Quantitative Methods for Business chapter 5 potx

Good visibility – pictorial presentation of data 5 Chapter objectives This chapter will help you to: ■ illustrate qualitative data using pictographs, bar charts andpie charts ■ portray q

Good visibility – pictorial presentation

of data

5

Chapter objectives

This chapter will help you to:

■ illustrate qualitative data using pictographs, bar charts andpie charts

■ portray quantitative data using histograms, cumulative quency charts and stem and leaf displays

fre-■ present bivariate quantitative data using scatter diagrams

■ display time series data using time series charts

■ use the technology: data presentation in EXCEL, MINITABand SPSS

■ become acquainted with business uses of pictorial data presentation

In the last chapter we looked at arranging and tabulating data, takingthe first steps in transforming raw data into information, bringingmeaning to the apparently meaningless In this chapter we will continuethis theme by considering various ways of portraying data in visual form.Used appropriately the diagrams and charts you will find here are veryeffective means of communicating the patterns and meaning con-tained in data, specifically the patterns and sequences in distributions

There are techniques that are very common in business documents sobeing able to understand what they mean is an important skill.

There are many different diagrams and charts that can be used to dothis, so it is important to know when to use them The techniques weuse depend on the type of data we want to present, in the same way asthe suitability of the methods of arranging data featured in the lastchapter depended on the type of data Essentially, the simpler the data,the simpler the presentational tools that can be used to represent them:simple nominal data restricted to a few categories can be shown effect-ively in the form of a simple bar chart whereas ratio data require themore rigorous scaling of something like a histogram

5.1 Displaying qualitative data

Section 4.4.1 of Chapter 4 covered the arrangement of qualitative data

in the form of summary tables As well as being a useful way of ing qualitative data, a summary table is an essential preliminary task topreparing a diagram to portray the data

display-A diagram is usually a much more effective way of communicatingdata because it is easier for the eye to digest than a table This will beimportant when you have to include data in a report or presentationbecause you want your audience to focus their attention on what youare saying They can do that more easily if they don’t have to work toohard to understand the form in which you have presented your data.Displaying qualitative data is fairly simple if there are few categories

of the attribute or characteristic being investigated With more egories, the task can be simplified by merging categories

cat-There are three types of diagram that you can use to show qualitative

data: pictographs, pie charts and bar charts We will deal with them in this

section in order of increasing sophistication

5.1.1 Pictographs

A pictograph is little more than a simple extension of a summary table.The categories of the attribute are listed as they are in a summary table,and we use symbols to represent the number of things in each category.The symbols you use in a pictograph should have a simple and directvisual association with the data

A pictograph like Figure 5.1 can be an effective way of presenting a ple set of qualitative data The symbols are a simple way of representing

sim-the number in each category and have sim-the extra advantage of sizing the context of the data.

empha-Pictographs do have some drawbacks that may put you off usingthem Unless you are artistically gifted and can create appropriateimages by hand, you will probably have to rely on computer software toproduce them for you Creating a pictograph using a PC can be a labori-ous process Spreadsheet and statistical packages cannot produce a pictograph for you directly from data, so symbols have to be graftedalongside text in a word processing package

If you do use pictographs you need to choose the symbols carefully.They should be easy to associate with the context of the data and not

so elaborate that the symbols themselves become the focus of attentionrather than the data they are supposed to represent

Example 5.1

The table below lists four racehorse trainers and the number of horses they trained thatwon races at a horse race meeting

Show this set of data in the form of a pictograph

Nadia Amazonka 5 Freddie Conn 3 Lavinia Loshart 1 Victor Sedlow 2

Trainer Number of winners

You may occasionally see a pictograph in academic and business ments; you are more likely to see them on television and in newspapers.The computer graphics software reporters and editors use is muchmore sophisticated than any that you are likely to have access to duringyour studies.

docu-5.1.2 Pie charts

The second method of displaying qualitative data that we will look at isthe pie chart Pie charts are used much more than pictographs in partbecause they can be produced using widely available computer software

