Quantitative Methods for Business chapter 4 doc

CHAPTER
Filling up – fuelling
quantitative analysis
4
Chapter objectives
This chapter will help you to:
■ understand key statistical terms
■ distinguish between primary and secondary data
■ recognize different types of data
■ arrange data using basic tabulation and frequency distributions
■ use the technology: arrange data in EXCEL, MINITAB and
SPSS.
In previous chapters we have concentrated on techniques or models
involving single values that are known with certainty. Examples of these
are break-even analysis and linear programming, which we looked at in
Chapter 2, and the Economic Order Quantity model featured in
Chapter 3. In break-even analysis the revenue per unit, the fixed cost
and the variable cost per unit are in each case a specified single value.
In linear programming we assume that both profit per unit and resource
usage are constant amounts. In the Economic Order Quantity model
the order cost and the stock-holding cost per unit are each known single
values. Because these types of models involve values that are fixed or
predetermined they are called deterministic models.
Deterministic models can be useful means of understanding and
resolving business problems. Their reliance on known single value
inputs makes them relatively easy to use but is their key shortcoming.
Companies simply cannot rely on a figure such as the amount of
Chapter 4 Filling up – fuelling quantitative analysis 109
raw material used per unit of production being a single constant value.
In practice, such an amount may not be known with certainty, because
it is subject to chance variation. Because of this company managers

may well need to study the variation and incorporate it within the models
they use to guide them.
Models that use input values that are uncertain rather than certain,
values that are subject to chance variation rather than known, are
called probabilistic models, after the field of probability, which involves
the measurement and analysis of chance. We shall be dealing with
probability in later chapters.
Before you can use probability to reflect the chance variation in busi-
ness situations you need to know how to get some idea of the variation.
To do this we have to start by ascertaining where relevant information
might be found. Having identified these sources you need to know
how to arrange and present what you find from them in forms that will
help you understand and communicate the variation. In order to do
this in appropriate ways it is important that you are aware of the different
types of data that you may meet.
The purpose of this chapter is therefore to acquaint you with some
essential preliminaries for studying variation. We will start with defin-
itions of some key terms, before looking into sources of data and con-
sidering the different types of data. Subsequently we shall look at basic
methods of arranging data.
4.1 Some key words you need to know
There are several important terms that you will find mentioned
frequently in this and subsequent chapters. They are:
Data The word data is a plural noun (the singular form is datum), which
means a set of known or given things, facts. Data can be numerical (e.g.
wages of employees) or non-numerical (e.g. job titles of employees).
Variable A variable is a quantity that varies, the opposite of a constant.
For example, the number of telephone calls made to a call centre per
hour is a variable, whereas the number of minutes in an hour is a
constant. Often a capital letter, usually X or Y, is used to represent a

variable.
Value A value is a specific amount that a variable could be. For
example the number of telephone calls made to a call centre per
hour could be 47 or 71.These are both possible values of the
variable ‘number of calls made’.
110 Quantitative methods for business Chapter 4
Observation or Observed value This is a value of a variable that has
actually occurred, i.e. been counted or measured. For example, if
58 telephone calls are made to a call centre in a particular hour that
is an observation or observed value of the variable ‘number of calls
made’.
An observation is represented by the lower case of the letter used
to represent the variable; for instance ‘x’ represents a single observed
value of the variable ‘X’.A small numerical suffix is added to
distinguish particular observations in a set; x
1
would represent the
first observed value, x
2
the second and so on.
Data set A data set consists of all the observations of all the variables
collected in the course of a study or investigation, together with the
variable names.
Random This describes something that occurs in an unplanned way,
by chance.
Random variable A random variable has observed values that arise by
chance.The number of new cars a car dealer sells during a month is a
random variable; whereas the number of days in a month is a variable
that is not random because its observed values are pre-determined.
Distribution The pattern exhibited by the observed values of a variable

when they are arranged in order of magnitude.A theoretical
distribution is one that has been deduced, rather than compiled from
observed values.
Population Generally this means the total number of persons residing in
a defined area at a given time. In quantitative methods a population
is the complete set of things or elements we want to investigate.These
may be human, such as all the people who have purchased a particular
product, or inanimate, such as all the cars repaired at a garage.
Sample A sample is a subset of a population, that is, a smaller number
of items picked from the population.A random sample is a sample that
has components chosen in a random way, on the basis that any single
item in the population has no more or less chance than any other to
be included in the sample.
A typical quantitative investigation of a business problem might involve
defining the population and specifying the variables to be studied.
Following this a sample of elements from the population is selected and
observations of the variables for each element in the sample recorded.
Once the data set has been assembled work can begin on arranging
and presenting the data so that the patterns of variation in the distri-
butions of values can be examined.
At this point you may find it useful to try Review Question 4.1 at the
end of the chapter.
Chapter 4 Filling up – fuelling quantitative analysis 111
4.2 Sources of data
The data that form the basis of an investigation might be collected at
first hand in response to a specific problem. This type of data, col-
lected by direct observation or measurement, is known as primary data.
The procedures used to gather primary data are surveys, experiments
and observational methods. A survey might involve asking consumers
their opinion of a product. A series of experiments might be conducted

