4
lzf/Shutterstock.com
Numerical Descriptive
Techniques
CHAPTER OUTLINE
4-1 Measures of Central Location
4-2 Measures of Variability
The Cost of One More Win in Major League Andrey Yurlov/Shutterstock.com
Baseball
DATA In the era of free agency, professional sports teams must compete for the
Xm04-00 services of the best players. It is generally believed that only teams whose
salaries place them in the top quarter have a chance of winning the champion-
ship. Efforts have been made to provide balance by establishing salary caps or some form of
equalization. To examine the problem, we gathered data from the 2015 baseball season. For
each team in major league baseball, we recorded the number of wins and the team payroll.
To make informed decisions, we need to know how the number of wins and the
team payroll are related. After the statistical technique is presented, we return to this
problem and solve it.
86
93453_ch04_hr_086-139.indd 86 1/31/17 3:43 PM
, N u m e r i c a l D e s c r i p ti v e T e c h ni q u e s
87
I
I n t ro d u c t i o n n Chapters 2 and 3, we presented several graphical techniques that describe data. In
this chapter, we introduce numerical descriptive techniques that allow the statistics
practitioner to be more precise in describing various characteristics of a sample or
population. These techniques are critical to the development of statistical inference.
As we pointed out in Chapter 2, arithmetic calculations can be applied to interval
data only. Consequently, most of the techniques introduced here may be used only to
numerically describe interval data. However, some of the techniques can be used for
ordinal data, and one of the techniques can be employed for nominal data.
When we introduced the histogram, we commented that there are several bits
of information that we look for. The first is the location of the center of the data. In
Section 4-1, we will present measures of central location. Another important char-
acteristic that we seek from a histogram is the spread of the data. The spread will be
measured more precisely by measures of variability, which we present in Section 4-2.
Section 4-3 introduces measures of relative standing.
In Section 3-3, we introduced the scatter diagram, which is a graphical method
that we use to analyze the relationship between two interval variables. The numerical
counterparts to the scatter diagram are called measures of linear relationship, and they are
presented in Section 4-4.
Section 4.5 features an application in finance and in Section 4.6 we compare the
information provided by graphical and numerical techniques. Finally, we complete this
chapter by providing guidelines on how to explore data and retrieve information.
S a m p l e S tat i s t i c or P o p u l at i o n P a r a m e t e r
Recall the terms introduced in Chapter 1: population, sample, parameter, and statistic.
A parameter is a descriptive measurement about a population, and a statistic is a descrip-
tive measurement about a sample. In this chapter, we introduce a dozen descriptive mea-
surements. For each one, we describe how to calculate both the population parameter
and the sample statistic. However, in most realistic applications, populations are very
large—in fact, virtually infinite. The formulas describing the calculation of parameters
are not practical and are seldom used. They are provided here primarily to teach the
concept and the notation. In Chapter 7, we introduce probability distributions, which
describe populations. At that time we show how parameters are calculated from prob-
ability distributions. In general, small data sets of the type we feature in this book are
samples.
93453_ch04_hr_086-139.indd 87 1/31/17 3:43 PM
,4-1 Measures of Central Location
4-1a Arithmetic Mean
There are three different measures that we use to describe the centre of a set of data. The
first is the best known, the arithmetic mean, which we’ll refer to simply as the mean.
Students may be more familiar with its other name, the average. The mean is computed by
summing the observations and dividing by the number of observations. We label the
observations in a sample , ,…, , where is the first observation, is the second
and so on until , where n is the sample size. As a result, the sample mean is denoted by
. In a population, the number of observations is labelled N and the population mean is
denoted by (Greek letter mu).
Mean
Population mean:
Sample mean:
Example 4.1
DATA Xm03-01 Mean Age of Online PC Gamers
Refer to Example 3.1. Find the mean age of the sample of Online PC Gamers.
Solution:
To calculate the mean, we add the observations and divide by the size of the
sample. Thus:
Excel
, There are several ways to command Excel to compute the mean. If we
simply want to compute the mean and no other statistics, we can use the
AVERAGE function.
