1. Some Preliminaries
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2. Basics of Analysis The process of data analysis Example 1: Observation
Gift Catalog Marketer
Mails 4 times a year to its customers
Company has I million customers on its file
Observation
Data
Information
Analysis
Encode
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3. Analyst takes a sample of 100,000 customers and notices the following.
Example 1
Cataloger would like to know if new customers buy more than old customers?
Classify New Customers as anyone who brought within the last twelve months.
Analyst takes a sample of 100,000 customers and notices the following.
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4. 5000 orders received in the last month
Example 1
5000 orders received in the last month
3000 (60%) were from new customers
2000 (40%) were from old customers
So it looks like the new customers are doing better
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5. Example 1 Is there any Catch here!!!!!
Data at this gross level, has no discrimination between customers within either group.
A customer who bought within the last 11 days is treated exactly similar to a customer who bought within the last 11 months.
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6. Example 1
Can we use some other variable to distinguish between old and new Customers?
Answer: Actual Dollars spent !
What can we do with this variable?
Find its Mean and Variation.
We might find that the average purchase amount for old customers is two or three times larger than the average among new customers
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7. Numerical Summaries of data
The two basic concepts are the center and the Spread of the data
Center of data
- Mean, which is given by
- Median
- Mode
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8. Numerical Summaries of data
Forms of Variation
Sum of differences about the mean:
Variance:
Standard Deviation: Square Root of Variance
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9. Confidence Intervals
In catalog eg, analyst wants to know average purchase amount of customers
He draws two samples of 75 customers each and finds the means to be $68 and $122
Since difference is large, he draws another 38 samples of 75 each
The mean of means of the 40 samples turns out to be $ 94.85
How confident should he be of this mean of means?
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10. Confidence Intervals
Analyst calculates the standard deviation of sample means, called Standard Error (SE). It is 12.91
Basic Premise for confidence Intervals
95 percent of the time the true mean purchase amount lies between plus or minus 1.96 standard errors from the mean of the sample means.
C.I. = Mean (+or-) (1.96) * Standard Error
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11. Confidence Intervals
However, if CI is calculated with only one sample then
Standard Error of sample mean
= Standard deviation of sample
Basic Premise for confidence Intervals with one sample
95 percent of the time the true mean lies between plus or minus 1.96 standard errors from the sample means.
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12. Example 2: Confidence Intervals for response rates
You are the marketing analyst for Online Apparel Company
You want to run a promotion for all customers on your database
In the past you have run many such promotions
Historically you needed a 4.5% response for the promotions to break-even
You want to test the viability of the current full-scale promotion by running a small test promotion
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13. Example 2: Confidence Intervals for response rates
Test 1,000 names selected at random from the full list.
You construct CI based on required rate of 4.5% and n=1000
Confidence Interval= Expected Response ± 1.96*SE
The SE=.00655, and CI is (.0322, .0578)
In our case C.I. = 3.22 % to 5.78%. Thus any response between 3.22 and 5.78 % supports hypothesis that true response rate is 4.5%
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14. Example 2: Confidence Intervals for response rates
The list is mailed and actually pulls in 3.5%
Thus, the true response rate maybe 4.5%
What if the actual rate pulled in were 5% ?
Regression towards mean: Phenomenon of test result being different from true result
Give more thought to lists whose cutoff rates lie within confidence interval
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15. Cross-Tabulation Frequency Distribution and © 2007 Prentice Hall 15

16. Chapter Outline 1) Frequency Distribution
2) Statistics Associated with Frequency Distribution
Measures of Location
Measures of Variability
Measures of Shape
3) Cross-Tabulations
Two Variable Case
Three Variable Case
General Comments on Cross-Tabulations
4) Statistics for Cross-Tabulation: Chi-Square
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17. Internet Usage Data Table 15.1
Respondent Sex Familiarity Internet Attitude Toward Usage of Internet
Number Usage Internet Technology Shopping Banking
Table 15.1
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18. 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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19. Frequency Distribution of Familiarity with the Internet
Table 15.2
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20. Frequency Histogram 2 3 4 5 6 7 1 Frequency Familiarity 8 Fig. 15.11
Frequency
Familiarity 8
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21. Statistics for 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. The mode is a good measure of location when the variable is inherently categorical or has otherwise been grouped into categories. X = i / n S 1
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22. Statistics for 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 the midpoint between the two middle values. The median is the 50th percentile.
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23. Statistics for Frequency Distribution: Measures of Variability
The range measures the spread of the data.
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.
The coefficient of variation is the ratio of the standard deviation to the mean expressed as a percentage, and is a unitless measure of relative variability. C V = s x / X
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24. Statistics for Frequency Distribution: Measures of Shape
Skewness. The tendency of the deviations from the mean to be larger in one direction than in the other. Tendency for one tail of the distribution to be heavier than the other.