on products to assess their quality. Observation might be used to ascertain
the hazards at building sites.
The advantages of using primary data are that they should match the
requirements of those conducting the investigation and they are up-to-
date. The disadvantages are that gathering such data is both costly and
time-consuming.
An alternative might be to find data that have already been collected
by someone else. This is known as secondary data. A company looking
for data for a specific study will have access to internal sources of sec-
ondary data, but as well as those there are a large number of external
sources; government statistical publications, company reports, aca-
demic and industry publications, and specialist information services
such as the Economist Intelligence Unit. The advantages of using sec-
ondary data are that they are usually easier and cheaper to obtain. The
disadvantages are that they could be out of date and may not be
entirely suitable for the purposes of the investigation.
4.3 Types of data
Collecting data is usually not an end in itself. When collected the data
will be in ‘raw’ form, a state that might lead someone to refer to it as
‘meaningless data’. Once it is collected the next stage is to begin trans-
forming it into information, literally to enable it to inform us about the
issue being investigated.
There is a wide range of techniques that you can use to organize, dis-
play and represent data. Selecting which ones to use depends on the type
of data you have. The nature of the raw material you are working with
determines your choice of tools. Scissors are fine for cutting paper but no
good for cutting wood. A saw will cut wood but is useless for cutting paper.
It is therefore essential that you understand the nature of the data you
want to analyse before embarking on the analysis, so in this section we will
look at several ways of distinguishing between different types of data.

There are different types of data because there are different ways in
which facts are gathered. Some data may exist because specific things
have characteristics that have been categorized whereas other data may
exist as a result of things being counted, or measured, on some sort of
scale.
Perhaps the most important way of contrasting data types is on
the basis of the scales of measurement used in obtaining them. The
acronym NOIR stands for Nominal, Ordinal, Interval, Ratio; the four
basic data types. Nominal is the ‘lowest’ form of data, which contains
the least amount of information. Ratio is the ‘highest’ form of data,
which contains the most amount of information.
The word nominal comes from the same Latin root as the word
name. Nominal data are data that consist solely of names or labels.
These labels might be numeric such as a bank account number, or they
might be non-numeric such as gender. Nominal data can be categor-
ized using the labels themselves to establish, for instance the number
of males and females. It is possible to represent and analyse nominal
data using proportions and modes (the modal category is the one that
contains the most observations), but carrying out more sophisticated
analysis such as calculating an average is inappropriate; for example,
adding a set of telephone numbers together and dividing by the number
there are to get an average would be meaningless.
Like nominal data, ordinal or ‘order’ data consist of labels that can
be used to categorize the data, but order data can also be ranked.
Examples of ordinal data are academic grades and finishing positions
in a horse race. An academic grade is a label (an ‘A’ grade student)
that also belongs to a ranking system (‘A’ is better than ‘B’). Because
ordinal data contain more information than nominal data we can use
a wider variety of techniques to represent and analyse them. As well as
proportions and modes we can also use order statistics, such as identifying

the middle or median observation. However, any method involving arith-
metic is not suitable for ordinal data because although the data can be
112 Quantitative methods for business Chapter 4
Example 4.1
Holders of a certain type of investment account are described as ‘wealthy’.
To investigate this we could use socio-economic definitions of class to categorize each
account holder, or we could count the number of homes owned by each account holder,
or we could measure the income of each account holder.
ranked the intervals between the ranks are not consistent. For instance,
the difference between the horse finishing first in a race and the one
finishing second is one place. The difference between the horse fin-
ishing third and the one finishing fourth is also one place, but this does
not mean that there is the same distance between the third- and fourth-
placed horses as there is between the first- and second-placed horses.
Interval data consist of labels and can be ranked, but in addition the
intervals are measured in fixed units so the differences between values
have meaning. It follows from this that unlike nominal and ordinal,
both of which can be either numeric or non-numeric, interval data are
always numeric. Because interval data are based on a consistent numer-
ical scale, techniques using arithmetical procedures can be applied to
them. Temperatures measured in degrees Fahrenheit are interval
data. The difference between 30° and 40° is the same as the difference
between 80° and 90°.
What distinguishes interval data from the highest data form, ratio
data, is that interval data are measured on a scale that does not have a
meaningful zero point to ‘anchor’ it. The zero point is arbitrary, for
instance 0° Fahrenheit does not mean a complete lack of heat, nor is it
the same as 0° Celsius. The lack of a meaningful zero also means that
ratios between the data are not consistent, for example 40° is not half
as hot as 80°. (The Celsius equivalents of these temperatures are 4.4°

and 26.7°, the same heat levels yet they have a completely different
ratio between them.)
Ratio-type data has all the characteristics of interval data – it consists
of labels that can be ranked as well as being measured in fixed amounts
on a numerical scale. The difference is that the scale has a meaningful
zero and ratios between observations are consistent. Distances are ratio
data whether we measure them in miles or kilometres. Zero kilometres
and zero miles mean the same – no distance. Ten miles is twice as far
as five, and their kilometre equivalents, 16 and 8, have the same ratio
between them.
Chapter 4 Filling up – fuelling quantitative analysis 113
Example 4.2
Identify the data types of the variables in Example 4.1.
The socio-economic classes of account holders are ordinal data because they are
labels for the account holders and they can be ranked.
The numbers of homes owned by account holders and the incomes of account hold-
ers are both ratio data. Four homes are twice as many as two, and £60,000 is twice as
much income as £30,000.
At this point you may find it useful to try Review Question 4.2 at the
end of the chapter.
Another important distinction you need to make is between qualita-
tive data and quantitative data. Qualitative data consist of categories or
types of a characteristic or attribute and are always either nominal or
ordinal. The categories form the basis of the analysis of qualitative
data. Quantitative data are based on counting ‘how many’ or measur-
ing ‘how much’ and are always of interval or ratio type. The numerical
scale used to produce the figures forms the basis of the analysis of
quantitative data.
There are two different types of quantitative data: discrete and contin-
uous. Discrete data are quantitative data that can take only a limited

