1. Chapter 16
Data Analysis: Frequency Distribution, Hypothesis Testing, and Cross-Tabulation

2. Relationship to Marketing
Figure Relationship of Frequency Distribution, Hypothesis Testing, and Cross-Tabulation to the Previous Chapters and the Marketing Research Process
Focus of this
Chapter
Relationship to
Previous Chapters
Approach to Problem
Field Work
Data Preparation
and Analysis
Report Preparation
and Presentation
Research Design
Problem Definition
Relationship to Marketing
Research Process
Frequency
General Procedure for Hypothesis Testing
Cross Tabulation
Research Questions and Hypothesis (Chapter 2)
Data Analysis Strategy (Chapter 15)
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3. Figure 16. 2 Frequency Distribution, Hypothesis
Figure Frequency Distribution, Hypothesis Testing, and Cross Tabulation: An Overview
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4. Opening Vignette What Would You Do?
Frequency Distribution
(Fig 16.3 & 16.4)(Tables 16.1 & 16.2)
Statistics Associated With Frequency Distribution (Fig 16.5)
Introduction to Hypothesis Testing
(Fig 16.6, 16.7, 16.8 & 16.9)
Be a DM! Be an MR! Experiential Learning
What Would You Do?
Cross Tabulation
(Tables 16.3, 16.4 & 16.5)
Statistics Associated With Cross Tabulation (Fig 16.1)
Cross Tabulation in Practice (Fig 16.11)
Application to Contemporary Issues (Figs 16.12, & 16.14)
International Social Media Ethics
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5. Frequency Distribution
In a frequency distribution, one variable is considered at a time.
A frequency distribution for a variable produces a table of frequency counts, percentages, and cumulative percentages for all the values associated with that variable.
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6. Figure 16.3 Conducting Frequency Analysis
Calculate the Frequency for Each Value of the Variable
Calculate the Percentage and Cumulative Percentage
Plot the Frequency Histogram
Calculate the Descriptive Statistics, Measures of Location, and Variability
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7. Usage and Attitude Toward Nike Shoes
Table 16.1
Usage and Attitude Toward Nike Shoes
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8. Cumulative Percentage
Table 16.2
Frequency Distribution of Attitude Toward Nike
Value Label
Value
Frequency
Percentage
Valid Percentage
Cumulative Percentage
Very unfavorable
Very favorable 1 2 3 4 5 6 7 9 8
11.1
13.3
17.8
20.0
8.9
2.2
11.4
13.6
18.2
20.5
9.1
Missing
25.0
38.6
52.3
70.5
90.9
100.0
Total 45
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9. Figure 16.4 Frequency Histogram
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10. Statistics Associated with Frequency Distribution Measures of Location
The mean, or average value, is the most commonly used measure of central tendency.
The mean, , is given by
Where,
Xi = Observed values of the variable X
n = Number of observations (sample size)
The mode is the value that occurs most frequently. It represents the highest peak of the distribution.
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11. Statistics Associated with Frequency Distribution Measures of Location
The median of a sample is the middle value when the data are arranged in ascending or descending order. If the number of data points is even, the median is usually estimated as the midpoint between the two middle values – by adding the two middle values and dividing their sum by 2. The median is the 50th percentile.
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12. Figure 16.5 Skewness of a Distribution
Skewed Distribution
Symmetric Distribution
Mean Median Mode (a)
Mean Median Mode (b)
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13. Statistics Associated with Frequency Distribution Measures of Variability
The range measures the spread of the data. It is simply the difference between the largest and smallest values in the sample.
Range = 𝑋 𝑙𝑎𝑟𝑔𝑒𝑠𝑡 − 𝑋 𝑠𝑚𝑎𝑙𝑙𝑒𝑠𝑡
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14. Statistics Associated with Frequency Distribution Measures of Variability
𝑆 𝑋 = 𝑖=1 𝑛 ( 𝑋 𝑖 − 𝑋 ) 2 𝑛−1
The variance is the mean squared deviation from the mean. The variance can never be negative.
The standard deviation is the square root of the variance.