A pie chart, like a pictograph, is designed to show how many thingsbelong to each category of an attribute It does this by representing theentire set of data as a circle or ‘pie’ and dividing the circle into segments

or ‘slices’ Each segment represents a category, and the size of the ment reflects the number of things in the category

seg-Just about every spreadsheet or statistical package can produce a piechart like Figure 5.2 either from the original data or from a summarytable You will find guidance on doing this using EXCEL, MINITAB andSPSS in the final section of this chapter These packages provide vari-ous ways of enhancing pie charts: colour and shading patterns, 3D effects,and detached or ‘exploded’ slices to emphasize a particular segment.With practice you will be able to use these options in creating piecharts, but don’t overdo it Remember that the pattern of the data iswhat you want to convey not your ability to use every possible gimmick

in the package

Pie charts are so widely used and understood that it is very tempting

to regard them as an almost universal means of displaying qualitative

data In many cases they are appropriate and effective, but in some ations they are not.

situ-Because the role of a pie chart is to show how different componentsmake up a whole, you should not use one when you cannot or do notwant to show the whole This may be because there are some valuesmissing from the data or perhaps there is an untidy ‘Other’ categoryfor data that do not fit in the main categories In leaving out any data,either for administrative or aesthetic reasons, you would not be pre-senting the whole, which is exactly what pie charts are designed to do.One reason that people find pie charts accessible is that the analogy

of cutting up a pie is quite an obvious one As long as the pie chartlooks like a pie it works However if you produce a pie chart that hastoo many categories it can look more like a bicycle wheel than a pie,and confuses rather than clarifies the data If you have a lot of cat-egories to present, say more than ten, either merge some of the cat-egories in order to reduce the number of segments in the pie chart orconsider an alternative way of presenting your data

can be produced using spreadsheet and statistical packages Howeverbecause there are several different varieties of bar charts, they aremore flexible tools We can use bar charts to portray not only simplecategorizations but also two-way classifications.

The basic function of a bar chart is the same as that of a pie chart,and for that matter a pictograph; to show the number or frequency

of things in each of a succession of categories of an attribute It sents the frequencies as a series of bars The height of each bar is indirect proportion to the frequency of the category; the taller the bar that represents a category, the more things there are in that category

repre-The type of bar chart shown in Figure 5.3 is called a simple bar chart

because it represents only one attribute If we had two attributes to display

we might use a more sophisticated type of bar chart, either a component bar chart or a stack bar chart.

The type of bar chart shown in Figure 5.4 is called a component bar chart because each bar is divided into parts or components The

Example 5.3

Produce a bar chart to display the data from Example 5.2

Crewe Doncaster Exeter Frome 0

10 20 30 40 50 60

alternative name for it, a stacked bar chart, reflects the way in whichthe components of each bar are stacked on top of one another.

A component bar chart is particularly useful if you want to emphasizethe relative proportions of each category, in other words to show the

balance within the categories of one attribute (in the case of Example 5.4 the depot) between the categories of another attribute (in Example 5.4

the type of call-out)

Example 5.4

The call-outs data in Example 5.2 have been scrutinized to establish how many call-outsfrom each depot concerned washing machines and how many concerned other appli-ances The numbers of the two call-out types from each depot are:

Display these data as a component bar chart

Crewe Doncaster Exeter Frome 0

If you want to focus on this balance exclusively and are not too concernedabout the absolute frequencies in each category you could use a com-ponent bar chart in which each bar is subdivided in percentage terms.

If you want to emphasize the absolute differences between the egories of one attribute (in Example 5.4 the depots) within the cat-

cat-egories of another (in Example 5.4 the types of call-out) you may find a

cluster bar chart more useful.