number of values because they are produced by counting in distinct or
‘discrete’ steps, or measuring against a scale made up of distinct steps.
There are three types of discrete data that you may come across.
First, data that can only take certain values because other values simply
cannot occur, for example the number of hats sold by a clothing
retailer in a day. There could be 12 sold one day and 7 on another, but
selling 9.3 hats in a day is not possible because there is no such thing as
0.3 of a hat. Such data are discrete by definition.
Secondly, data that take only certain values because those are the
ones that have been established by long-standing custom and practice,
for example bars in the UK sell draught beer in whole and half pints.
You could try asking for three-quarters of a pint, but the bar staff would
no doubt insist that you purchase the smaller or larger quantity. They
simply would not have the equipment or pricing information to hand
to do otherwise.
There are also data that only take certain values because the people
who have provided the data or the analysis have decided, for conveni-
ence, to round values that do not have to be discrete. This is what you
are doing when you give your age to the last full year. Similarly, the tem-
peratures given in weather reports are rounded to the nearest degree,
and the distances on road signs are usually rounded to the nearest
mile. These data are discrete by convention rather than by definition.
They are really continuous data.
Discrete data often but not always consist of whole number values.
The number of visitors to a website will always be a whole number, but
shoe sizes include half sizes. In other cases, like the UK standard sizes
of women’s clothing, only some whole numbers occur.
The important thing to remember about discrete data is that there
are gaps between the values that can occur, that is why this type of data
is sometimes referred to as discontinuous data. In contrast, continuous

114 Quantitative methods for business Chapter 4
data consist of numerical values that are not restricted to specific
numbers. Such data are called continuous because there are no gaps
between feasible values. This is because measuring on a continuous
scale such as distance or temperature yields continuous data.
The precision of continuous data is limited only by how precisely the
quantities are measured. For instance, we measure both the length of bus
journeys and athletic performances using the scale of time. In the first
case a clock or a wristwatch is sufficiently accurate, but in the second case
we would use a stopwatch or an even more sophisticated timing device.
The terms discrete variable and continuous variable are used in describing
data sets. A discrete variable has discrete values whereas a continuous
variable has continuous values.
At this point you may find it useful to try Review Questions 4.3 and
4.4 at the end of the chapter.
In most of your early work on analysing variation you will probably be
using data that consist of observed values of a single variable. However
you may need to analyse data that consist of observed values of two
variables in order to find out if there is a connection between them. For
instance, we might want to ascertain how cab fares are related to journey
times.
In dealing with a single variable we apply univariate analysis, whereas
in dealing with two variables we apply bivariate analysis. The prefixes
uni- and bi- in these words convey the same meanings as they do in
other words like unilateral and bilateral. You may also find reference to
multivariate analysis, which involves exploring relationships between
more than two variables.
Chapter 4 Filling up – fuelling quantitative analysis 115
Example 4.3
A motoring magazine describes cars using the following variables:

Type of vehicle – Hatchback/Estate/MPV/Off-Road/Performance
Number of passengers that can be carried
Fuel type – petrol/diesel
Fuel efficiency in miles per gallon
Which variables are qualitative and which quantitative?
The type of car and fuel type are qualitative; the number of passengers and the fuel
efficiency are quantitative.
Which quantitative variables are discrete and which continuous?
The number of passengers is discrete; the fuel efficiency is continuous.
116 Quantitative methods for business Chapter 4
You may come across data referred to as either hard or soft. Hard data
are facts, measurements or characteristics arising from situations that
actually exist or were in existence. Temperatures recorded at a weather
station and the nationalities of tourists are examples of hard data. Soft
data are about beliefs, attitudes and behaviours. Asking consumers
what they know about a product or how they feel about an advertise-
ment will yield soft data. The implication of this distinction is that hard
data can be subjected to a wider range of quantitative analysis. Soft
data is at best ordinal and therefore offers less scope for quantitative
analysis.
A further distinction you need to know is between cross-section and
time series data. Cross-section data are data collected at the same point
in time or based on the same period of time. Time series data consist
of observations collected at regular intervals over time. The volumes of
wine produced in European countries in 2002 are cross-section data
whereas the volumes of wine produced in Italy in the years 1992 to
2002 are time series data.
At this point you may find it useful to try Review Question 4.5 at the
end of the chapter.
4.4 Arrangement of data

