Case 13.2 Integrated Case: The Global Motors Survey Differences Analysis

CHAPTER



14

Learning Objectives

• Tolearnwhatismeantbyan

“association”betweentwo

variables



Making Use of

Associations Tests

Significant Associations Can Help

Managers Make Better Decisions



• Toexaminevariousrelationships

thatmaybeconstruedas

associations



Sima Vasa is Partner and Chief Executive

Officer of Paradigm Sample. She has 20 years



• Tounderstandwhereandhow

cross-tabulationswithChisquareanalysisareapplied



of experience in building and growing market

research businesses specifically in the technology space. Positions she has held include:



• Tobecomefamiliarwiththe

useandinterpretationof

correlations

• Tolearnhowtoobtainand

interpretcross-tabulations,Chisquarefindings,andcorrelations

withSPSS



Senior Partner at Momentum Market Intelligence, President of NPD Techworld at the

Sima Vasa,

Partner and CEO,

Paradigm Sample™



NPD Group, Vice President of the Technology

Division at NPD, and a member of the IBM

Market Intelligence Group. During her tenure



at NPD, Vasa spearheaded a $19 million business unit revenue increase



“Where We are”

1 Establish the need for marketing



research.

2 Define the problem.

3 Establish research objectives.

4 Determine research design.

5 Identify information types and



sources.

6 Determine methods of accessing



data.

7 Design data collection forms.

8 Determine the sample plan and size.

9 Collect data.







10 Analyze data.

11 Prepare and present the final



research report.



in three years to $40 million by refocusing priorities and aggressively

expanding the business portfolio.

As humans we are always searching for associations in life. What types

of recreational activities give you the most enjoyment? What types of foods

are tasteful to you? By knowing these associations, we can add enjoyment to our lives. Marketing managers are also searching for associations.

Which advertising copy will be associated with the highest awareness level

of the advertised brand? Which type of salesperson compensation and

reward packages will result in the highest level of satisfaction and the lowest turnover? By understanding these associations, marketers can achieve

higher sales and net profits while keeping costs low. This helps them earn

a higher ROA, which gives the owners of the firm a higher RONW.

Marketing managers learn these associations through trial and error,

just as you learn which activities give you the most pleasure or which

foods you prefer. However, sometimes the learning process takes too

long or being wrong is too costly in a business situation. Trial and error

is not always the best way to learn. In these situations a manager can

obtain information and examine it to see if there are associations. But,



what if information collected yesterday shows there is

an association, as in when ad copy B has the highest

awareness scores? How do we know that pattern of association will occur if we collect the information tomorrow? Or, next month? Is that pattern of association a

true pattern that exists in the population or only in the

one sample of information we collected? Fortunately,



Text and Images: By permission, Sima Vasa, Paradigm Sample™



the statisticians have given us ways to answer these questions. You will learn how to find

significant associations by reading this chapter.

Paradigm Sample provides market research insight, through innovative data access to

high-value audiences based on exclusive panels and partner panels along with proven mobile

technologies. To learn about Paradigm Sample’s innovative IdeaShifters and GlobalShifters

panels, refer to www.paradigmsample.com.



T



his chapter illustrates the usefulness of statistical analyses beyond simple descriptive

measures, statistical inference, and differences tests. Often, as we have described in the

opening comments of this chapter, marketers are interested in relationships among variables. For example, Frito-Lay wants to know what kinds of people and under what circumstances these people choose to buy Cheetos, Fritos, Lay’s potato chips, and any of the other

items in the Frito-Lay line. The Chevrolet Division of General Motors wants to know what

types of individuals would respond favorably to the various style changes proposed for the

Cruze. A newspaper wants to understand

the lifestyle characteristics of its subscribers so that it can modify or change

sections in the newspaper to better suit

its audience. Furthermore, the newspaper

desires information about various types

of subscribers so it can communicate this

information to its advertisers, helping

them in copy design and advertisement

placement within the various newspaper

sections. For all of these cases, statistical

procedures called associative analyses

are available to help identify answers to

these questions. Associative analyses

determine whether a stable relationship

exists between two variables; they are the

central topic of this chapter.

We begin the chapter by describing

the four different types of relationships

possible between two variables. Then

Photo: trekandshoot/Fotolia



Associative analyses

determine whether a

stable relationship exists

between two variables.