The type of bar chart shown in Example 5.6 is called a cluster bar chartbecause it uses a group or cluster of bars to show the composition ofeach category of one characteristic by categories of a second charac-teristic For instance in Figure 5.6 the bars for Crewe show how the call-outs from the Crewe depot are composed of call-outs for washingmachines and call-outs for other appliances

At this point you may find it useful to try Review Questions 5.1 to 5.3

at the end of the chapter

Example 5.5

Produce a component bar chart for the data in Example 5.4 in which the sections of thebars represent the percentages of call-outs by appliance type

Washing machine Other appliance

Frome Exeter

Doncaster Crewe

Example 5.6

Produce a cluster bar chart to portray the data from Example 5.4

Washing machine Other appliance

Frome Exeter

Doncaster Crewe

5.2 Displaying quantitative data

Quantitative data are more sophisticated data than qualitative data andtherefore the methods used to present quantitative data are generallymore elaborate The exception to this is where you want to representthe simplest type of quantitative data, discrete quantitative variablesthat have very few feasible values You can treat the values in these data

as you would categories in qualitative data, using them to construct abar chart or pie chart.

the histogram This is a special type of bar chart where each bar or block

represents the frequency of a class of values rather than the frequency

of a single value Because they are composed in this way histograms are

sometimes called block diagrams.

You can see that in Figure 5.8 there are no gaps between the blocks

in the histogram The classes on which it is based start with ‘0–19’ then

‘20–39’ and so on When plotting such classes you may be tempted toleave gaps to reflect the fact that there is a numerical gap between theend of the first class and the beginning of the next but this would be

Figure 5.7 shows these data in the form of a bar chart

Figure 5.7

Number of customers by number of refills

3 2

1 0

8 7 6 5 4 3 2 1 0

Number of refills

wrong because the gap would be meaningless as it is simply not sible to receive say 19.2 messages.

pos-A histogram is a visual tool that displays the pattern or distribution

of observed values of a variable The larger the size of the block thatrepresents a class, the greater the number of values that has occurred

in that class Because the connection between the size of the blocksand the frequencies of the classes is the key feature of the diagram thescale along the vertical or ‘Y’ axis must start at zero, as in Figure 5.8

Example 5.8

In Example 4.7 the numbers of email messages received by 22 office workers werearranged in the following grouped frequency distribution

Show this grouped frequency distribution as a histogram

60 40

20 0

As long as the classes in a grouped frequency distribution are of thesame width it is simply the heights of the blocks of the histogram thatreflect the frequencies of observed values in the classes If the classeshave different widths it is important that the areas of the blocks areproportional to the frequencies of the classes The best way of ensuring

this is to represent the frequency density rather than the frequency of the

classes The frequency density is the frequency of values in a classdivided by the width of the class It expresses how densely the valuesare packed in the class to which they belong

In this distribution the classes have different widths, but an additional complication

is that the first class has no numerical beginning and the last class has no numericalend, they are both ‘open-ended’ classes

Before we can proceed we need to ‘close’ these classes In the case of the first classthis is straightforward; we can simply express it as ‘0 to 14’ The last class poses more of

a problem If we knew the age of the oldest person we could use that as the end of the

class, but as we don’t we have to select an arbitrary yet plausible end of the class In

keeping with the style of some of the other classes we could use ‘65 to 84’

The amended classes with their frequency densities are:

Using frequency densities in Figure 5.9 means that the height of theblock representing the ‘15 to 24’ class is increased to reflect the fact that

it is narrower than the other classes Despite having only one quarter of

the frequency of the ‘25–44’ class the height of the block representing the

‘15–24’ class is half the height of the block representing the ‘25–44’ class The class is half the width of the classes to the right of it, so to keep the area

in proportion to the frequency the height of the block has to be doubled.