Arranging or classifying data in some sort of systematic manner is the
vital first stage you should take in transforming the data into informa-
tion, and hence getting it to ‘talk to you’. The way you approach this
depends on the type of data you wish to analyse.
4.4.1 Arranging qualitative data
Dealing with qualitative data is quite straightforward as long as the
number of categories of the characteristic being studied is relatively
small. Even if there are a large number of categories, the task can be
made easier by merging categories.
The most basic way you can present a set of qualitative data is to tabu-
late it, to arrange it in the form of a summary table. A summary table
consists of two parts, a list of categories of the characteristic, and the
number of things that fall into each category, known as the frequency of
the category. Compiling such a table is simply a matter of counting
how many elements in the study fall into each category.
Chapter 4 Filling up – fuelling quantitative analysis 117
In Table 4.1 the outlet types are qualitative data. The ‘Other’ cat-
egory, which might contain several different types of outlet, such as
hypermarkets and market stalls, has been created in order to keep the
summary table to manageable proportions.
Notice that for each category, the number of outlets as a percentage
of the total, the relative frequency of the category, is listed on the right
hand side. This is to make it easier to communicate the contents; say-
ing 30.8% of the outlets are shoe shops is more effective than saying
12/39ths of them were shoe shops, although they are different ways of
saying the same thing.
You may want to use a summary table to present more than one attrib-
ute. Such a two-way tabulation is also known as a contingency table because
it enables us to look for connections between the attributes,in other
words to find out whether one attribute is contingent upon another.

Example 4.4
Suppose we want to find how many different types of retail outlet in an area sell trainers.
We could tour the area or consult the telephone directory in order to compile a list
of outlets, but the list itself may be too crude a form in which to present our results.
By listing the types of outlet and the number of each type of outlet we find we can
construct a summary table:
Table 4.1
The number of outlets selling trainers by type of outlet
Relative
Type of outlet Frequency frequency (%)
Shoe shops 12 30.8
Sports shops 11 28.2
Department stores 6 15.4
Other 10 25.6
Total number of outlets 39 100.0
Example 4.5
Four large retailers each operate their own loyalty scheme. Customers can apply for
loyalty cards and receive points when they present them whilst making purchases.
These points are accumulated and can subsequently be used to obtain gifts or
discounts.
118 Quantitative methods for business Chapter 4
At this point you may find it useful to try Review Questions 4.6 to 4.8
at the end of the chapter.
4.4.2 Arranging quantitative data
The nature of quantitative data is different to qualitative data and
therefore the methods used to arrange quantitative data are rather dif-
ferent. However, the most appropriate way of arranging some quanti-
tative data is the same as the approach we have used to arrange
qualitative data.
This applies to the analysis of a discrete quantitative variable that has

a very few feasible values. You simply treat the values as you would the
categories of a characteristic and tabulate the data to show how often
each value occurs. When quantitative data are tabulated, the resulting
table is called a frequency distribution because it demonstrates how
frequently each value in the distribution occurs.
A survey of usage levels of loyalty cards provided the information in the following
table:
Table 4.2
Number of transactions by loyalty card use
Transactions
Retailer With card Without card Total
Aptyeka 236 705 941
Botinky 294 439 733
Crassivy 145 759 904
Total 675 1903 2578
Example 4.6
The UREA department store offers free refills when customers purchase hot beverages
in its cafe. The numbers of refills taken by 20 customers were:
0131 2022010310121102
Chapter 4 Filling up – fuelling quantitative analysis 119
At this point you may find it useful to try Review Questions 4.9 to
4.11 at the end of the chapter.
We can present the data in Example 4.6 in the form of a simple table
because there are only a very limited number of values. Unfortunately
this is not always the case, even with discrete quantitative data.
For instance, if Example 4.6 included customers who spent all day in
the café and drank 20 or so cups of coffee each then the number of
refills might go from none to 30. This would result in a table with far
too many rows to be of use.
To get around this problem we can group the data into fewer cat-

egories or classes by compiling a grouped frequency distribution. This
shows the frequency of observations in each class.
These figures can be tabulated as follows:
Table 4.3
Number of hot beverage refills taken
Number of refills Number of customers
06
17
25
32
Total number of customers 20
Example 4.7
The numbers of email messages received by 22 office workers in one day were:
501425 8103352124515 7
5 98 13 31 52 6 75 17 20 12 64
Produce a grouped frequency distribution to present these data.
Number of messages
received Frequency
0–19 11
20–39 4
40–59 4
60–79 2
80–99 1
Total frequency 22
120 Quantitative methods for business Chapter 4
In order to compile a grouped frequency distribution you will need to
exercise a little judgement because there are many sets of classes that
could be used for a specific set of data. To help you, there are three rules:
1 Don’t use classes that overlap.
2 Don’t leave gaps between classes.

3 The first class must begin low enough to include the lowest
observation and the last class must finish high enough to
include the highest observation.
In Example 4.7 it would be wrong to use the classes 0–20, 20–40, 40–60
and so on because a value on the very edge of the classes like 20 could be
put into either one, or even both, of two classes. Although there are
numerical gaps between the classes that have been used in Example 4.7,
they are not real gaps because no feasible value could fall into them. The
first class finishes on 19 and the second begins on 20, but since the num-
ber of messages received is a discrete variable a value like 19.6, which
would fall into the gap, simply will not occur. Since there are no observed
values lower than zero or higher than 99, the third rule is satisfied.
We could sum up these rules by saying that anyone looking at a
grouped frequency distribution should be in no doubt where each feas-
ible value belongs. Every piece of data must have one and only one
place for it to be. To avoid any ambiguity whatsoever, you may like to
use the phrase ‘and under’ between the beginning and end of each
class. The classes in Example 4.7 could be rewritten as:
0 and under 20
20 and under 40 … and so on.
It is especially important to apply these rules when you are dealing with
continuous quantitative data. Unless you decide to use ‘and under’ or
a similar style of words, it is vital that the beginning and end of each
class is specified to at least the same degree of precision as the data.
Example 4.8
The results of measuring the contents (in millilitres) of a sample of 30 bottles of ‘Nogat’
nail polish labelled as containing 10 ml were:
10.30 10.05 10.06 9.82 10.09 9.85 9.98 9.97 10.28 10.01 9.92
10.03 10.17 9.95 10.23 9.92 10.05 10.11 10.02 10.06 10.21 10.04
10.12 9.99 10.19 9.89 10.05 10.11 10.00 9.92