379



380



Chapter 14 • Making Use of assoCiations tests



we describe cross-tabulations and indicate how a cross-tabulation can be used to determine

whether a statistically significant association exists between the two variables. From crosstabulations, we move to a general discussion of correlation coefficients, and we illustrate the

use of Pearson product moment correlations. As in our previous analysis chapters, we show

the SPSS steps to perform these analyses and the resulting output.



Types of Relationships Between Two Variables



A relationship is a

consistent and systematic

linkage between the levels

or labels for two variables.



A nonmonotonic

relationship means two

variables are associated

but only in a general

sense.



As you learned in Chapter 8, every scale has unique descriptors, called levels or labels, that

identify the different demarcations of that scale. The term levels implies that the variable is

either an interval or a ratio scale; while the term labels implies that the level of measurement is

not scale, typically nominal. An example of a simple label is “yes” or “no,” as in if a respondent

is labeled as a buyer (yes) or nonbuyer (no) of a particular product or service. Of course, if the

researcher measures how many times a respondent bought a product, the level would be the

number of times, and the scale would satisfy the assumptions of a ratio scale.

A relationship is a consistent and systematic linkage between the levels for two scale

variables or between the labels for two nominal variables. This linkage is statistical, not

necessarily causal. A causal linkage is one in which it is certain one variable affected the

other; with a statistical linkage, there is no certainty because some other variable might

have had some influence. Nonetheless, statistical linkages or relationships often provide

insights that lead to understanding even though we do not know it there is a cause-andeffect relationship. For example, if we found a relationship that most marathon runners

purchased “Vitaminwater,” we understand that the ingredients are important to marathoners. Associative analysis procedures are useful because they determine if there is a

consistent and systematic relationship between the presence (label) or amount (level) of

one variable and the presence (label) or amount (level) of another variable. There are four

basic types of relationships between two variables: nonmonotonic, monotonic, linear, and

curvilinear. A discussion of each follows.



nOnmOnOtOnic reLatiOnships

A nonmonotonic relationship is one in which the presence (or absence) of the label for

one variable is systematically associated with the presence (or absence) of the label for

another variable. The term nonmonotonic means essentially that there

is no discernible direction to the relationship, but a relationship exists.

Drink Orders at McDonald’s

For example, McDonald’s, Burger King, and Wendy’s all know from

100%

experience that morning customers typically purchase coffee whereas

20%

80%

noon customers typically purchase soft drinks. The relationship is in

no way exclusive—there is no guarantee that a morning customer will

90%

60%

always order coffee or that an afternoon customer will always order a

80%

40%

soft drink. In general, though, this relationship exists, as can be seen in

Figure 14.1. The nonmonotonic relationship is simply that the morn20%

ing customer tends to purchase breakfast foods such as eggs, biscuits,

10%

0%

and coffee, and the afternoon customers tend to purchase lunch items

Breakfast

Lunch

such as burgers, fries, and soft drinks. So, the “morning” label is asCoffee

Soft Drink

sociated with the “coffee” label while the “noon” label is associated

with “soft drink” label. In other words, with a monotonic relationship,

FIGURE 14.1

when you find the presence of one label for a variable, you tend to find the presence of

Example of a Nonanother specific label of another variable: breakfast diners typically order coffee. But the

monotonic Relationassociation is general, and we must state each one by spelling it out verbally. In other

ship Between Drink

words, we know only the general pattern of presence or nonpresence with a nonmonotonic

Orders and Meal

relationship. We show you how to meaure nonmonotonic relationships using Chi-square

analysis later in this chapter.

Type at McDonald’s



types of relationships Between two VariaBles



mOnOtOnic reLatiOnships

In monotonic relationships, the researcher can assign

a general direction to the association between two variables. There are two types of monotonic relationships:

increasing and decreasing. Monotonic increasing relationships are those in which one variable increases as the

other variable increases. As you would guess, monotonic

decreasing relationships are those in which one variable

increases as the other variable decreases. You should note

that in neither case is there any indication of the exact

amount of change in one variable as the other changes.

Monotonic means that the relationship can be described

only in a general directional sense. Beyond this, precision in the description is lacking. For example, if a company increases its advertising, we would expect its sales

to increase, but we do not know the amount of the sales

increase. Monotonic relationships are not in the scope of

this textbook, so we will simply mention them here.