In Figure 5.9 there are no gaps between the classes, although it might

be tempting to insert them as each class finishes on the number beforethe next class begins This would be wrong because, for instance, peopleare considered to be 14 years old right up until the day before their fifteenth birthday

The pattern of the distribution shown in Figure 5.9 is broadly

bal-anced or symmetrical There are two large blocks in the middle and

smaller blocks to the left and right of the ‘bulge’ From this we wouldconclude that the majority of observed values occur towards the middle

of the age range, with only a few relatively young and old customers

In contrast, if you look back at Figure 5.8, the histogram showing thenumbers of email messages received, you will see an asymmetrical or

skewed pattern The block on the left-hand side is the largest and the size

of the blocks gets smaller to the right of it It could be more accurately

described as right or positively skewed From Figure 5.8 we can conclude

Figure 5.9

Histogram of ages of customers opening bank accounts

85 65

45 25

15 0

that the majority of office workers receive a relatively modest number

of email messages and only a few office workers receive large numbers

of email messages

You may come across distributions that are left or negatively skewed.

In these the classes on the left-hand side have smaller frequencies andthose on the right-hand side have larger frequencies

In Figure 5.10 there are no gaps between the classes because thereare no numerical gaps between the classes; they are seamless

Example 5.10

Raketa Airlines say they allow their passengers to take up to 5 kg of baggage with theminto the cabin The weights of cabin baggage taken onto one flight were recorded andthe following grouped frequency distribution compiled from the data:

Portray this distribution in the form of a histogram

4 3

2 1

5.2.2 Cumulative frequency graphs

An alternative method of presenting data arranged in a grouped

fre-quency distribution is the cumulative frefre-quency graph This diagram shows the way in which the data accumulates through the distribution

from the first to the last class in the grouped frequency distribution Ituses the same horizontal axis as you would use to construct a histogram

to present the same data, but you have to make sure that the vertical axis,which must begin at zero, extends far enough to cover the total frequency

of the distribution

To plot a cumulative frequency graph you must begin by workingout the cumulative frequency of each class in the grouped frequencydistribution The cumulative frequency of a class is the frequency of the

class itself added to the cumulative, or combined frequency of all the

preceding classes

The cumulative frequency of the first class is simply the frequency ofthe first class because it has no preceding classes The cumulative fre-quency of the second class is the frequency of the second class added tothe frequency of the first class The cumulative frequency of the thirdclass is the frequency of the third class added to the cumulative frequency

of the second class, and so on

Note that the cumulative frequency of the last class in the distribution

in Example 5.11 is 22, the total frequency of values in the distribution.This should always be the case Once we have included the values inthe final class in the cumulative total we should have included everyvalue in the distribution

The cumulative frequency figures represent the number of valuesthat have been accumulated by the end of a class A cumulative fre-quency graph is a series of single points each of which represents thecumulative frequency of its class plotted above the very end of its class.There should be one plotted point for every class in the distribution.The final step is to connect the points with straight lines.

If you look carefully at Figure 5.11 you will see that the line begins atzero on the horizontal axis, which is the beginning of the first class,and zero on the vertical axis This is a logical starting point It signifiesthat no values have been accumulated before the beginning of the firstclass The line then climbs steeply before flattening off The steepclimb represents the concentration of values in the first class, whichcontains half of the values in the distribution The flatter sections tothe right represent the very few values in the later classes

The line in Figure 5.12 starts with a gentle slope then rises moresteeply before finishing with a gentle slope This signifies that the firstclasses contain few values, the middle classes contain many values, andthe final classes contain few values This is a symmetrical distribution,whereas the distribution shown in Figure 5.11 is a skewed distribution

It may be more convenient to plot a cumulative relative frequency

graph, in which the points represent the proportions of the total ber of values that occur in and prior to each class This is particularly use-ful if the total number of values in the distribution is an awkward number.You will find further discussion of cumulative frequency graphs in thenext chapter because they offer an easy way of finding the approximatevalues of medians, quartiles and other order statistics

num-At this point you may find it useful to try Review Questions 5.4 to 5.9

at the end of the chapter

10.4 10.3

10.2 10.1 10.0

9.9 9.8

Example 5.14

The size of cash payments made by 119 customers at a petrol station is summarized inthe following grouped frequency distribution Plot a cumulative relative frequency graph

5.2.3 Stem and leaf displays

Histograms and cumulative frequency graphs are effective and widelyused means of presenting quantitative data Until fairly recently theycould be described as unrivalled However, there is an alternative way of

presenting quantitative data in visual form, the stem and leaf display.

