}   
   
Every day we make decisions that may be personal, business related, or of some other kind.  Usually these decisions are 
made under conditions of uncertainty. Many times, the situations or problems we face in the real world have no precise 
or definite solution. 
   
Statistical \pron{st£e£ht£ist£ikl} methods help us to make scientific and intelligent decisions in such situations. 
Decisions made by using statistical methods are called \nuvo{educated guesses}.  Decisions made without using 
statistical (or scientific) methods are pure guesses and, hence, may prove to be unreliable. 
   
Like almost all fields of study, statistics has two aspects: \emph{theoretical} \pron{£f£Q\-£hre\-t£ikl} and 
\emph{applied}. Theoretical or mathematical statistics deals with the development, derivation, and proof of statistical 
theorems, formulas, rules, and laws. Applied statistics involves the applications of those theorems, formulas, rules, 
and laws to solve real-world problems. 
   
Broadly speaking, applied statistics can be divided into two areas: \nuvo{descriptive} \pron{d£i£hskr£ipt£iv} statistics 
and \nuvo{inferential} \pron{£b£inf£e£hren£sl} statistics. Descriptive statistics consists of methods for organising, 
displaying, and describing data by using tables, graphs, and summary \pron{£hs£vm£er£i} 
measures.\index{statistics!descriptive}\index{statistics!inferential} 
   
Suppose we have information on the test scores of students enrolled in a statistics class. In statistical terminology, 
the whole set of numbers that represents the scores of students is called a \emph{data}\,\footnote{\emph{plural of} 
  datum \pron{£hd£qt£em}.}  \pron{£hd£qt£e} \emph{set}, the name of each student is called an \nuvo{element} 
\pron{£hel£im£ent} or \nuvo{individual} \pron{£b£ind£i£hv£i£d£Rl}, and the score of each student is called an 
\nuvo{observation} \pron{£b£obz£e£hv£q£sn}.\index{data!set}\index{datum\cfnote}\index{set!data} 
   
A data set in its original form is usually very large. Consequently, such data set is not very helpful in drawing 
conclusions or making decisions. It is easier to draw conclusions from summary tables and diagrams \pron{£hd£p£egr£am} 
than from the original version of a data set. Therefore, we reduce data to a manageable size by constructing tables, 
drawing graphs, or calculating summary measures such as averages \pron{£h£av(£e)r£i£d}. 
   
In statistics, the collection of all elements of interest is called a \nuvo{population} or \nuvo{universe}. The 
selection of a few elements from this population is called a \nuvo{sample} \pron{£hs£Ampl}. 
   
A major portion of statistics deals with making decisions, inferences, predictions, and forecasts about populations 
based on results obtained from samples. The area of statistics that deals with such decision-making procedures is 
referred to as inferential statistics. This branch of statistics is also called inductive \pron{£in£hd£vkt£iv} reasoning 
\pron{£hr£Iz(£e)n£i£n} or inductive statistics. 
   
A population consists of all elements --- individuals, items \pron{£h£pt£em}, or objects --- whose characteristics are 
being studied. The population that is being studied is also called the target \pron{£ht£Ag£it} population. 
   
\enlargethispage{\baselineskip}
The collection of information from the elements of a population or a sample is called a \nuvo{survey} \pron{£hs£Ev£q}. A 
survey that includes every element of the target population is called a \nuvo{census} \pron{£hsens£es}. Often the size 
of the target population is large. Hence, in practice, a census is rarely taken because it is very expensive and time 
consuming. In many cases, it is even impossible to identify each element of the target population. Usually, to conduct a 
survey, we select a sample and collect the required information from the elements included in that sample. We then make 
decisions based on this sample information. Such a survey is called a \emph{sample survey}.\index{survey!sample} 
   
The purpose of conducting a sample survey is to make decisions about the corresponding population. It is important that 
the results obtained from a sample survey closely match the results that we would obtain by conducting a census. 
Otherwise, any decision based on a sample survey will not apply to the corresponding population. 
   