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15. Figure 16.6 A General Procedure for Hypothesis Testing
Step 1
Step 2
Step 3
Step 4
Step 5
Step 6
Step 7
Step 8
Formulate H0 and H1
Select Appropriate Test
Choose Level of Significance, 
Collect Data and Calculate Test Statistic
Determine Probability
Associated with Test
Statistic (TSCAL)
Determine Critical
Value of Test Statistic
TSCR
Compare with Level of
Significance, 
Determine if TSCAL falls
into (Non) Rejection Region
Reject or Do Not Reject H0
Draw Marketing Research Conclusion a) b)
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16. A General Procedure for Hypothesis Testing Step 1: Formulate the Hypothesis
A null hypothesis is a statement of the status quo, one of no difference or no effect. If the null hypothesis is not rejected, no changes will be made.
An alternative hypothesis is one in which some difference or effect is expected. Accepting the alternative hypothesis will lead to changes in opinions or actions.
The null hypothesis refers to a specified value of the population parameter (e.g. μ, σ, π), not a sample statistic (e.g. 𝑋 ).
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17. A General Procedure for Hypothesis Testing Step 1: Formulate the Hypothesis (Cont.)
A null hypothesis may be rejected, but it can never be accepted based on a single test. In classical hypothesis testing, there is no way to determine whether the null hypothesis is true.
In marketing research, the null hypothesis is formulated in such a way that its rejection leads to the acceptance of the desired conclusion. The alternative hypothesis represents the conclusion for which evidence is sought.
𝐻 0 : 𝜋≤0.40
𝐻 1 : 𝜋>0.40
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18. A General Procedure for Hypothesis Testing Step 1: Formulate the Hypothesis (Cont.)
The test of the null hypothesis is a one-tailed test, because the alternative hypothesis is expressed directionally.
If that is not the case, then a two-tailed test would be required, and the hypotheses would be expressed as:
𝐻 0 : 𝜋=0.40
𝐻 1 : 𝜋≠0.40
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19. A General Procedure for Hypothesis Testing Step 2: Select an Appropriate Test
The test statistic measures how close the sample has come to the null hypothesis and follows a well-known distribution, such as the normal, t, or chi-square.
In our example, the z statistic, which follows the standard normal distribution, would be appropriate.
𝜎 𝑝 = 𝜋(1−𝜋) 𝑛
𝑧= 𝑝−𝜋 𝜎 𝑝
where
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20. A General Procedure for Hypothesis Testing Step 3: Choose a Level of Significance
Type I Error
Type I error occurs when the sample results lead to the rejection of the null hypothesis when it is in fact true.
The probability of type I error () is also called the level of significance.
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21. A General Procedure for Hypothesis Testing Step 3: Choose a Level of Significance (Cont.)
Type II Error
Type II error occurs when, based on the sample results, the null hypothesis is not rejected when it is in fact false.
The probability of type II error is denoted by .
Unlike , which is specified by the researcher, the magnitude of  depends on the actual value of the population parameter (proportion).
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22. A General Procedure for Hypothesis Testing Step 3: Choose a Level of Significance (Cont.)
Power of a Test
The power of a test is the probability (1 - ) of rejecting the null hypothesis when it is false and should be rejected.
Although  is unknown, it is related to . An extremely low value of  (e.g., = 0.001) will result in intolerably high  errors.
Therefore, it is necessary to balance the two types of errors.
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23. Figure 16.7 Type I Error (α) and Type II Error (β)
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24. Chosen Level of Significance α=0.05
Figure 16.8
Probability of z With a One-Tailed Test
Chosen Confidence Level = 95%
Chosen Level of Significance α=0.05
z = 1.645
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25. A General Procedure for Hypothesis Testing Step 4: Collect Data and Calculate Test Statistic
𝜎 𝑝 = 𝜋(1−𝜋) 𝑛 = (0.40)(0.60) 500 =0.0219
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26. A General Procedure for Hypothesis Testing Step 4: Collect Data and Calculate Test Statistic (Cont.)
The test statistic z can be calculated as follows:
𝑧= 𝑝−𝜋 𝜎 𝑝
= 0.44−
=1.83
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27. A General Procedure for Hypothesis Testing Step 5: Determine the Probability (Critical Value)
Using standard normal tables (Table 2 of the Statistical Appendix), the probability of obtaining a z value of 1.83 can be calculated (see Figure 15.5).