Chapter 4 Filling up – fuelling quantitative analysis 121
When you construct a grouped frequency distribution you will also
need to decide how many classes to use and how wide they are. These
are related issues: the fewer the number of classes the wider each one
needs to be. It is a question of balance. You should avoid having a very
few very wide classes because they will only convey a crude impression
of the distribution. On the other hand, if you have very many narrow
classes you will be conveying too much detail. So, what is too few and
what is too many? As a guide, take the square root of the number of
observations in the set of data. In Example 4.8 there are 30 observa-
tions. The square root of 30 is 5.48, so we should round down to 5 or
up to 6 because we can only have whole numbers of classes. We have
actually used six classes for these data, which according to this guide is
about right.
Once you have some idea of the number of classes, the width of the
classes has to be decided. It is useful if all the classes have the same
width, especially if the frequency distribution is going to be the basis
for further work on the data.
The set of classes you use must cover all the observations from lowest
to highest, so to help you decide the width of classes, subtract the low-
est observation from the highest observation to give you the difference
between the two, known as the range of the values. Divide this by the
number of classes you want to have and the result will be the minimum
class width you must use. If you look back at Example 4.7 the range of
observations is 93 (98 minus 5) which, when divided by 5 gives 18.6. So
if we want a set of five classes of equal width to cover the range from 5
to 98, each class must be at least 18.6 wide.
This number, 18.6, is not particularly ‘neat’, so to make our grouped
frequency distribution easier to interpret we can round it up. The most
obvious number to take is 20, so 5 classes 20 units wide will be sufficient

Arrange these figures in a grouped frequency distribution.
Nail polish (ml) Frequency
9.80–9.89 3
9.90–9.99 7
10.00–10.09 11
10.10–10.19 5
10.20–10.29 3
10.30–10.39 1
Total frequency 30
122 Quantitative methods for business Chapter 4
to cover the range. In fact because these classes will combine to
cover a range of 100, whereas the range of our data is 93 we have some
flexibility when it comes to deciding where the first class should start.
The first class must begin at or below the lowest observation in the
set, in Example 4.7 this means it must start at 5 or below. Because 5 is a
fairly ‘neat’ round number it would make a perfectly acceptable start
for our first class, which would then be ‘5–24’, the second class would
be 25–44’ and so on. But what if the first observed value was 3 or 7?
Starting a set of classes with such a value would result in a grouped fre-
quency distribution that would look rather ungainly. If we start the
classes at a round number lower than the lowest value in the distribu-
tion, for instance zero in Example 4.7, we can guarantee that the
resulting set of classes will have ‘neat’ beginnings.
Grouped frequency distributions are very useful for comparing two
or more sets of data because the classes provide a common framework.
The best way of using grouped frequency distributions in this way is
to calculate the relative frequencies of the number of observations in
every class for each set of data.
Example 4.9
A rival brand of nail polish, Pallyets, also comes in 10 ml bottles. The contents in milli-

litres of a sample of 26 bottles of this product were:
10.19 9.92 10.22 10.39 9.95 10.15 10.12 10.25 9.94
9.88 9.92 10.23 9.86 10.34 10.37 10.38 10.34 10.08
10.23 10.05 9.86 9.92 10.35 10.07 9.93 10.14
Classify these data using the classes from Example 4.8 and work out the relative
frequencies for both distributions.
Relative Relative
Nail polish Frequency frequency (%) Frequency frequency
(ml) (Nogat) (Nogat) (Pallyets) (%)
9.80–9.89 3 10.0 3 11.5
9.90–9.99 7 23.3 6 23.1
10.00–10.09 11 36.7 3 11.5
10.10–10.19 5 16.7 4 15.4
10.20–10.29 3 10.0 4 15.4
10.30–10.39 1 3.3 6 23.1
Total 30 100.0 26 100.0
Chapter 4 Filling up – fuelling quantitative analysis 123
The use of relative frequencies in Example 4.9, given in percentages
to one place of decimals, makes direct comparison of the two sets of
data much easier. Saying for instance that 3.3% of the sample of Nogat
and 34.6% of the sample of Pallyets contained 10.3 ml or more is more
straightforward than comparing 1/30 with 6/26.
At this point you may find it useful to try Review Questions 4.12 to
4.20 at the end of the chapter.
4.5 Using the technology: arranging data
in EXCEL, MINITAB and SPSS
4.5.1 EXCEL
The PivotTable facility in EXCEL enables you to compile simple tabu-
lations. Click on Data at the top of the screen and you will find it listed
as PivotTable and PivotChart Report on the pull-down menu. Prior to