A sample that represents the characteristics of the population as closely as possible is called a \emph{representative} 
\pron{£brepr£i£hzent£et£iv} \emph{sample}.\index{sample!representative} 
   
A sample may be \nuvo{random} \pron{£hr£and£em} or non-random. In a \emph{random sample}, each element of the population 
has some chance of being included in the sample. However, in a non-random sample this may not be the case. If the chance 
of being selected is the same for each element of the population, the sample is called a \emph{simple random 
  sample}.\index{sample!random}\index{sample!simple random} 
   
A sample may be selected with or without \nuvo{replacement}. In \emph{sampling with replacement}, each time we select an 
element from the population, we put it back in the population before we select the next one. Thus, in sampling with 
replacement, the population contains the same number of items each time a selection is made. Consequently, we may select 
the same item more than once in such a sample.\index{sampling!replacement!with} 
   
\emph{Sampling without replacement} occurs when the selected element is not replaced in the population. In this case, 
each time we select an item, the size of the population is reduced by one element. Thus, we cannot select the same item 
more than once. Most of the times, samples taken in statistics are without 
replacement.\cite{mann:1995}\index{sampling!replacement!without} 
   
\subsection{Basic terms}   
   
An \nuvo{element} or \nuvo{member} of a sample or population is a specific subject or object (for example, a person, 
firm, item, state, or country) about which the information is collected.  A \nuvo{variable} is a characteristic under 
study that assumes different values for different elements.  The value of a variable for an element is called an 
\nuvo{observation} or \nuvo{measurement}.  A \emph{data set} is a collection of observations on one or more 
variables.\index{data!set}\index{set!data} 
   
\enlargethispage{-\baselineskip} 
   
\subsection{Types of variables}   
   
A variable that can be measured numerically \pron{nj£U£hmer£ik(£e)l£i} is called a \nuvo{quantitative} 
\pron{£hkw£ont£it£et£iv} variable. The data collected on a quantitative variable are called quantitative data.%
\index{data!quantitative}\index{variable!quantitative} 
   
Incomes, heights, gross sales, prices of homes, number of cars owned, and accidents are examples of quantitative
variables since each of them can be expressed numerically. Such quantitative variables can be classified as either
\nuvo{discrete} \pron{d£i£hskr£It} variables or \nuvo{continuous} \pron{k£en£ht£inj£u£es}
variables.\index{variable!discrete}\index{variable!continuous}
   
The values that a certain quantitative variable can assume may be countable or not. For example, we can count the number 
of cars owned by a family but we cannot count the income of the family. The variable the values of which are countable 
is called a discrete variable.  There are no possible intermediate values between consecutive values of a discrete 
variable. 
   
A continuous variable can assume any numerical value over a certain interval or intervals. 
   
The time taken to complete an examination is an example of a continuous variable because it can assume any value, let us 
say, between 30 and 60 minutes. The time taken may be $42{.}6$ minutes, $42{.}67$ minutes, or $42{.}674$ minutes. 
(Theoretically, we can measure time as precisely as we want.) Similarly, the height of a person can be measured to the 
tenth of an inch or to the hundredth of an inch. However, neither time nor height can be counted in a discrete fashion. 
Note that any variable that involves money is considered a continuous variable. 
   
Variables that cannot be measured numerically but can be divided into different categories \pron{£hk£at£eg£er£i} are 
called \nuvo{categorical} \pron{£bk£at£e£hg£or£ikl} or \nuvo{qualitative} \pron{£hkw£ol£it£et£iv} variables.  The data 
collected on such variables are called qualitative data.%
\index{category}\index{variable!qualitative}\index{variable!categorical}\index{data!qualitative} 
   
Based on the time over which they are collected, data can be classified as either \nuvo{cross-section} or 
\nuvo{time-series} data. Data collected on different elements at the same point of time or for the same period of time 
are called cross-section data. The information on incomes of 100 families for the year 1994 is an example of 
cross-section data.\index{data!cross-section} 
   
Data collected on the same element for the same variable at different points in time or for different periods of time 
are called time-series data. Information on GB exports for the years 1975 to 1993 is an example of time-series 
data.\index{data!time-series} 
   
When data are collected, the information obtained from each member of a population or sample is recorded in the sequence 
in which it becomes available. This sequence of data recording is random and unranked. Such data are called \nuvo{raw} 
\pron{r£O} \emph{data}.\index{data!raw} 
   
\section{Organising and graphing qualitative data}   
   
\subsection{Frequency distributions}   
   
A sample of 100 students enrolled at a university were asked what they intended to do after graduation. Forty-four said 
they wanted to work for private companies; 16 said they wanted to work for the federal government; 23 wanted to work for 
state or local governments; and 17 intended to start their own businesses. 
   