The shaded area between -  and 1.83 is Therefore, the area to the right of z = 1.83 is =
Alternatively, the critical value of z, which will give an area to the right side of the critical value of 0.05, is between 1.64 and 1.65 and equals
Note, in determining the critical value of the test statistic, the area to the right of the critical value is either  or /2. It is  for a one-tail test and /2 for a two-tail test.
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28. A General Procedure for Hypothesis Testing Steps 6 & 7: Compare the Probability (Critical Value) and Making the Decision
If the probability associated with the calculated or observed value of the test statistic (TSCAL) is less than the level of significance (), the null hypothesis is rejected.
The probability associated with the calculated or observed value of the test statistic is This is the probability of getting a p value of 0.44 when p = This is less than the level of significance of Hence, the null hypothesis is rejected.
Alternatively, if the absolute calculated value of the test statistic |(TSCAL)| is greater than the absolute critical value of the test statistic |(TSCR)|, the null hypothesis is rejected.
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29. A General Procedure for Hypothesis Testing Steps 6 & 7: Compare the Probability (Critical Value) and Making the Decision (Cont.)
The calculated value of the test statistic z = 1.83 lies in the rejection region, beyond the value of Again, the same conclusion to reject the null hypothesis is reached.
Note that the two ways of testing the null hypothesis are equivalent but mathematically opposite in the direction of comparison.
If the probability of TSCAL < significance level () then reject H0 but if |TSCAL | > |TSCR | then reject H0.
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30. A General Procedure for Hypothesis Testing Step 8: Marketing Research Conclusion
The conclusion reached by hypothesis testing must be expressed in terms of the marketing research problem.
In our example, we conclude that there is evidence that the proportion of customers preferring the new plan is significantly greater than Hence, the recommendation would be to introduce the new service plan.
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31. A Broad Classification of Hypothesis Testing Procedures
Figure 16.9
A Broad Classification of Hypothesis Testing Procedures
Hypothesis
Testing
Test of Association
Test of Difference
Means
Proportions
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32. Cross-Tabulation
While a frequency distribution describes one variable at a time, a cross-tabulation describes two or more variables simultaneously.
Cross-tabulation results in tables that reflect the joint distribution of two or more variables with a limited number of categories or distinct values, e.g., Table 16.3.
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33. A Cross-Tabulation of Gender and Usage of Nike Shoes
Table 16.3
A Cross-Tabulation of Gender and Usage of Nike Shoes
Usage
GENDER
Female
Male
Raw Total
Light Users 14 5 19
Medium Users 10
Heavy Users 11 16
Column Total 24 21
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34. Two Variables Cross-Tabulation
Since two variables have been cross classified, percentages could be computed either column wise, based on column totals (Table 16.4), or row wise, based on row totals (Table 16.5).
The general rule is to compute the percentages in the direction of the independent variable, across the dependent variable. The correct way of calculating percentages is as shown in Table 16.4.
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35. Usage of Nike Shoes by Gender
Table 16.4
Usage of Nike Shoes by Gender
Usage
GENDER
Female
Male
Light Users
58.4%
23.8%
Medium Users
20.8%
Heavy Users
52.4%
Column Total
100.0%
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36. Gender by Usage of Nike Shoes
Table 16.5
Gender by Usage of Nike Shoes
Usage
GENDER
Female
Male
Raw Total
Light Users
73.7%
26.3%
100.0%
Medium Users
50.0%
Heavy Users
31.2%
68.8%
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37. Statistics Associated with Cross-Tabulation Chi-Square
To determine whether a systematic association exists, the probability of obtaining a value of chi-square as large or larger than the one calculated from the cross-tabulation is estimated.
An important characteristic of the chi-square statistic is the number of degrees of freedom (df) associated with it. That is, df = (r - 1) x (c -1).
The null hypothesis (H0) of no association between the two variables will be rejected only when the calculated value of the test statistic is greater than the critical value of the chi-square distribution with the appropriate degrees of freedom, as shown in Figure
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38. Level of Significance, α
Figure Chi-Square Test of Association
Level of Significance, α
 2
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39. Statistics Associated with Cross-Tabulation Chi-Square (Cont.)