using it the data you want to tabulate should be entered into a column
under a suitable variable name. In Example 4.10 we will use the facility
to produce a table for the data in Example 4.6.
Example 4.10
In Example 4.6 the number of hot beverage refills taken by 20 customers were:
013120 22010310121102
■ Enter the variable name (Refills) in the first cell of a column and the figures into
the cells beneath.
■ Click Data and select PivotTable and PivotChart Report from the pull-down menu.
■ In the PivotTable and PivotChart Wizard Step 1 of 3 window that appears the
default settings should be Microsoft Excel list or database under the question
Where is the data that you want to analyze?, and PivotTable under the question
What kind of report do you want to create?. If these are not the default settings
select them by clicking on the buttons to their left. Click the Next> button.
■ In the PivotTable and PivotChart Wizard Step 2 of 3 window that appears the cur-
sor should be positioned in the window to the right of Range:. Specify the range of
cells containing the data by clicking at the top of the column where the data are
located and dragging the mouse down to cover all the entries in the column. When
you release the mouse button the cell range appears in the window. Make sure you
have included the variable name in the specified range. Click the Next> button.
■ In the PivotTable and PivotChart Wizard Step 3 of 3 window that appears select
Existing worksheet to locate the table the package will produce in the worksheet
124 Quantitative methods for business Chapter 4
The same EXCEL facility can be used to create a two-way or contin-
gency table. Such a table might be helpful to the researchers who had
gathered the data used in Example 4.10 if they were interested in whether
female and male customers appeared to take similar numbers of refills
and they had noted the gender of each customer in their sample.
you can see. The cell location in the window beneath Existing worksheet is the
position the package will use to locate the table in your worksheet. If it obscures

any of the existing entries in your worksheet simply click on a cell a suitable
distance away and the table location will be altered automatically.
■ Click the Layout button to the bottom left of the PivotTable and PivotChart
Wizard Step 3 of 3 window. The PivotTable and PivotChart – Layout window that
appears shows the framework that will be used to construct the table. It should
also have a button to the left with the name of the variable, Refills, on it. Click on
this button and drag it to the area of the table framework labelled ROW. This will
ensure that the values from the Refills column will be used as the rows in the
table. The Refills button now appears at the top of the ROW area and in its ori-
ginal position. Click on the button in its original position and drag it to the DATA
area of the table framework, which tells the package to use the data in the Refills
column to compile the table. When you do this you should see a button labelled
Sum of Refills in the DATA area. This means that the package will add up the val-
ues in the Refills column rather than count them. Double left click on Sum of
Refills and the PivotTable Field window appears. Select Count from the list of
options under Summarize by then click OK. The button in the DATA area should
now be labelled Count of Refills. Click OK in the PivotTable and PivotChart –
Layout window.
■ Click on the Finish button in the PivotTable and PivotChart Step 3 of 3 window.
The following table should appear in the worksheet:
The small PivotTable window that appears in the worksheet at the same time as the
table is of no immediate use and can be deleted.
Count of Refills
Refills Total
06
17
25
32
Grand total 20
Chapter 4 Filling up – fuelling quantitative analysis 125

4.5.2 MINITAB
The MINITAB package has a Tables facility that you can use to compile
summary tables. You will find it listed as Tables on the Stat pull-down
menu. The Tables sub-menu includes Tally for simple tables and Cross
Example 4.11
The refills data from Example 4.6 and the genders of the customers are:
0 13 120 2 2 0 1 0 31 01 21 10 2
F FMFFM M MF F MFF FMFMMMM
■ Enter the variable name Gender at the top of a column next to the column in
which you have located the refills figures. Enter the gender of each customer, F
or M as appropriate, alongside the number of refills taken by the customer.
■ Follow the procedure outlined in Example 4.10. When you reach the PivotTable
and PivotChart Wizard Step 2 of 3 window click and drag across the columns con-
taining the variable names and data for both Refills and Gender so that the cells
specified in the window to the right of Range: cover two columns.
■ When you reach the PivotTable and PivotChart Wizard Step 3 of 3 window click
the Layout button and in the PivotTable and PivotChart – Layout window you
should see two buttons on the right, one labelled Refills and the other labelled
Gender. Click and drag the Refills button into both the ROW and DATA areas of
the table framework, then click and drag the Gender button to the COLUMN
area of the table framework. Double left click on Sum of Refills and select Count
from the list of options under Summarize by in the PivotTable Field window then
click OK. The button in the DATA area should now be labelled Count of Refills.
Click OK in the PivotTable and PivotChart – Layout window.
■ Click on the Finish button in the PivotTable and PivotChart Step 3 of 3 window.
The following table should appear in the worksheet:
Count of Refills Gender
Refills F M Grand total
0336
1437

2235
3112
Grand total 10 10 20
126 Quantitative methods for business Chapter 4
Tabulation for two-way tables. In Example 4.12 we will outline the
procedures for using these tools using the data from Example 4.6.
Example 4.12
The refills data from Example 4.6 and the genders of the customers are:
013120 22010310121102
FFMF FM MMFFMFFFMFMMMM
■ Enter the variable name (Refills) in the unnumbered grey cell at the top of a col-
umn of the worksheet and enter the figures into the column cells beneath.
■ Click on Stat at the top of the screen and select Tables from the pull-down menu
that appears. Click on Tally in the sub-menu.
■ In the Tally window that appears you will see the name Refills on the left-hand side
with the number of the column where the values of the variable are stored. Double
left click on Refills and it will appear in the window below Variables:, which tells the
package that you want a table compiled from the values in that column.
■ Ensure that the Counts option under Display is ticked then click OK and the
following table should appear in the session window in the upper part of the
screen:
The letter N in this output represents the total number of observations counted.
To obtain a two-way table showing gender and numbers of refills:
■ Enter the variable name Gender in the unnumbered grey cell at the top of a col-
umn next to the column in which you have entered the refills figures. Enter the
gender of each customer, F or M as appropriate, alongside the number of refills
taken by the customer.
■ Select Tables from the Stat pull-down menu, and select Cross Tabulation from
the Tables sub-menu.
■ In the Cross Tabulation window both Refills and Gender variable names are