The table $1$ lists the type of employment and the number of students who intend to engage in each type of employment. 
In this table, the variable is the type of employment, which is a qualitative variable. The categories listed in the 
first column of the table are mutually exclusive. In other words, each of the 100 students belongs to one and only one 
of these categories. 
   
The number of students who belong to a certain category is called the \nuvo{frequency} \pron{£hfr£Ikw£ens£i} of this 
category.  A \emph{frequency distribution} \pron{£bd£istr£i£hbj£U£sn} exhibits how the frequencies are distributed over 
various categories.  The next table is called a \emph{frequency distribution table} or simply a \emph{frequency 
  table}.\index{table!frequency}%
\index{distribution!frequency!table}\index{distribution!frequency}\index{frequency!table}\index{table} 
   
\begin{longtable}{l c}   
  \multicolumn{2}{l}{\bfseries\sffamily Table 1}\\ \hline\hline 
  Type of employement & Number of students\\ 
  Private companies & 44\\ 
  Federal government & 16\\ 
  State or local government & 23\\ 
  Own business & 17\\ 
  & $\text{Sum} = 100$\\ \hline 
\end{longtable}   
   
\begin{Def}   
  A \emph{frequency distribution} for qualitative data lists all categories and the number of elements that belong to 
  each of the categories. 
\end{Def}   
   
\subsection{Relative frequency and percentage distributions}   
   
\begin{Def}   
  The \emph{relative frequency} of a category is obtained by dividing the frequency of that category by the sum of all 
  frequencies.\index{frequency!relative} 
\end{Def}   
   
Thus, the relative frequency shows what fractional part or proportion of the total frequency belongs to the 
corresponding category. A relative frequency distribution lists the relative frequencies for all categories. 
   
   
\begin{Def}   
  The \nuvo{percentage} for a category is obtained by multiplying the relative frequency of that category by 100. A 
  percentage distribution lists the percentages for all categories. 
\end{Def}   
   
A sample was taken of 25 high school seniors who were planning to go to college. Each of the students was asked which of 
the following majors he or she intended to choose: business, economics, management information systems (MIS), behavioral 
sciences (BS), other. The responses are given in the table below. 
   
   
\begin{longtable}{l c c >{\hfill}p{1.6cm} p{.6cm}}   
  \multicolumn{4}{l}{\bfseries\sffamily Table 2}\\ \hline\hline 
  Major & Frequency ($f$)& Relative &  & \\ 
  &                & Frequency & \multicolumn{2}{c}{Percentage}\\ 
  Business & 6 & $.24$ & 24& \\ 
  Economics & 3 & $.12$ & 12& \\ 
  MIS & 6 & $.24$ & 24& \\ 
  BS & 2 & $.08$ & 8& \\ 
  Others & 8 & $.32$ & 32& \\ 
  & $\text{sum} = 25$ &$\text{sum} = 1.00$ & \multicolumn{2}{c}{$\text{sum} = 100$}\\ \hline 
\end{longtable}   
   
   
\subsection{Graphical representation of qualitative data}   
   
All of us have heard the saying `a picture is worth a thousand words'. A graphic display can reveal at a glance the main 
characteristics of a data set. The \nuvo{bar graph} \pron{£hb£A£bgr£Af} and the \nuvo{pie chart} \pron{£hp£p£b£t£At} are two 
types of graphs used to display qualitative data. 
   
\subsubsection{Bar graph}   
   
To construct a bar graph (also called a \nuvo{bar chart}), we mark the various categories on the horizontal axis as in 
figure \ref{chap:statistics}--1. Note that all categories are represented by intervals of the same width. 
   
\enlargethispage{\baselineskip}
We mark the frequencies on the vertical axis.  Then we draw one bar for each category such that the height of the bar 
represents the frequency of the corresponding category. We leave a small gap between adjacent bars. 
   
\begin{fgre}
  \IG{stat01}
\end{fgre}
   
The bar graphs for relative frequency and percentage distributions can be drawn simply by marking the relative 
frequencies or percentages, instead of the frequencies, on the vertical axis. 
   
