Quantitative Methods for Business chapter 4 doc

C H A P T E R Filling up – fuelling quantitative analysis 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 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 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 business situations you need to know how to get some idea of the variation To 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 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 definitions of some key terms, before looking into sources of data and considering 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 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; x1 would represent the first observed value, x2 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 distributions 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 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, collected 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-todate 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 secondary data, but as well as those there are a large number of external sources; government statistical publications, company reports, academic and industry publications, and specialist information services such as the Economist Intelligence Unit The advantages of using secondary 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 transforming 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, display 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 112 Quantitative methods for business Chapter 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 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 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 categorized 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 arithmetic is not suitable for ordinal data because although the data can be Chapter Filling up – fuelling quantitative analysis 113 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 finishing 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 fourthplaced 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 numerical 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 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 holders are both ratio data Four homes are twice as many as two, and £60,000 is twice as much income as £30,000 114 Quantitative methods for business Chapter 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 qualitative 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 measuring ‘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 continuous 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 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 otherwise There are also data that only take certain values because the people who have provided the data or the analysis have decided, for convenience, to round values that not have to be discrete This is what you are doing when you give your age to the last full year Similarly, the temperatures 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 Chapter Filling up – fuelling quantitative analysis 115 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 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 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 in other words like unilateral and bilateral You may also find reference to multivariate analysis, which involves exploring relationships between more than two variables 116 Quantitative methods for business Chapter 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 advertisement 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 information, 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 tabulate 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 Filling up – fuelling quantitative analysis 117 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 Type of outlet Shoe shops Sports shops Department stores Other Total number of outlets Frequency 12 11 10 39 Relative frequency (%) 30.8 28.2 15.4 25.6 100.0 In Table 4.1 the outlet types are qualitative data The ‘Other’ category, 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; saying 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 attribute 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 4.5 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 120 Quantitative methods for business Chapter 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: Don’t use classes that overlap Don’t leave gaps between classes 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 number 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 feasible 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: 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 10.03 10.12 10.05 10.17 9.99 10.06 9.95 10.19 9.82 10.23 9.89 10.09 9.92 10.05 9.85 10.05 10.11 9.98 9.97 10.28 10.01 9.92 10.11 10.02 10.06 10.21 10.04 10.00 9.92 Chapter Filling up – fuelling quantitative analysis 121 Arrange these figures in a grouped frequency distribution Nail polish (ml) 9.80–9.89 9.90–9.99 10.00–10.09 10.10–10.19 10.20–10.29 10.30–10.39 Total frequency Frequency 11 30 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 observations The square root of 30 is 5.48, so we should round down to or up to 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 lowest 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 gives 18.6 So if we want a set of five classes of equal width to cover the range from 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 classes 20 units wide will be sufficient 122 Quantitative methods for business Chapter 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 or below Because 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 or 7? Starting a set of classes with such a value would result in a grouped frequency distribution that would look rather ungainly If we start the classes at a round number lower than the lowest value in the distribution, 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 millilitres of a sample of 26 bottles of this product were: 10.19 9.88 10.23 9.92 9.92 10.05 10.22 10.23 9.86 10.39 9.86 9.92 9.95 10.34 10.35 10.15 10.37 10.07 10.12 10.38 9.93 10.25 9.94 10.34 10.08 10.14 Classify these data using the classes from Example 4.8 and work out the relative frequencies for both distributions Nail polish (ml) 9.80–9.89 9.90–9.99 10.00–10.09 10.10–10.19 10.20–10.29 10.30–10.39 Total Frequency (Nogat) 11 30 Relative frequency (%) (Nogat) 10.0 23.3 36.7 16.7 10.0 3.3 100.0 Frequency (Pallyets) 4 26 Relative frequency (%) 11.5 23.1 11.5 15.4 15.4 23.1 100.0 Chapter 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 tabulations 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: ■ ■ ■ ■ ■ 2 1 1 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 of 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 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 of window that appears the cursor 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 of window that appears select Existing worksheet to locate the table the package will produce in the worksheet 124 Quantitative methods for business ■ ■ Chapter 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 of 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 original 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 this you should see a button labelled Sum of Refills in the DATA area This means that the package will add up the values 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 of window The following table should appear in the worksheet: Count of Refills Refills Grand total Total 20 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 The same EXCEL facility