This is one of a number of newer techniques known collectively as

Exploratory Data Analysis (EDA) If you want to know more about EDA,

the books by Tukey (1977), and Velleman and Hoaglin (1981) provide

The role of a stem and leaf display is the same as the role of a histogram, namely to show the pattern of a distribution But unlike ahistogram, a stem and leaf display is constructed using the actual data

as building blocks, so as well as showing the pattern of a distribution it isalso a list of the observations that make up that distribution It is a veryuseful tool for making an initial investigation of a set of data as it por-trays the shape of the distribution, identifies unusual observations andprovides the basis for judging the suitability of different types of averages.The basis of a stem and leaf display is the structure of numbers, thefact that a number is made up of units, tens, hundreds and so on Forinstance the number 45 is composed of two digits, the 4 tens and the

5 units Using the analogy of a plant, the stem of the number 45 is thenumber on the left-hand side, 4 (the number of tens) and the leaf isthe number on the right hand side, 5 (the number of units) A stem on

a plant can have different leaves; in the same way the numerical stem 4can have different numerical leaves The number 48 has the same stem

as the number 45, but a different leaf, 8

To produce a stem and leaf display for a set of data we have to list theset of stem digits that appear in the data and then record each observa-tion by putting its leaf digit alongside its stem digit When we have donethis for every observed value in the set of data the result is a series of ‘stemlines’ each of which consists of a stem digit and the leaf digits of all theobservations sharing that particular stem The final stage in the process

is to arrange the leaf digits on each stem line in order of magnitude.The message ‘leaf unit 1’ on the final version of the stem and leafdisplay in Example 5.15 has the same role as the scale on the horizon-tal or ‘X’ axis of a histogram, in that it specifies the order of magnitude

Produce a stem and leaf display for these data

Every value consists of two digits: tens and units The tens are the stem digits and theunits are the leaf digits The lowest value is 28 and the highest is 74 so the first stem linewill be for the stem digit 2, and the last one for the stem digit 7 The first stem line willhave a leaf digit for the lowest value, the 8 from 28 The second stem line, for the stemdigit 3, will have two leaf digits, the 8 from 38 and the 9 from 39, and so on

of the data Without this message someone might look at the display,see that the highest value in the distribution has the stem digit 7 andthe leaf digit 4, but be unclear whether the value is 0.74, 7.4, 74, 740,

7400, or any other number with a 7 followed by a 4 It is only when youknow that the leaf digits are units in this display that you can be surethe stem digit 7 and the leaf digit 4 represents the number 74

Although a stem and leaf display may appear a little strange it is atool that is well worth learning to use because it has two advantagesover a histogram: particular values can be highlighted and two distri-butions can be shown in one display A histogram simply cannot do theformer because it consists of blocks rather than data It is possible to

Example 5.16

Five of the Musor restaurants whose seating capacities are given in Example 5.15 are incity centre locations The seating capacities for these five restaurants are shown in boldtype below:

We can embolden these values in the stem and leaf display

This is a stem and leaf display, but it is not yet finished We need to rearrange the leafdigits so they are listed from the smallest to the largest

plot a histogram showing two distributions but the result is cumbersomeand you would do better to plot two separate histograms.

To show two distributions in one stem and leaf display you simply listthe leaf digits for one distribution to the left of the list of stem digits andthe leaf digits for the other distribution to the right of the stem digits

By looking at the display in Example 5.17 you can see that in generalthe restaurants in the Bristol area have larger seating capacities thanthose in East Anglia

You can see from the display in Example 5.15 that the city centre restaurants are amongthose with larger seating capacities