\subsubsection{Pie chart}   
   
A pie chart is more commonly used to display percentages, although it can be used to display frequencies or relative 
frequencies. The whole pie (or circle) represents the total sample or population. The pie is divided into different 
portions that represent the percentages of the population or sample belonging to different categories. 
   
\begin{fgre}
  \IG{stat02}
\end{fgre}
   
\section{Organising and graphing quantitative data}   
   
The next table gives the heights (in inches) of a random sample of 30 National Basketball Association players.  The 
first column of this table lists the classes, which represent the quantitative variable, the height. 

\newpage
   
For quantitative data, an interval that includes all the values that fall within two numbers, the \emph{lower} and 
\emph{upper limits} (or \emph{class boundaries} \pron{£hb£Wn\-d(£e)\-r£i}) is called a \nuvo{class}.  Note that the 
classes always represent a variable. As we can observe, the classes are non-overlapping; that is, each value belongs to 
one and only one class. We have to understand that 72--74 stands for \intferouv{72}{75} as the lower limit of the next 
class is 75. So the boundaries of the first class are 72 and \textbf{75}. 
   
The third column in the table lists the number of players who have their heights within each class.  For example, three 
players have an height of 72 inches to less than 75 inches. The numbers listed in the third column of this table are 
called the \emph{frequencies}. The sum of all the frequencies is denoted by $§Sf$.  The frequencies are denoted by 
$f$.\index{boundary}\index{frequency}\index{limit!lower}\index{$\Sigma f$} \index{limit!upper} 
   
\begin{longtable}{c c r<{\quad}}   
  \multicolumn{3}{l}{\bfseries\sffamily Table 3}\\ \hline\hline 
  {Height} & class & \multicolumn{1}{c}{Frequency} \\ 
  {(in inches)} & midpoint &\multicolumn{1}{c}{($f$)}\\ 
  72--74& 73.5& 3 \\ 
  75--77& 76.5& 5 \\  % rectification du 2003-04-06
  78--80& 79.5& 7 \\  % rectification du 2003-04-06
  81--83& 82.5& 10\\ 
  84--86& 85.5& 5 \\ 
  & & $§Sf = 30$\\ \hline 
\end{longtable}   
   
An \emph{array} is an arrangement of raw numerical data in ascending or descending order of magnitude. When summarising 
large masses of raw quantitative data, it is often useful to distribute the data into \emph{classes} and to determine 
the number of individuals belonging to each class, called the \emph{class frequency}. A tabular arrangement of data by 
classes together with the corresponding class frequency is called a \emph{frequency distribution}, or \emph{frequency 
  table}. Data organised and summarised in that way are often called \emph{grouped data}. The data presented in table 3 
are an illustration of a frequency distribution table for quantitative data.\index{data!grouped} 
   
A symbol defining a class, such as 72--74 or \intferouv{72}{75}, is called a \emph{class interval}.  The \emph{class 
  width} (or \emph{class size}) is equal to the difference between the two class boundaries.  Here we have $75-72=3$ so 
the width of the first class is 3. We can observe that all classes in table 3 have the same size.%
\index{width}\index{size}\index{class!width}\index{class!size}\index{class!interval}\index{interval!class} 
   
The \emph{class mark} is the midpoint of the class interval. It is also called the \emph{class midpoint}.%
\index{class!midpoint}\index{class!mark}\index{midpoint}\index{mark} For purposes of further mathematical analysis, all 
observations belonging to a given class interval are assumed to coincide with the class mark. The mark is obtained by 
halving the sum of the two boundaries of a class. So the midpoint of the first class is $\dps \dfrac{75+72}{2}=73{.}5$. 
   