The chi-square statistic (2) is used to test the statistical significance of the observed association in a cross-tabulation. The expected frequency for each cell can be calculated by using a simple formula:
𝑓 𝑒 = 𝑛 𝑟 𝑛 𝑐 𝑛
where = total number in the row
= total number in the column
= total sample size
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40. Expected Frequency
For the data in Table 16.3, for the six cells from left to right and top to bottom
= (24 x 19)/45 = = (21 x 19)/45 =8.9
= (24 x 10)/45 = = (21 x 10)/45 =4.7
= (24 x 16)/45 = = (21 x 16)/45 =7.5
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41. χ 2 = 𝑎𝑙𝑙 𝑐𝑒𝑙𝑙𝑠 ( 𝑓 0 − 𝑓 𝑒 ) 2 𝑓 𝑒 = (14 -10.1)2 + (5 – 8.9)2
χ 2 = 𝑎𝑙𝑙 𝑐𝑒𝑙𝑙𝑠 ( 𝑓 0 − 𝑓 𝑒 ) 2 𝑓 𝑒
= ( )2 + (5 – 8.9)2
+ (5 – 5.3)2 + (5 – 4.7)2
+ (5 – 8.5)2 + (11 – 7.5)2 =
= 6.33
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42. Statistics Associated with Cross-Tabulation Chi-Square (Cont.)
The chi-square distribution is a skewed distribution whose shape depends solely on the number of degrees of freedom. As the number of degrees of freedom increases, the chi-square distribution becomes more symmetrical.
Table 3 in the Statistical Appendix contains upper-tail areas of the chi-square distribution for different degrees of freedom. For 2 degrees of freedom, the probability of exceeding a chi-square value of is 0.05.
For the cross-tabulation given in Table 16.3, there are (3-1) x (2-1) = 2 degrees of freedom. The calculated chi-square statistic had a value of Since this is greater than the critical value of 5.991, the null hypothesis of no association is rejected indicating that the association is statistically significant at the 0.05 level.
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43. Statistics Associated with Cross-Tabulation Phi Coefficient
The phi coefficient () is used as a measure of the strength of association in the special case of a table with two rows and two columns (a 2 x 2 table).
The phi coefficient is proportional to the square root of the chi-square statistic:
It takes the value of 0 when there is no association, which would be indicated by a chi-square value of 0 as well. When the variables are perfectly associated, phi assumes the value of 1 and all the observations fall just on the main or minor diagonal.
 = 𝜒 2 𝑛
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44. Statistics Associated with Cross-Tabulation Contingency Coefficient
While the phi coefficient is specific to a 2 x 2 table, the contingency coefficient (C) can be used to assess the strength of association in a table of any size.
The contingency coefficient varies between 0 and 1.
The maximum value of the contingency coefficient depends on the size of the table (number of rows and number of columns). For this reason, it should be used only to compare tables of the same size.
𝐶= 𝜒 2 𝜒 2 +𝑛
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45. Statistics Associated with Cross-Tabulation Cramer’s V
Cramer's V is a modified version of the phi correlation coefficient, , and is used in tables larger than 2 x 2. or
𝑉=  2 min 𝑟−1 ,(𝑐−1) .
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46. Cross-Tabulation in Practice
While conducting cross-tabulation analysis in practice, it is useful to proceed along the following steps:
Test the null hypothesis that there is no association between the variables using the chi-square statistic. If you fail to reject the null hypothesis, then there is no relationship.
If H0 is rejected, then determine the strength of the association using an appropriate statistic (phi-coefficient, contingency coefficient, or Cramer's V), as discussed earlier.
If H0 is rejected, interpret the pattern of the relationship by computing the percentages in the direction of the independent variable, across the dependent variable. Draw marketing conclusions.
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47. Figure 16.11 Conducting Cross-Tabulation Analysis
Construct the Cross-Tabulation Data
Calculate the Chi-Square Statistic, Test the Null Hypothesis of No Association
Reject H0?
Interpret the Pattern of Relationship by Calculating Percentages in the Direction of the Independent Variable
No Association
Determine the Strength of Association Using an Appropriate Statistic NO
YES
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48. Copyright © 2011 Pearson Education, Inc.