Kurtosis is a measure of the relative peakedness or flatness of the frequency distribution curve. The kurtosis of a normal distribution is zero.
-kurtosis>0, then dist is more peaked than normal dist.
-kurtosis<0, then dist is flatter than a normal distribution.
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25. Skewness of a Distribution
Fig. 15.2
Skewed Distribution
Symmetric Distribution
Mean Median Mode (a)
Mean Median Mode (b)
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26. 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 15.3.
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27. Gender and Internet Usage
Table 15.3
Gender
Row
Internet Usage
Male
Female
Total
Light (1) 5 10 15
Heavy (2)
Column Total 1
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28. Two Variables Cross-Tabulation
Since two variables have been cross-classified, percentages could be computed either columnwise, based on column totals (Table 15.4), or rowwise, based on row totals (Table 15.5).
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 15.4.
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29. Internet Usage by Gender
Table 15.4
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30. Gender by Internet Usage
Table 15.5
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31. Introduction of a Third Variable in Cross-Tabulation
Refined Association between the Two Variables
No Association between the Two Variables
No Change in the Initial Pattern
Some Association between the Two Variables
Fig. 15.7
Introduce a Third Variable
Original Two Variables
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32. 3 Variables Cross-Tab: Refine an Initial Relationship
As can be seen from Table 15.6, 52% (31%) of unmarried (married) respondents fell in the high-purchase category
Do unmarried respondents purchase more fashion clothing?
A third variable, the buyer's sex, was introduced
As shown in Table 15.7,
- 60% (25%) of unmarried (married) females fell in the high-purchase category
- 40% (35%) of unmarried (married) males fell in the high-purchase category.
Unmarried respondents are more likely to fall in the high purchase category than married ones, and this effect is much more pronounced for females than for males.
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33. Purchase of Fashion Clothing by Marital Status
Table 15.6
Purchase of
Fashion
Current Marital Status
Clothing
Married
Unmarried
High
31%
52%
Low
69%
48%
Column
100%
Number of
respondents
700
300
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34. Purchase of Fashion Clothing by Marital Status and Gender
Table 15.7
Purchase of
Fashion
Clothing
Sex
Male
Female
Married
Not
High
35%
40%
25%
60%
Low
65%
75%
Column
totals
100%
Number of
cases
400
120
300
180
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35. 3 Variables Cross-Tab: Initial Relationship was Spurious
Table 15.8 shows that 32% (21%) of those with (without) college degrees own an expensive automobile
Income may also be a factor
In Table 15.9, when the data for the high income and low income groups are examined separately, the association between education and ownership of expensive automobiles disappears,
Initial relationship observed between these two variables was spurious.
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36. Ownership of Expensive Automobiles by Education Level
Table 15.8
Own Expensive
Automobile
Education
College Degree
No College Degree
Yes No
Column totals
Number of cases
32%
68%
100%
250
21%
79%
750
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37. Ownership of Expensive Automobiles by Education Level and Income Levels
Table 15.9
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38. 3 Variables Cross-Tab: Reveal Suppressed Association
Table shows no association between desire to travel abroad and age.
In Table 15.11, sex was introduced as the third variable.
Controlling for effect of sex, the suppressed association between desire to travel abroad and age is revealed for the separate categories of males and females.
Since the association between desire to travel abroad and age runs in the opposite direction for males and females, the relationship between these two variables is masked when the data are aggregated across sex as in Table
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39. Desire to Travel Abroad by Age
Table 15.10
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40. Desire to Travel Abroad by Age and Gender
Table 15.11
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41. Three Variables Cross-Tabulations No Change in Initial Relationship
Consider the cross-tabulation of family size and the tendency to eat out frequently in fast-food restaurants as shown in Table No association is observed.
When income was introduced as a third variable in the analysis, Table was obtained. Again, no association was observed.
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42. Eating Frequently in Fast-Food Restaurants by Family Size
Table 15.12
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43. Eating Frequently in Fast Food-Restaurants by Family Size and Income
Table 15.13
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44. Statistics Associated with Cross-Tab: Chi-Square
H0: there is no association between the two variables
Use chi-square statistic
H0 will be rejected when the calculated value of the test statistic is greater than the critical value of the chi-square distribution
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45. Statistics Associated with Cross-Tab: Chi-Square
compares the observed cell frequencies (fo) to the frequencies to be expected when there is no association between variables (fe)
The expected frequency for each cell can be calculated by using a simple formula:
nr=total number in the row
nc=total number in the column
n=total sample size
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46. Statistics for Cross-Tab: Chi-Square
From Table 3 in the Statistical Appendix, the probability of exceeding a chi-square value of is 0.05.
The calculated chi-square is Since this is less than the critical value of 3.841, the null hypothesis can not be rejected
Thus, the association is not statistically significant at the 0.05 level.
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