   
\subsection{Constructing frequency distribution table}   
   
While constructing a frequency distribution table, we have to make the following three decisions: 
\begin{itemize}   
\item \textsf{Number of classes}\quad Usually the number of classes for a frequency distribution table varies from 5 to 
  20, depending mainly on the number of observations in the data set. It is preferable to have more classes as the size 
  of a data set increases. 
\item \textsf{Class width}\quad Although it is not uncommon to have classes of different sizes, most of the time, it is 
  preferable to have the same width for all classes. To determine the class width when all classes are of the same size, 
  first find the difference between the largest and the smallest values in the data. Then the approximate width of a 
  class is obtained by dividing that difference by the number of desired classes. 
\item \textsf{Lower limit of the first class or the starting point}\quad Any convenient number, which is equal to or 
  less than the smallest value in the data set, can be used as the lower limit of the first class.\index{point!starting} 
\end{itemize}     
   
   
\subsection{Graphing grouped data}   
   
Grouped data can be displayed by using a \nuvo{histogram} \pron{£hh£ist£e£bgr£am} or a \nuvo{polygon} 
\pron{£hp£ol£ig£en}. 
   
\subsubsection{Histograms}   
   
To draw a histogram, we first mark classes on the horizontal axis. Class boundaries are used to mark classes on the 
horizontal axis. Next, we draw a bar for each class so that its area represents the frequency of this class. The bars in 
a histogram are drawn adjacent to each other without leaving any gap between them. 
   
%\input{psfig/007003.fgs} 
\begin{fgre}
  \IG{stat03}
\end{fgre}
   
\subsubsection{Polygons}   
   
%\input{psfig/007004.fgs} 
\begin{fgre}
  \IG{stat04}
\end{fgre}
   
A polygon is another device that can be used to present quantitative data in graphic form. To draw a frequency polygon, 
we first mark a dot above the midpoint of each class at a height equal to the frequency of that class. Next, we mark two 
more classes, one at each end, and mark their midpoints. These two classes have zero frequencies. In the last step, we 
join the adjacent dots with straight lines. The resulting line graph is called a frequency polygon or simply a polygon. 

{\small   
A polygon with relative frequencies (respectively: percentages) marked on the vertical axis is called a relative 
frequency (respectively: percentage) polygon.\index{polygon!frequency!relative} \index{polygon!percentage} 
}   
\enlargethispage{\baselineskip} 
   
The symbol `\raisebox{.5ex}{\rule{4.5pt}{.6pt}\rule[-5.1pt]{.6pt}{11pt} \rule[-5.1pt]{.6pt}{11pt}\rule{4.5pt}{.6pt}}'
%%%`--| |--' % correction du 2003-15-07
used in the horizontal axis of figure \ref{chap:statistics}--4 represents a break, called the 
\nuvo{truncation} \pron{tr£v£n£hk£q£sn}, in the axis. It indicates that the entire axis is not shown in this figure.  As 
can be noticed, the zero to 69 portion of the axis has been omitted. 
   
\subsubsection{Warning}   
   
Describing data using graphs helps to give us insights into the main characteristics of the data.  However, graphs, 
unfortunately, can also be used, intentionally or unintentionally, to distort the facts and to deceive the reader. 
Following are two ways of manipulate graphs to convey a particular opinion or impression. 
\begin{itemize}   
\item Changing the scale either on one or on both axes, that is, shortening or stretching one or both axes. 
\item Truncating the frequency axis, that is, starting the frequency axis at a number greater than zero. 
\end{itemize}   
   
When interpreting a graph, we should be very cautious. We should observe carefully whether the frequency axis has been 
truncated or whether any axis has been unnecessarily shortened or stretched. 
   
\subsection{Cumulative frequency distributions}   
   
Consider the example about the heights of 30 NBA players. Suppose we want to know how many players are 80 inches tall or 
shorter. Such a question can be answered using a \nuvo{cumulative} \pron{£hkj£Umj£ul£et£iv} \emph{frequency 
  distribution}. Each class in a cumulative frequency distribution table gives the total number of values that fall 
below a certain value. A cumulative frequency distribution is constructed for quantitative data 
only.\index{frequency!cumulative}\index{distribution!frequency!cumulative} 
   
\begin{longtable}{c r}   
  \multicolumn{2}{l}{\bfseries\sffamily Table 4}\\ \hline\hline 
  {Height} & \multicolumn{1}{c}{Cumulative}\\ 
  {(in inches)} &\multicolumn{1}{c}{Frequency}\\ 
  72--74& 3 \\ 
  75--77& 8 \\ 
  78--80& 15 \\ 
  81--83& 25\\ 
  84--86& 30 \\ \hline 
\end{longtable}   
   