381



That higher SPF sunscreen blocks more ultraviolet rays is a

nonmonotonic relationship, meaning that the relationship is

not exact.

Photo: Yuri Arcurs/Shutterstock



Linear reLatiOnships

Next, we turn to a more precise relationship—and one that is very easy to envision. A linear

relationship is a “straight-line association” between two scale variables. Here, knowledge

of the amount of one variable will automatically yield knowledge of the amount of the other

variable as a consequence of applying the linear or straight-line formula that is known to exist

between them. In its general form, a straight-line formula is as follows:



A monotonic relationship

means you know the

general direction

(increasing or decreasing)

of the relationship

between two variables.



Formula for a straight line

y = a + bx

where

y = the dependent variable being estimated or predicted

a = the intercept

b = the slope

x = the independent variable used to predict the dependent variable

The terms intercept and slope should be familiar to you, but if they are a bit hazy, do not

be concerned as we describe the straight-line formula in detail in the next chapter. We also

clarify the terms independent and dependent in Chapter 15.

It should be apparent that a linear relationship is much more precise and contains a great

deal more information than does a nonmontonic or a monotonic relationship. By simply substituting the values of a and b, an exact amount can be determined for y given any value of x. For

example, if Jack-in-the-Box estimates that every customer will spent about $9 per lunch visit,

it is easy to use a linear relationship to estimate how many dollars of revenue will be associated

with the number of customers for any given location. The following equation would be used:

Straight-line formula example

y = 0 + 9 times number of customers

In this example, x is the number of customers. If 100 customers come to a Jack-inthe-Box location, the associated expected total revenues would be $0 plus $9 times 100, or

$900. If 200 customers are expected to visit the location, the expected total revenue would

be $0  plus  $9 times 200, or $1,800. To be sure, the Jack-in-the-Box location would not



A linear relationship means

the two variables have a

“straight-line” relationship.



382



Chapter 14 • Making Use of assoCiations tests



Linear relationships are

quite precise.



A curvilinear relationship

means some smooth curve

pattern describes the

association.



derive exactly $1,800 for 200 customers, but the linear relationship shows what is expected

to happen, on average. In subsequent sections of the textbook, we describe correlation and

regression analysis, both of which rely on linear relationships.

curviLinear reLatiOnships

Finally, curvilinear relationships are those in which one variable is associated with another

variable, but the relationship is described by a curve rather than a straight line. In other words,

the formula for a curved relationship is used rather than the formula for a straight line. Many

curvilinear patterns are possible. The relationship may be an S-shape, a J-shape, or some

other curved-shape pattern. Curvilinear relationships are beyond the scope of this text; nonetheless, it is important to list them as a type of relationship that can be investigated through the

use of special-purpose statistical procedures.



Characterizing Relationships Between Variables

Depending on its type, a relationship can usually be characterized in three ways: by its

presence, direction, and strength of association. We need to describe these before taking up

specific statistical analyses of associations between two variables.



The presence of a

relationship between two

variables is determined by

a statistical test.



Direction means that you

know if the relationship is

positive or negative, while

pattern means you know

the general nature of the

relationship.



Strength means you

know how consistent the

relationship is.



presence

Presence refers to the finding that a systematic relationship exists between the two variables

of interest in the population. Presence is a statistical issue. By this statement, we mean that the

marketing researcher relies on statistical significance tests to determine if there is sufficient

evidence in the sample to support the claim that a particular association is present in the population. The chapter on statistical inference introduced the concept of a null hypothesis. With

associative analysis, the null hypothesis states there is no association (relationship) present in

the population and the appropriate statistical test is applied to test this hypothesis. If the test

results reject the null hypothesis, then we can state that an association (relationship) is present in the population (at a certain level of confidence). We describe the statistical tests used in

associative analysis later in this chapter.

DirectiOn (Or pattern)

You have seen that in the cases of monotonic and linear relationships, associations may be

described with regard to direction. For a linear relationship, if b (slope) is positive, then the

linear relationship is increasing; if b is negative, then the linear relationship is decreasing.

So the direction of the relationship is straightforward with linear relationships.