49. Copyright © 2011 Pearson Education, Inc.

50. Copyright © 2011 Pearson Education, Inc.

51. International Marketing Research
Report preparation can be complicated by the need to prepare reports for management in different countries and in different languages.
In oral presentations, the presenter should be sensitive to cultural norms. for example, telling jokes, which is frequently done in the united states, is not appropriate in all cultures.
Different recommendations might be made for implementing the research findings in different countries.
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52. Marketing Research & Social Media
Social media, particularly blogs and Twitter, can play a crucial role in disseminating the results and the report of a marketing research project and the decisions made by the company based on the findings.
Blogs can also provide an avenue for a company to obtain consumer reaction to the research findings as well as their feedback on the decisions made and actions taken by the company based on the research findings.
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53. Marketing Research & Social Media (Cont.)
Social media research results can be effectively presented using charts and graphs such as the Twitter trends’ statistics graph.
Social media community members’ stories can often be effective illustrations of statistical findings when used in reports or executive presentations.
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54. Ethics in Social Media
Ethical issues include ignoring pertinent data when drawing conclusions or making recommendations, not reporting relevant information (such as low response rates), deliberately misusing statistics, falsifying figures, altering research results, and misinterpreting the results with the objective of supporting a personal or corporate viewpoint.
The researchers should prepare reports that accurately and fully document the details of all the procedures and findings.
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55. Ethics in Social Media (Cont.)
Clients also have the responsibility for full and accurate disclosure of the research findings and are obligated to employ these findings honorably.
ethical issues also arise when client firms, such as tobacco companies, use marketing research findings to formulate questionable marketing programs.
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56. SPSS Windows: Frequencies
The main program in SPSS is FREQUENCIES. It produces a table of frequency counts, percentages, and cumulative percentages for the values of each variable. It gives all of the associated statistics.
If the data are interval scaled and only the summary statistics are desired, the DESCRIPTIVES procedure can be used.
The EXPLORE procedure produces summary statistics and graphical displays, either for all of the cases or separately for groups of cases. Mean, median, variance, standard deviation, minimum, maximum, and range are some of the statistics that can be calculated.
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57. SPSS Windows: Frequencies
To select these frequencies procedures, click the following:
Analyze > Descriptive Statistics > Frequencies . . . or
Analyze > Descriptive Statistics > Descriptives . . .
Analyze > Descriptive Statistics > Explore . . .
We illustrate the detailed steps using the data of Table 16.1.
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58. SPSS Detailed Steps: Frequencies
Select ANALYZE on the SPSS menu bar.
Click DESCRIPTIVE STATISTICS, and select FREQUENCIES.
Move the variable "Attitude toward Nike <attitude>" to the VARIABLE(S) box.
Click STATISTICS.
Select MEAN, MEDIAN, MODE, STD. DEVIATION, VARIANCE, and RANGE.
Click CONTINUE.
Click CHARTS.
Click HISTOGRAMS, then click CONTINUE.
Click OK.
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59. SPSS Windows: Cross-tabulations
The major cross-tabulation program is CROSSTABS. This program will display the cross-classification tables and provide cell counts, row and column percentages, the chi-square test for significance, and all the measures of the strength of the association that have been discussed. To select these procedures, click the following:
Analyze > Descriptive Statistics > Crosstabs
We illustrate the detailed steps using the data of Table 16.1.
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60. SPSS Detailed Steps: Cross-Tabulations
Select ANALYZE on the SPSS menu bar.
Click DESCRIPTIVE STATISTICS, and select CROSSTABS.
Move the variable "User Group <usergr>" to the ROW(S) box.
Move the variable "Sex <sex>" to the COLUMN(S) box.
Click CELLS.
Select OBSERVED under COUNTS, and select COLUMN under PERCENTAGES.
Click CONTINUE.
Click STATISTICS.
Click CHI-SQUARE, PHI, and CRAMER'S V.
Click OK.