listed in the space on the right-hand side. Click on the upper variable name and
drag down to cover both variable names then click the Select button below. They
will appear in the window under Classification variables:.
Refills Count
06
17
25
32
N ϭ 20
Chapter 4 Filling up – fuelling quantitative analysis 127
4.5.3 SPSS
The SPSS package has a Tables of Frequencies facility that can pro-
duce simple and two-way tables. You can find it in the Custom Tables
option listed on the Analyze pull-down menu. Example 4.13 below
outlines how it can be used to produce tables for the data from
Example 4.6.
■ Click the space to the left of Counts under Display then click OK and the follow-
ing table should appear in the session window in the upper part of the screen:
Gender
Refills F M All
0336
1437
2235
3112
All 10 10 20
Example 4.13
The refills data from Example 4.6 and the genders of the customers are:
0131202 2 0103101 2 1 1 0 2
FFMFFMMMFFMFFFMF MMMM
■ On entering SPSS you will be presented with a window with the question What

would you like to do? at the top. Click the button to the left of Type in data then
click the OK button.
■ Enter the refills observations into the cells of a column of the worksheet.
■ Click the Variable View tab at the bottom left of the screen. On the left of the
screen that appears you will see a column headed Name. Type Refills over the
default name that appears there. Click on the Data View tab at the bottom left of
the screen and you will return to the data worksheet.
■ Click on Analyze at the top of the screen and from the pull-down menu select
Custom Tables. Click on Tables of Frequencies.
■ In the Tables of Frequencies window that appears you will see the variable name
refills highlighted in the space on the left. Click the ᭤ button to the left of
Frequencies for: and the refills name should be switched to the space below
128 Quantitative methods for business Chapter 4
Frequencies for:. Click OK and you should see the following table in the output
viewer screen:
To obtain a two-way table showing gender and numbers of refills:
■ Put the gender data into a column adjacent to the one you used to store the
refills data. Enter the gender of each customer, F or M as appropriate, alongside
the number of refills taken by the customer.
■ Click the Variable View tab at the bottom left of the screen. In the column headed
Name type Gender over the default name given to the new row. Click on the
Data View tab at the bottom left of the screen to return to the data worksheet.
■ Click Analyze at the top of the screen and select Custom Tables and click on
Tables of Frequencies.
■ In the Tables of Frequencies window the refills variable name is highlighted in
the space on the left. Click the ᭤ button to the left of Frequencies for: to select
refills name.
■ Click on the gender variable name in the space on the left then click the ᭤ but-
ton to the left of In Each Table and gender should now appear in the space
below In Each Table:. Click OK and you should see the following table in the

output viewer screen:
Count
0.00 6
1.00 7
2.00 5
3.00 2
Count
FM
0.00 3 3
1.00 4 3
2.00 2 3
3.00 1 1
Review questions
Answers to the following questions, including fully worked solutions to
the Key questions marked with an asterisk (*), are on pages 636–637.
4.1 Match the definitions listed below on the right-hand side to the
words listed on the left-hand side.
Chapter 4 Filling up – fuelling quantitative analysis 129
(a) distribution (i) something that occurs by chance
(b) element (ii) a subset of a population
(c) random (iii) a complete set of things to study
(d) sample (iv) a value of a variable that has occurred
(e) population (v) a systematic arrangement of data
(f) observation (vi) a single member of a population
4.2 Identify the type of scale of measurement (nominal, ordinal,
interval or ratio) appropriate for each of the following types of
data.
(a) Star ratings of hotels
(b) Sales revenues of companies
(c) Grades of officers in armed forces

(d) House numbers in a street
(e) Prices of cars
(f) Classes of accommodation on passenger flights
(g) Passport numbers
(h) Numbers in a rating scale on a questionnaire
(i) Index numbers such as the FTSE100 (‘Footsie’)
4.3 Indicate which of the variables below will have discrete values
and which will have continuous values.
(a) Time taken to answer telephone calls
(b) Clothing sizes for female apparel
(c) Age of consumers
(d) Calories in foodstuffs
(e) Shoe sizes
(f) Visitors to a theme park
(g) Interest rates
(h) Transactions in a supermarket
4.4 Indicate which of the variables below are qualitative, discrete
quantitative or continuous quantitative.
(a) Duration of telephone calls
(b) Modes of travel to work
(c) The alcohol contents of beers
(d) Sizes of theatre audiences
(e) Places of birth of passport applicants
(f) Numbers of websites found in a search
4.5 Select which of the statements listed below on the right-hand
side best describes each of the terms on the left-hand side.
(a) time series data (i) concern attitudes and beliefs
(b) nominal data (ii) are limited to distinct numerical
values
(c) hard data (iii) consist of values of two variables