For nonmonotonic relationships, positive or negative direction is inappropriate, because

we can only describe the pattern verbally.1 It will soon become clear to you that the scaling

assumptions of variables having nonmonotonic association negate the directional aspects

of the relationship. Nevertheless, we can verbally describe the pattern of the association

as we have in our examples using presence or absence, and that statement substitutes

for direction.

strength Of assOciatiOn

Finally, when present—that is, when statistically significant—the association between two

variables can be envisioned as to its strength, commonly using words such as “strong,”

“moderate,” “weak,” or some similar characterization. That is, when a consistent and systematic association is found to be present between two variables, it is then up to the marketing

researcher to ascertain the strength of association. Strong associations are those in which

there is a high probability of the two variables exhibiting a dependable relationship, regardless of the type of relationship being analyzed. A low degree of association, on the other hand,



Cross-taBUlations



TablE 14.1



383



Step-by-Step Procedure for Analyzing Relationships



step

1. Choose variables to analyze.

2. Determine the scaling assumptions of

the chosen variables.

3. Use the correct relationship analysis.



4. Determine if the relationship is present.

5. If present, determine the direction of

the relationship.

6. If present, assess the strength of the

relationship.



Description

Identify which variables you think might be related.

For purposes of this chapter, both must be either scale

(interval or ratio) or categorical (nominal) variables.

For two nominal variables (distinct categories),

use cross-tabulation; for two scale variables, use

correlation.

If the analysis shows the relationship is statistically

significant, it is present.

A linear (scale variables) relationship will be either

increasing or decreasing; a nonmonotonic relationship

(nominal scales) will require looking for a pattern.

With correlation, the size of the coefficient denotes

strength; with cross-tabulation, the pattern is subjectively assessed.



is one in which there is a low probability of the two variables’ exhibiting a dependable relationship. The relationship exists between the variables, but it is less evident.

There is an orderly procedure for determining the presence, direction, and strength of a

relationship, which is outlined in Table 14.1. As can be seen in the table, you must first decide what type of relationship can exist between the two variables of interest: nonmontonic

or linear. As you will learn in this chapter, the answer to this question depends on the scaling

assumptions of the variables; as we illustrated earlier, nominal scales can embody only imprecise, pattern-like relationships, but scale variables (interval or ratio scales) can incorporate

very precise and linear relationships. Once you identify the appropriate relationship type as

either nonmonotonic or linear, the next step is to determine whether that relationship actually

exists in the population you are analyzing. This step requires a statistical test, and again, we

describe the proper tests beginning with the next section of this chapter.

When you determine that a true relationship does exist in the population by means of the

correct statistical test, you then establish its direction or pattern. Again, the type of relationship dictates how you describe its direction. You might have to inspect the relationship in a

table or graph, or you might need only to look for a positive or negative sign before the computed statistic. Finally, the strength of the relationship remains to be judged. Some associative

analysis statistics, such as correlations, indicate the strength in a straightforward manner—

that is, just by their absolute size. With nominal-scaled variables, however, you must inspect

the pattern to judge the strength. We describe this procedure—the use of cross-tabulations—

next, and we describe correlation analysis later in this chapter.



Based on scaling

assumptions, first

determine the type of

relationship, and then

perform the appropriate

statistical test.



Cross-Tabulations

Cross-tabulation and the associated Chi-square value we are about to explain are used to assess

if a nonmonotonic relationship exists between two nominally scaled variables. Remember that

nonmonotonic relationships are those in which the presence of the label for one nominally scaled

variable coincides with the presence or absence of the label for another nominally scaled variable

such as lunch buyers ordering soft drinks with their meals. (Actually, cross-tabulation can be used

for any 2 variables with well-defined labels, but it is best demonstrated with nominal variables.)

crOss-tabuLatiOn anaLysis

When investigating the relationship between two nominally scaled variables, we typically

use “cross-tabs,” or the use of a cross-tabulation table, defined as a table in which data is



A cross-tabulation consists

of rows and columns

defined by the categories

classifying each variable.