Instructions
Type or import the data into one or more columns. (Open Xm03-01) Type
into any empty cell:
For Example 4.1, we would type into any empty cell:
The active cell would store the mean as 23.31.
lzf/Shutterstock.com
Numerical Descriptive
Techniques
CHAPTER OUTLINE
4-1 Measures of Central Location
4-2 Measures of Variability
The Cost of One More Win in Major League Andrey Yurlov/Shutterstock.com
Baseball
DATA In the era of free agency, professional sports teams must compete for the
Xm04-00 services of the best players. It is generally believed that only teams whose
salaries place them in the top quarter have a chance of winning the champion-
ship. Efforts have been made to provide balance by establishing salary caps or some form of
equalization. To examine the problem, we gathered data from the 2015 baseball season. For
each team in major league baseball, we recorded the number of wins and the team payroll.
To make informed decisions, we need to know how the number of wins and the
team payroll are related. After the statistical technique is presented, we return to this
problem and solve it.
86
93453_ch04_hr_086-139.indd 86 1/31/17 3:43 PM
, N u m e r i c a l D e s c r i p ti v e T e c h ni q u e s
87
I
I n t ro d u c t i o n n Chapters 2 and 3, we presented several graphical techniques that describe data. In
this chapter, we introduce numerical descriptive techniques that allow the statistics
practitioner to be more precise in describing various characteristics of a sample or
population. These techniques are critical to the development of statistical inference.
As we pointed out in Chapter 2, arithmetic calculations can be applied to interval
data only. Consequently, most of the techniques introduced here may be used only to
numerically describe interval data. However, some of the techniques can be used for
ordinal data, and one of the techniques can be employed for nominal data.
When we introduced the histogram, we commented that there are several bits
of information that we look for. The first is the location of the center of the data. In
Section 4-1, we will present measures of central location. Another important char-
acteristic that we seek from a histogram is the spread of the data. The spread will be
measured more precisely by measures of variability, which we present in Section 4-2.
Section 4-3 introduces measures of relative standing.
In Section 3-3, we introduced the scatter diagram, which is a graphical method
that we use to analyze the relationship between two interval variables. The numerical
counterparts to the scatter diagram are called measures of linear relationship, and they are
presented in Section 4-4.
Section 4.5 features an application in finance and in Section 4.6 we compare the
information provided by graphical and numerical techniques. Finally, we complete this
chapter by providing guidelines on how to explore data and retrieve information.
S a m p l e S tat i s t i c or P o p u l at i o n P a r a m e t e r
Recall the terms introduced in Chapter 1: population, sample, parameter, and statistic.
A parameter is a descriptive measurement about a population, and a statistic is a descrip-
tive measurement about a sample. In this chapter, we introduce a dozen descriptive mea-
surements. For each one, we describe how to calculate both the population parameter
and the sample statistic. However, in most realistic applications, populations are very
large—in fact, virtually infinite. The formulas describing the calculation of parameters
are not practical and are seldom used. They are provided here primarily to teach the
concept and the notation. In Chapter 7, we introduce probability distributions, which
describe populations. At that time we show how parameters are calculated from prob-
ability distributions. In general, small data sets of the type we feature in this book are
samples.
93453_ch04_hr_086-139.indd 87 1/31/17 3:43 PM
,4-1 Measures of Central Location
4-1a Arithmetic Mean
There are three different measures that we use to describe the centre of a set of data. The
first is the best known, the arithmetic mean, which we’ll refer to simply as the mean.
Students may be more familiar with its other name, the average. The mean is computed by
summing the observations and dividing by the number of observations. We label the
observations in a sample , ,…, , where is the first observation, is the second
and so on until , where n is the sample size. As a result, the sample mean is denoted by
. In a population, the number of observations is labelled N and the population mean is
denoted by (Greek letter mu).
Mean
Population mean:
Sample mean:
Example 4.1
DATA Xm03-01 Mean Age of Online PC Gamers
Refer to Example 3.1. Find the mean age of the sample of Online PC Gamers.
Solution:
To calculate the mean, we add the observations and divide by the size of the
sample. Thus:
Excel
, There are several ways to command Excel to compute the mean. If we
simply want to compute the mean and no other statistics, we can use the
AVERAGE function.
Instructions
Type or import the data into one or more columns. (Open Xm03-01) Type
into any empty cell:
For Example 4.1, we would type into any empty cell:
The active cell would store the mean as 23.31.