can be used to create a two-way or contingency 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 Chapter Filling up – fuelling quantitative analysis 125 Example 4.11 The refills data from Example 4.6 and the genders of the customers are: F ■ ■ ■ ■ F M F F M M M F F M F F 1 F M F M M M M 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 of window click and drag across the columns containing 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 of 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 of window The following table should appear in the worksheet: Count of Refills Refills Grand total Gender F M Grand total 10 3 10 20 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 126 Quantitative methods for business Chapter 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: F ■ ■ ■ ■ F M F F M 2 M M F F M F F F M F 1 M M M M Enter the variable name (Refills) in the unnumbered grey cell at the top of a column 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: Refills Count N ϭ 20 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 column 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: Chapter ■ Filling up – fuelling quantitative analysis 127 Click the space to the left of Counts under Display then click OK and the following table should appear in the session window in the upper part of the screen: Gender Refills All F M All 10 3 10 20 4.5.3 SPSS The SPSS package has a Tables of Frequencies facility that can produce 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 Example 4.13 The refills data from Example 4.6 and the genders of the customers are: F ■ ■ ■ ■ ■ F M F F 2 M M M F F M F F F M F M M M M 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 Frequencies for: Click OK and you should see the following table in the output viewer screen: Count 0.00 1.00 2.00 3.00 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 ᭤ button 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 F 0.00 1.00 2.00 3.00 M 3 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 Filling up – fuelling quantitative analysis 4.2 4.3 4.4 4.5 129 (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 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’) 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 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 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 (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 4.7 4.8 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 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 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 previous retail experience A total of 45 applicants had previous retail experience but not in the clothing sector Of the males only had no previous retail experience Chapter Filling up – fuelling quantitative analysis 4.9* 131 Use the information given to construct a contingency table showing the breakdown of applicants by gender and experience The numbers of people in 35 passenger cars travelling along a road during the morning rush hour were: 11212 13511 21114 12111 41211 41123 23141 4.10 Compile a frequency distribution for this set of data 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: 2 4.11 5 2 4 4 4 Arrange these data into a frequency distribution 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 4.12 Produce a frequency distribution for these figures The number of business trips abroad taken in the last year by each of a sample of 41 executives were: 11 15 11 4 10 12 14 11 14 14 17 10 12 12 13 11 10 14 9 (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 32 4.14 35 38 35 26 27 40 26 21 30 39 36 23 36 33 27 33 24 28 23 28 31 35 Construct a grouped frequency distribution for these data The numbers of laptops sold during a week in each of the 37 outlets of a chain of computer dealers were: 14 22 17 15 12 18 11 23 10 13 17 132 Quantitative methods for business 25 15 4.15 13 13 14 Chapter 20 17 18 13 13 16 15 15 23 15 10 14 15 Present these data in the form of a grouped frequency distribution 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 4.16 Compile a grouped frequency distribution for these figures 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 4.17 Create a frequency distribution to present these data The hourly wages (in £s) of 32 jobs offered by an employment agency are: 6.28 4.41 5.73 6.48 4.90 4.52 5.11 5.94 5.82 7.14 7.28 8.15 7.04 4.67 6.90 5.85 5.65 5.50 4.12 5.27 5.25 6.43 4.65 5.37 4.24 6.45 4.70 5.09 4.82 6.23 5.40 5.26 Construct a grouped frequency distribution for these figures 4.18* The monthly membership fees in £s for 22 health clubs are: 32 28 43 78 44 60 22 63 73 32 69 67 48 41 67 65 33 56 48 48 67 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 56 50 51 44 55 32 32 31 39 55 42 21 28 36 55 24 Arrange these data into a grouped frequency distribution using the same classes as in (a) Chapter Filling up – fuelling quantitative analysis 4.19 133 (c) Work out the relative frequencies for every class in the distributions and use them to compare the percentage of clubs charging £50 or more with the percentage of health centres charging £50 or more A company has plants in Manchester and Southampton Thirty staff from Manchester and 24 staff from Southampton attended a training course at the end of which they were each given a proficiency test The marks for the test (out of 100) for the staff from Manchester were: 49 67 35 53 72 50 70 49 58 35 62 47 55 87 54 38 64 39 44 67 44 79 70 40 67 57 64 85 63 34 52 89 59 38 The marks for the staff from Southampton were: 51 85 29 4.20 65 78 49 74 75 68 24 46 81 45 28 66 51 75 22 39 70 (a) Classify both sets of data into grouped frequency distributions using classes 10 marks wide (b) Find the relative frequency for each class of the distributions (c) The company considers marks of less than 40 in this test to be unsatisfactory Using your relative frequency figures compare the proportions of staff obtaining unsatisfactory results from the two locations A leisure company operates three amusement arcades in the UK: at Redcar, Skegness and Torquay As part of a performance review the duration in minutes of the period spent in the arcades by each of a sample of customers visiting was recorded The durations of visits made by 21 customers visiting the Redcar arcade were: 23 21 23 27 39 27 72 34 73 31 13 32 44 74 43 74 37 44 21 36 The figures for 18 customers visiting the Skegness arcade were: 31 35 51 45 69 48 12 25 53 28 32 36 28 60 66 36 30 The figures for 20 customers visiting the Torquay arcade were: 19 11 15 37 21 10 10 20 14 10 134 Quantitative methods for business Chapter (a) Classify both sets of data into grouped frequency distributions (b) Calculate the relative frequency for each class of all three distributions (c) The company expects customers to spend at least 20 minutes on visits to their arcades Use your relative frequency figures to compare the performances of the arcades in this respect ... employment agency are: 6.28 4. 41 5.73 6 .48 4. 90 4. 52 5.11 5. 94 5.82 7. 14 7.28 8.15 7. 04 4.67 6.90 5.85 5.65 5.50 4. 12 5.27 5.25 6 .43 4. 65 5.37 4. 24 6 .45 4. 70 5.09 4. 82 6.23 5 .40 5.26 Construct a grouped... 53 72 50 70 49 58 35 62 47 55 87 54 38 64 39 44 67 44 79 70 40 67 57 64 85 63 34 52 89 59 38 The marks for the staff from Southampton were: 51 85 29 4. 20 65 78 49 74 75 68 24 46 81 45 28 66 51... sale at a car hypermarket are: 48 60 1720 2350 2770 3 340 42 40 48 50 43 90 3870 2790 3 740 2230 1690 2750 1390 49 90 3660 1900 5200 43 90 3690 1760 48 00 1730 2 040 40 70 2670 4. 17 Create a frequency distribution