384



Chapter 14 • Making Use of assoCiations tests



compared using a row and column format. A cross-tabulation table is sometimes referred to

as an “r × c” (or r-by-c) table because it is comprised of rows by columns. The intersection

of a row and a column is called a cross-tabulation cell. As an example, let’s take a survey where there are two types of individuals: buyers of Michelob Light beer and nonbuyers

of Michelob Light beer. There are also two types of occupations: professional workers who

might be called “white colar” employees and manual workers who are sometimes referred

to as “blue colar” workers. There is no requirement that the number of rows and columns

are equal; we are just using a 2 × 2 cross-tabulation to keep the example as simple as possible. A cross-tabulation table for our Michelob Light beer survey is presented in Table 14.2.

The columns are in vertical alignment and are indicated in this table as either “Buyer” or

“Nonbuyer” of Michelob Light, whereas the rows are indicated as “White Collar” or “Blue

Collar” for occupation. Additionally, there is a “Totals” column and row.



A cross-classification

table can have four types

of numbers in each cell:

frequency, raw percentage,

column percentage, and

row percentage.

Raw percentages are cell

frequencies divided by the

grand total.



types Of frequencies anD percentages

in a crOss-tabuLatiOn tabLe

Look at the frequencies table in Table 14.2A. It includes plus (+) and equal (=) signs to help

you learn the terminology and to understand how the numbers are computed. The frequencies table contains the raw numbers determined from the preliminary tabulation.2 The upper left-hand cell number is a frequency cell that counts people in the sample who are both

white-collar workers and buyers of Michelob Light (152), and the cell frequency to its right

identifies the number of individuals who are white-collar workers who do not buy Michelob

Light (8). These cell numbers represent raw counts or frequencies—that is, the number of

respondents who possess the quality indicated by the row label as well as the quality indicated

by the column label. The cell frequencies can be summed to determine the row totals and

column totals. For example, Buyer/White Collar (152) and Nonbuyer/White Collar (8) sum to

160, while Buyer/White Collar (152) and Buyer/Blue Collar (14) sum to 166. Similarly, the

row and column totals sum to equal the grand total of 200. Take a few minutes to be familiar

with the terms and computations in the frequencies table as they will be referred to in the

following discussion.

Table 14.2B illustrates how at least three different sets of percentages can be computed

for cells in the table. These three percentages tables are: the raw percentages table, the column

percentages table, and the row percentages table.

The first table in Table 14.2B shows that the raw frequencies can be converted to raw

percentages by dividing each by the grand total. The raw percentages table contains the percentages of the raw frequency numbers just discussed. The grand total location now has 100%

TablE 14.2a



Cross-Tabulation Frequencies Table for a Michelob Light

Survey

frequencies table

type of Buyer



 

White Collar

Buyer/White Collar

Cell Frequency



Occupational Status



Buyer

152



+



+



+

Blue Collar

Totals

Column Totals



14

=

166



nonbuyer

=

8



+

+



totals

160

+



26

=

34



=

=



Grand Total



40

=

200



Row

Totals



Cross-taBUlations



TablE 14.2b



Cross-Tabulation Percentages Tables for a Michelob Light Survey



Raw Percentages Table

 

 



 

White Collar

Buyer/White Collar

Cell Raw Percent



 

Occupational Status



 

Blue Collar



 

 



 

Totals



Buyer

76%

(152/200)

+

7%

(14/200)

=

83%

(166/200)



+



+



+



nonbuyer

4%

(8/200)

+

13%

(26/200)

=

17%

(34/200)



=



=



=



totals

80%

(160/200)

+

20%

(40/200)

=

100%

(200/200)



Column Percentages Table

 

 



 

White Collar



Buyer/White Collar

  Column Percent



Occupational Status



 

Blue Collar



 

 

 



 

 

Totals



Row Percentages Table

 

 

  Buyer/White Collar



White Collar



Occupational Status



Blue Collar



 



Totals



Row Percent



Buyer

92%

(152/166)

+

8%

(14/166)

=

100%

(166/166)

Buyer

95%

(152/160)

35%

(14/40)

83%

(166/200)



nonbuyer

24%

(8/34)

+

76%

(26/34)

=

100%

(34)



+

+

+



nonbuyer

5%

(8/160)

65%

(26/40)

17%

(34/200)



(or 200/200) of the grand total. Above it are 80% and 20% for the raw percentages of whitecollar occupational respondents and blue-collar occupational respondents, respectively, in the

sample. Compute a couple of the cells just to verify that you understand how they are derived.

For instance 152 ÷ 200 = 76%. Our + and = signs indicate how the totals are computed.