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61. Excel: Frequencies
The Data > Data Analysis > Histogram function computes the descriptive statistics. The output produces the mean, standard error, median, mode, standard deviation, variance, range, minimum, maximum, sum, count, and confidence level. Frequencies can be selected under the histogram function. A histogram can be produced in bar format.
We illustrate the detailed steps using the data of Table 16.1.
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62. Excel Detailed Steps: Frequencies
Select DATA tab.
In the ANALYSIS Group, select DATA ANALYSIS.
The DATA ANALYSIS Window pops up.
Select HISTOGRAM from the DATA ANALYSIS Window.
Click OK.
The HISTOGRAM pop-up window appears on screen.
The HISTOGRAM window has two portions:
a. INPUT
b. OUTPUT OPTIONS
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63. Excel Detailed Steps: Frequencies (Cont.)
Input portion asks for two inputs.
a. Click in the INPUT RANGE box and select (highlight) all 45
rows under ATTITUDE. $D$2: $D$46 should appear
in the INPUT RANGE box.
b. Do not enter anything into the BIN RANGE box.
(Note that this input is optional; if you do not provide a
reference, the system will take values based on
range of the data set).
In the OUTPUT portion of pop-up window, select the following options:
a. NEW WORKBOOK
b. CUMULATIVE PERCENTAGE
c. CHART OUTPUT
Click OK.
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64. Excel: Cross-Tabulations
The Insert > Pivot Table function performs cross-tabulations in Excel. To do additional analysis or customize data, select a different summary function, such as maximum, minimum, average, or standard deviation. In addition, a custom calculation can be selected to analyze values based on other cells in the data plane. The Chi-square Test can be accessed under Formulas > Insert Function > ChiTest.
We illustrate the detailed steps using the data of Table 16.3
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65. Excel Detailed Steps: Cross-Tabulations
Select Insert (Alt + N).
Click on Pivot Table. The Pivot Table window pops up.
Select columns A to D and rows 1 to 46. “$A$1:$D$46” should appear in the range box.
Select NEW WORKSHEET in the CREATE PIVOT TABLE window.
Click OK.
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66. Excel Detailed Steps: Cross-Tabulations (Cont.)
6. Drag variables into the layout on the left in the following format:
SEX
USERGR CASENO (Double-click CASENO
and select Count)
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67. Excel Detailed Steps: Cross-Tabulations (Cont.)
Right click on SUM OF CASENO in the lower-right corner. Select “VALUE FIELD SETTINGS”.
Under “SUMMARIZE VALUE FIELD BY” select COUNT.
Click OK.
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68. Exhibit 16.1 Other Computer Programs: Frequencies
MINITAB
The main function is STAT > DESCRIPTIVE STATISTICS. The output values include the mean, median, mode, standard deviation, minimum, maximum, and quartiles. A histogram in a bar chart or graph can be produced from the GRAPH > HISTOGRAM selection.
SAS
The SUMMARY STATISTICS task provides summary statistics including basic summary statistics, percentile summary statistics, and more advanced summary statistics including confidence intervals, t statistics, coefficient of variation, and sums of squares. It also provides graphical displays including histograms. The ONE-WAY FREQUENCIES task can be used to generate frequency tables as well as binomial and chi-square tests.
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69. Exhibit 16.2 Other Computer Programs: Cross-Tabulations
MINITAB
In Minitab, cross-tabulations and chi-square are under the STAT > TABLES function. Each of these features must be selected separately under the Tables function.
SAS
The major cross-tabulation task is called TABLE ANALYSIS. This task will display the cross-classification table and provide cell counts, row and column percentages, the chi-square test for significance, and all the measures of strength of the association that have been discussed.
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70. Acronym: Frequencies
The statistics associated with frequencies can be summarized by the acronym FREQUENCIES: F requency histogram R ange E stimate of location: mean Q uotients: percentages U ndulation: variance E stimate of location: mode N umbers or counts C umulative percentage I ncorrect and missing values E stimate of location: median S hape of the distribution
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71. Acronym: Tabulate
The salient characteristics of cross-tabulations can be summarized by the acronym TABULATE: T wo variables at a time A ssociation and not causation is measured B ased on cell count of at least five U derstood easily by managers L imited number of categories A ssociated statistics T wo ways to calculate percentages E xpected cell frequencies
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