130 Quantitative methods for business Chapter 4
(d) discrete data (iv) are collected at regular
intervals
(e) cross-sectional data (v) are factual
(f) bivariate data (vi) are based on a scale with an
arbitrary zero
(g) soft data (vii) are only labels
(h) interval data (viii) relate to a specific point or
period of time
4.6* A bus company operates services to and from the East, North,
South and West of a city. A recent report from the Chief
Executive contains the following summary of their operations.
(All figures have been rounded to the nearest thousand.)
The total number of passenger journeys made on our services was
430,000. Of these, 124,000 were to and from the North, 63,000 to
and from the South, and 78,000 to and from the East. Passengers
used bus passes to pay for 158,000 of the total number of journeys:
43,000 on northern services, 51,000 on western services, and 35,000
on eastern services. Passengers who did not use a bus pass paid for
their journeys in cash.
Construct a two-way tabulation with rows for the city areas
and columns for the method of payment. Work out the figures
that are not quoted in the summary by using the information
provided.
4.7 A hotel had 1360 bookings for accommodation in a month. Of
these 940 were for one night. Business bookings amounted to
813 of the total number, all but 141 being for one night. Leisure
bookings amounted to a further 362, the remaining bookings
being associated with functions (weddings etc.). Only 23 of
these latter bookings were for more than one night.

Draw up a two-way table for these figures with rows for the
types of booking and columns for the length of stay. Deduce the
figures that are not given by using the information provided.
4.8 A total of 127 people applied for several jobs at a new clothing
retail outlet. Seventy-four applicants were female, and of these
32 had previous experience of clothing retail and 19 had no pre-
vious retail experience. A total of 45 applicants had previous
retail experience but not in the clothing sector. Of the males
only 9 had no previous retail experience.
Chapter 4 Filling up – fuelling quantitative analysis 131
Use the information given to construct a contingency
table showing the breakdown of applicants by gender and
experience.
4.9* The numbers of people in 35 passenger cars travelling along a
road during the morning rush hour were:
1 1 2 1 2 1 3 5 1 1 2 1 1 1 4 1 2 1 1 1 4 1 2 1 1 4 1 1 2 3
2 3 1 4 1
Compile a frequency distribution for this set of data.
4.10 The ‘To Let’ column in the accommodation pages of a local
newspaper contains details of 20 houses available to rent. The
numbers of bedrooms in these properties are:
2352424443
2532344324
Arrange these data into a frequency distribution.
4.11 The ages of 28 applicants for a graduate management trainee
post are:
21 23 21 21 23 21 24 22 21 24 21 26 23 22 21 22 23 21 22 21
22 25 21 22 21 22 21 24
Produce a frequency distribution for these figures.
4.12 The number of business trips abroad taken in the last year by

each of a sample of 41 executives were:
3 11 1 10 14 14 12 6 1 10 7
11 9 2 7 11 17 12132 014
6 4 3 12 14 8 7119 6 9
15 0 4 9 7 10 4 5
(a) Arrange these data into a frequency distribution.
(b) Classify these data into a grouped frequency distribution
using the classes 0–2, 3–5, 6–8, 9–11, 12–14 and 15–17.
4.13* The speeds (in miles per hour) of 24 cars travelling along a
road that has a 30 mph speed limit were:
31 35 35 27 26 30 36 23 36 33 27 31
32 38 26 40 21 39 33 24 28 23 28 35
Construct a grouped frequency distribution for these data.
4.14 The numbers of laptops sold during a week in each of the 37
outlets of a chain of computer dealers were:
614 22171512181123101317 8
132 Quantitative methods for business Chapter 4
25 13 0 13 20 18 13 16 15 0 15 14
15 9 7 14 17 13 3 15 7 23 10 15
Present these data in the form of a grouped frequency
distribution.
4.15 The rates of growth in revenue (%) of 25 companies over a
year were:
4.22 3.85 10.23 5.11 7.91 4.60 8.16 5.28 3.98 2.51 9.95
6.98 6.06 9.24 3.29 9.75 0.11 11.38 1.41 4.05 1.93 5.16
1.99 12.41 7.73
Compile a grouped frequency distribution for these figures.
4.16 The prices (in £s) of 27 second-hand ‘Krushenia’ cars on sale at
a car hypermarket are:
4860 1720 2350 2770 3340 4240 4850 4390 3870

2790 3740 2230 1690 2750 1390 4990 3660 1900
5200 4390 3690 1760 4800 1730 2040 4070 2670
Create a frequency distribution to present these data.
4.17 The hourly wages (in £s) of 32 jobs offered by an employment
agency are:
6.28 4.90 4.52 5.11 5.94 5.82 7.14 7.28 8.15 7.04
4.41 4.67 6.90 5.85 5.65 5.50 4.12 5.27 5.25 6.43
5.73 4.65 5.37 4.24 6.45 4.70 5.09 4.82 6.23 5.40
6.48 5.26
Construct a grouped frequency distribution for these
figures.
4.18* The monthly membership fees in £s for 22 health clubs are:
32 43 44 22 73 69 48 67 33 56 67
28 78 60 63 32 67 41 65 48 48 77
(a) Arrange these data into a grouped frequency distribution.
Use classes £10 wide starting at £20.
(b) The monthly membership fees in £s for 17 fitness centres
in local authority leisure centres are:
27 50 44 32 31 55 21 36 24
56 51 55 32 39 42 28 55
Arrange these data into a grouped frequency distribution
using the same classes as in (a).