Two additional cross-tabulation tables can be presented, and these are more valuable in

revealing underlying relationships. The column percentages table divides the raw frequencies by its column total raw frequency. That is, the formula is as follows:

Formula for a column

cell percent



Column cell percent =



Cell frequency

Total of cell frequencies in that column



For instance, it is apparent that of the nonbuyers, 24% are white-collar and 76% are bluecollar. Note the reverse pattern for the buyers group: 92% of white-collar respondents are

Michelob Light buyers and 8% are blue-collar buyers. You are beginning to see the nonmonotonic relationship; in the presence of white collar we have the presence of buying.



totals

80%

(160/200)

+

20%

(40/200)

=

100%

(200/200)



=

=

=



totals

100%

(160/160)

100%

(40/40)

100%

(200/200)



385



386



Chapter 14 • Making Use of assoCiations tests



The row percentages table presents the data with the row totals as the 100% base for

each. That is, a row cell percentage is computed as follows:

Formula for a row

cell percent

Row (column) percentages

are row (column) cell

frequencies divided by

the row (column) total.



Row cell percent =



cell frequency

total of cell frequencies in that row



Now, it is possible to see that, of the white-collar respondents, 95% are buyers and 5%

are nonbuyers. As you compare the row percentages table to the column percentages table,

you should detect the relationship between occupational status and Michelob Light beer preference. Can you state it at this time?

Unequal percentage concentrations of individuals in a few cells, as we have in this example, illustrates the possible presence of a nonmonotonic association. If we had found that

approximately 25% of the sample had fallen in each of the four cells, no relationship would

be found to exist—it would be equally probable for a white or blue collar worker to be either

a buyer or nonbuyer. However, the large concentrations of individuals in two particular cells

here suggests that there is a high probability that a buyer of Michelob Light beer is also a

white-collar worker, and there is also a tendency for nonbuyers to work in blue-collar occupations. In other words, there is probably an association between occupational status and the

beer-buying behavior of individuals in the population represented by this sample. However,

as noted in step 4 of our procedure for analyzing relationships (Table 14.1), we must test the

statistical significance of the apparent relationship before we can say anything more about it.

The test will tell us if this pattern would occur again if we repeated the study.



Chi-Square Analysis

Chi-square analysis

assesses the statistical

significance of

nonmonotonic

associations in crosstabulation tables.



Expected frequencies

are calculated based on

the null hypothesis of

no association between

the two variables under

investigation.



Chi-square (χ2) analysis is the examination of frequencies for two nominal-scaled variables in a

cross-tabulation table to determine whether the variables have a statistically significant nonmonotonic relationship.3 The formal procedure for Chi-square analysis begins when the researcher formulates a statistical null hypothesis that the two variables under investigation are not associated in

the population. Actually, it is not necessary for the researcher to state this hypothesis in a formal

sense, for Chi-square analysis always implicitly takes this hypothesis into account. In other words,

whenever we use Chi-square analysis explicitly with a cross-tabulation, we always begin with the

assumption that no association exists between the two nominal-scaled variables under analysis.4

ObserveD anD expecteD frequencies

The statistical procedure is as follows. The cross-tabulation table in Table 14.2A contains

observed frequencies, which are the actual cell counts in the cross-tabulation table. These

observed frequencies are compared to expected frequencies, which are defined as the theoretical frequencies that are derived from the null hypothesis of no association between the two

variables. The degree to which the observed frequencies depart from the expected frequencies

is expressed in a single number called the Chi-square statistic. The computed Chi-square

statistic is then compared to a table Chi-square value (at a chosen level of significance) to

determine whether the computed value is significantly different from zero.