\enlargethispage{\baselineskip} 
   
The \emph{cumulative relative frequencies} are obtained by dividing the cumulative frequencies by the total number of 
observations in the data set. The \emph{cumulative percentages} are obtained by multiplying the cumulative relative 
frequencies by 100.\index{frequency!cumulative!relative}\index{percentage!cumulative} 
   
\subsubsection{Ogives}   
   
When plotted on a diagram, the cumulative frequencies give a curve that is called an \nuvo{ogive} \pron{£h£w£da£iv}.  To 
draw the ogive, the variable (height) is marked on the horizontal axis and the cumulative frequencies on the vertical 
axis. Then, the dots are marked above the upper boundaries of various classes at the heights equal to the corresponding 
cumulative frequencies. The ogive is obtained by joining consecutive points with straight-line segments. Note that the 
ogive starts at the lower boundary of the first class and ends at the upper boundary of the last class. 
   
%\input{psfig/007005.fgs} 
\begin{fgre}
  \IG{stat05}
\end{fgre}
   
\section{Measures of central tendency}   
   
We often represent a data set by numerical summary measures, usually called the typical values. A measure of central 
tendency or average gives the centre of a histogram or a frequency distribution curve. There are three main types of 
average ---~the mean, the mode, and the median; and the first type may be subdivided into arithmetic mean, geometric mean, 
and harmonic mean. These are all different types of representative value and, being recognised as such, the noun average 
is rarely used in statistical work when referring to actual values. 
   
\subsection{Mean}   
   
The simplest form of \nuvo{mean} \pron{m£In} is the arithmetic \pron{£b£ar£i£f£hmet£ik} mean.\index{mean!arithmetic} It 
is the most frequently used \emph{measure of central tendency} \pron{£htend£ens£i}.\index{measure!central tendency} It 
is the easiest to calculate.  One just adds up all the values, divides by the total number of values, and the result is 
the arithmetic mean. 
   
To calculate the mean for grouped data, first find the midpoint of each class and then multiply the midpoints by the 
frequencies of the corresponding classes. The sum of those products gives an approximation for the sum of all values. To 
find the value of the mean, divide the sum by the total number of observations in the data. 
   
\subsection{Outliers or extreme values}   
   
Values that are very small or very large relative to the majority of the values in a data set are called outliers 
\pron{£h£Wt£bl£p£e£Z} or extreme values.\index{outlier}\index{value!extreme} 
   
A major shortcoming of the mean as a measure of central tendency is that it is very sensitive to outliers. 
   
\subsection{Median}   
   
The \nuvo{median} \pron{£hm£Id£i£en} is the value of the middle term in a data set that has been ranked in increasing 
order. 
   
To determine the median for ungrouped data: 
\begin{itemize}   
\item if the number of observations $n$ in a data set is odd, then the median is given by the value of the middle term 
  in the ranked data that is the $({n+1}/{2})$th value; 
\item if the number $n$ is even, then the median is given by the average of the values of the two middle terms the 
  $({n}/{2})$th and $({n}/{2}+1)$th. 
\end{itemize}   
   
To determine the median of grouped data we use the ogive. The median is the abscissa of the point the ordinate of which 
is half of the total number of observations. 
   
\begin{fgre}
  \IG{stat06}
\end{fgre}
   
\subsection{Mode}   
   
Mode is a French word that means fashion --- an item that is most popular or common. In statistics, the \nuvo{mode} 
\pron{m£wd} represents the most common value in a data set; it is the value that occurs with the highest frequency in a 
data set. 
   
\subsection{Measure of dispersion}   
   
The \nuvo{range} \pron{r£qn£d} is the simplest \emph{measure of dispersion} \pron{d£i£hsp£E£sn}.\index{measure!of 
  dispersion} It is the difference between the largest and the smallest values in a data set. 
   