The expected frequencies are those that would be found if there were no association between the two variables. Remember, this is the null hypothesis. About the only “difficult” part

of Chi-square analysis is in the computation of the expected frequencies. The computation is

accomplished using the following equation:

Formula for an expected

cross-tabulation

cell frequency



Expected cell frequency =



cell column total * cell row total

grand total



Chi-sqUare analysis



387



The application of this equation generates a number for each cell that would occur if no

association existed. Returning to our Michelob Light beer example, you were told that 160

white-collar and 40 blue-collar consumers had been sampled, and it was found that there

were 166 buyers and 34 nonbuyers of Michelob Light. The expected frequency for each cell,

assuming no association, calculated with the expected cell frequency is as follows:

Calculations of

expected cell

frequencies

using the

Michelob Beer

example



160 * 166

= 132.8

200

160 * 34

White-collar nonbuyer =

= 27.2

200

40 * 166

Blue-collar buyer =

= 33.2

200

40 * 34

Blue-collar nonbuyer =

= 6.8

200

White-collar buyer =



Notes:

Buyers total = 166

Nonbuyers total = 34

White-collar total = 160

Blue-collar total = 40

Grand total = 200



the cOmputeD χ2 vaLue

Next, compare the observed frequencies to these expected frequencies. The formula for this

computation is as follows:

Chi-square formula



x2 = a

n



i- 1



1Observedi - Expectedi2 2

Expectedi



Where

Observedi = observed frequency in cell i

Expectedi = expected frequency in cell i

n = number of cells

Applied to our Michelob beer example,

Calculation of

Chi-square value

(Michelob example)



Notes:

x =

+

Observed frequencies

132.8

27.2

are in Table 14.2A.

2

2

114 - 33.22

126 - 6.82

+

+

= 81.64 Expected frequencies

33.2

6.8

are computed above.

2



1152 - 132.82 2



18 - 27.22 2



You can see from the equation that each expected frequency is compared (via subtraction) to

the observed frequency and squared to adjust for any negative values and to avoid the cancellation effect. This value is divided by the expected frequency to adjust for cell size differences, and

these amounts are summed across all of the cells. If there are many large deviations of observed

frequencies from the expected frequencies, the computed Chi-square value will increase; but if

there are only a few slight deviations from the expected frequencies, the computed Chi-square

number will be small. In other words, the computed Chi-square value is really a summary indication of how far away from the expected frequencies the observed frequencies are found to be. As

such, it expresses the departure of the sample findings from the null hypothesis of no association.

the chi-square DistributiOn

Now that you’ve learned how to calculate a Chi-square value, you need to know if it is statistically significant. In previous chapters, we described how the normal curve or z distribution, the F distribution, and Student’s t distribution, all of which exist in tables, are used



The computed Chi-square

value compares observed

to expected frequencies.



The Chi-square statistic

summarizes how far

away from the expected

frequencies the observed

cell frequencies are found

to be.



388



Chapter 14 • Making Use of assoCiations tests



Do these blue-collar workers want their boss to buy them a

Michelob Light for doing a great job? Cross-tabulation can

answer this question.

Photo: AISPIX by Image Source/Shutterstock



Formula for Chi-Square

degrees of freedom



by a computer statistical program to determine level of

significance. Chi-square analysis requires the use of a

different distribution. The Chi-square distribution is

skewed to the right, and the rejection region is always

at the right-hand tail of the distribution. It differs from

the normal and t distributions in that it changes its shape

depending on the situation at hand, and it does not have

negative values. Figure 14.2 shows examples of two

Chi-square distributions.

The Chi-square distribution’s shape is determined by

the number of degrees of freedom. The figure shows that

the more the degrees of freedom, the more the curve’s

tail is pulled to the right. In other words, the more the

degrees of freedom, the larger the Chi-square value must

be to fall in the rejection region for the null hypothesis.

It is a simple matter to determine the number of degrees of freedom. In a cross-tabulation table, the degrees

of freedom are found through the following formula:

Degrees of freedom = 1r - 12 1c - 12



Where:

r = the number of rows

c = the number of columns

The Chi-square

distribution’s shape

changes depending on

the number of degrees

of freedom.



A table of Chi-square values contains critical points that determine the break between

acceptance and rejection regions at various levels of significance. It also takes into account

the numbers of degrees of freedom associated with each curve. That is, a computed Chisquare value says nothing by itself—you must consider the number of degrees of freedom in

the cross-tabulation table because more degrees of freedom are indicative of higher critical

Chi-square table values for the same level of significance. The logic of this situation stems

from the number of cells. With more cells, there is more opportunity for departure from the

Chi-Square Curve for

4 Degrees of Freedom



Chi-Square Curve for

6 Degrees of Freedom



FIGURE 14.2 The

Chi-Square Curve’s

Shape Depends on its

Degrees of Freedom



0



Rejection Region is the

Right-Hand End of Curve