\section{Samples}   
   
{\footnotesize\setlength{\columnsep}{10pt}\setlength{\columnseprule}{.1pt} 
\begin{multicols}{2}   
  The basic idea of sampling is probably almost as old as mankind and, for all we know, possibly preceded it since some 
  lower animals appear to have the knack of tasting a proportion of the food offered to them and of rejecting the whole 
  meal on the strength of the one portion tasted!  Sampling is based %correction 2002-01-17
  on choice and selectiveness. Early man had but few 
  possessions and he did not need to count them nor to compare one with another. But as soon as the barter system began 
  to assume larger scale proportions, the two parties to the exchange, say, of coconuts and local barley-wine, would not 
  have tasted every coconut and every drop of wine. Instead they would have tested a few units of each commodity and 
  would have assumed that the rest of the units were up to the standard of the sample units which they had selected. The 
  purchaser of the wine, for instance, could not have tested all the wine without becoming highly intoxicated and 
  probably insensible to the whole transaction. He might, perhaps, have tasted a little from each skin of wine, but in 
  doing this he would still have been sampling on the theory that the remainder of the wine in each skin was exactly the 
  same as the small amounts tasted, although his judgement might have suffered as he proceeded.  
     
  It is equally impossible today to inspect every item separately. Electric light bulbs, for instance, are tested for 
  the longevity of useful life by lighting them until the elements break.  If every bulb was tested in this way, there 
  would not be any bulbs left to put into the lampholders at home. One hundred per cent inspection like this destroys 
  the article, and some form of sampling is necessary to avoid this wholesale destruction while at the same time 
  producing evidence that the bulk from which the sample is drawn is being maintained at a specified level of quality. 
  Other populations are so large that it would be physically impossible to gather data in respect of every member. 
  Correct sampling methods therefore make it possible to gather information which would otherwise be 
  unobtainable.\cite{reich:1961} 
\end{multicols}}   
   
We are often interested in knowing the proportion of a population that possesses a certain characteristics, for example, 
the proportion of Englishmen who own an electric kettle. The \emph{population proportion},\index{proportion!population} 
denoted by $p$, is the ratio of the number of elements in a population with a specific characteristic (here electric 
kettle owners) to the total number of elements in the population. As the population is too large to be studied directly, 
we may use a sample. Let $n$ be the \emph{sample size},\index{sample!size}\index{size!sample} that is the number of 
elements in the sample. The \emph{sample proportion},\index{proportion!sample} denoted by $\hat{p}$ (read $p$ hat) is 
the ratio of number of elements in the sample with a specific characteristic to the total number of elements in the 
sample. The value of $\hat{p}$ is an \nuvo{estimate} of $p$.\index{$\hat{p}$}\index{hat}
   
For different samples may lead to different values of $\hat{p}$, we say that $\hat{p}$ is subject to variability. 
Nonetheless, if the sample fulfils certain conditions, $\hat{p}$ is a good approximation of $p$. More precisely, if the 
sample is a simple random sample and if its size is great enough (not less than 25) then: 
   
\begin{prop}   
  More than 93\% of all the possible samples lead to a value of $\hat{p}$ which is between $p- \dfrac{1}{\sqrt{n}}$ and 
  $p+\dfrac{1}{\sqrt{n}}$. 
\end{prop}   
   
Practically a random sample is considered as simple if the population is large enough or if the sampling is made without 
replacement. 
   
Some consequences follow immediately: 
\begin{itemize}   
\item The bigger the sample, the more precise the estimate. 
\item $\dps p-\dfrac{1}{\sqrt{n}}\leq\hat{p}\leq p+\dfrac{1}{\sqrt{n}} \Iff \hat{p}-\dfrac{1}{\sqrt{n}}\leq p\leq 
  \hat{p}+\dfrac{1}{\sqrt{n}}$. 
\end{itemize}   
   
The second consequence is of pratical interest for when we perform a sampling the population proportion $p$ is not 
known. What we know, after the survey, is the value of $\hat{p}$. We can then be 93\% certain that the true value of $p$ 
is somewhere in \interfer{\hat{p}-\dfrac{1}{\sqrt{n}}}{\hat{p}+\dfrac{1}{\sqrt{n}}} 
which is known as the 93\% \emph{confidence interval}, 93\% being the \emph{confidence level}.%   
\index{confidence!interval}\index{confidence!level} 
   
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