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Transcript of © 2007 Prentice Hall16-1 Some Preliminaries. © 2007 Prentice Hall16-2 Basics of Analysis The...
© 2007 Prentice Hall 16-1
Some Preliminaries
© 2007 Prentice Hall 16-2
Basics of Analysis
The process of data analysis
Example 1: Gift Catalog Marketer Mails 4 times a year to its customers Company has I million customers on its file
Observation Data Information
Encode Analysis
© 2007 Prentice Hall 16-3
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.
© 2007 Prentice Hall 16-4
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
© 2007 Prentice Hall 16-5
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.
© 2007 Prentice Hall 16-6
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
© 2007 Prentice Hall 16-7
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
n
xx
n
ii
1
© 2007 Prentice Hall 16-8
Numerical Summaries of data
Forms of Variation
Sum of differences about the mean:
Variance:
Standard Deviation: Square Root of Variance
1
)(1
2
n
xxn
ii
)(1
n
ii xx
© 2007 Prentice Hall 16-9
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?
© 2007 Prentice Hall 16-10
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
© 2007 Prentice Hall 16-11
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.
n
© 2007 Prentice Hall 16-12
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
© 2007 Prentice Hall 16-13
Example 2: Confidence Intervals for response rates
Test 1,000 names selected at random from a new list.
To break-even the list must be expected to have a response rate of 4.5 percent
Confidence Interval= Expected Response (+/-) 1.96*SE
= p(+/-) 1.96*SE
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%
© 2007 Prentice Hall 16-14
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
© 2007 Prentice Hall 16-15
Frequency Distribution and Cross-Tabulation
© 2007 Prentice Hall 15
© 2007 Prentice Hall 16-16
Chapter Outline1) Frequency Distribution
2) Statistics Associated with Frequency Distribution
i. Measures of Location
ii. Measures of Variability
iii. Measures of Shape
3) Cross-Tabulations
i. Two Variable Case
ii. Three Variable Case
iii. General Comments on Cross-Tabulations
4) Statistics for Cross-Tabulation: Chi-Square
© 2007 Prentice Hall 16-17
Internet Usage Data Respondent Sex Familiarity Internet Attitude Toward Usage of InternetNumber Usage Internet Technology Shopping Banking 1 1.00 7.00 14.00 7.00 6.00 1.00 1.002 2.00 2.00 2.00 3.00 3.00 2.00 2.003 2.00 3.00 3.00 4.00 3.00 1.00 2.004 2.00 3.00 3.00 7.00 5.00 1.00 2.00 5 1.00 7.00 13.00 7.00 7.00 1.00 1.006 2.00 4.00 6.00 5.00 4.00 1.00 2.007 2.00 2.00 2.00 4.00 5.00 2.00 2.008 2.00 3.00 6.00 5.00 4.00 2.00 2.009 2.00 3.00 6.00 6.00 4.00 1.00 2.0010 1.00 9.00 15.00 7.00 6.00 1.00 2.0011 2.00 4.00 3.00 4.00 3.00 2.00 2.0012 2.00 5.00 4.00 6.00 4.00 2.00 2.0013 1.00 6.00 9.00 6.00 5.00 2.00 1.0014 1.00 6.00 8.00 3.00 2.00 2.00 2.0015 1.00 6.00 5.00 5.00 4.00 1.00 2.0016 2.00 4.00 3.00 4.00 3.00 2.00 2.0017 1.00 6.00 9.00 5.00 3.00 1.00 1.0018 1.00 4.00 4.00 5.00 4.00 1.00 2.0019 1.00 7.00 14.00 6.00 6.00 1.00 1.0020 2.00 6.00 6.00 6.00 4.00 2.00 2.0021 1.00 6.00 9.00 4.00 2.00 2.00 2.0022 1.00 5.00 5.00 5.00 4.00 2.00 1.0023 2.00 3.00 2.00 4.00 2.00 2.00 2.0024 1.00 7.00 15.00 6.00 6.00 1.00 1.0025 2.00 6.00 6.00 5.00 3.00 1.00 2.0026 1.00 6.00 13.00 6.00 6.00 1.00 1.0027 2.00 5.00 4.00 5.00 5.00 1.00 1.0028 2.00 4.00 2.00 3.00 2.00 2.00 2.00 29 1.00 4.00 4.00 5.00 3.00 1.00 2.0030 1.00 3.00 3.00 7.00 5.00 1.00 2.00
Table 15.1
© 2007 Prentice Hall 16-18
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.
© 2007 Prentice Hall 16-19
Frequency Distribution of Familiaritywith the Internet
Table 15.2
Valid Cumulative Value label Value Frequency (N) Percentage percentage percentage Not so familiar 1 0 0.0 0.0 0.0 2 2 6.7 6.9 6.9 3 6 20.0 20.7 27.6 4 6 20.0 20.7 48.3 5 3 10.0 10.3 58.6 6 8 26.7 27.6 86.2 Very familiar 7 4 13.3 13.8 100.0 Missing 9 1 3.3 TOTAL 30 100.0 100.0
© 2007 Prentice Hall 16-20
Frequency Histogram
Fig. 15.1
2 3 4 5 6 70
7
4
3
2
1
6
5
Frequ
en
cy
Familiarity
8
© 2007 Prentice Hall 16-21
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.
Statistics for Frequency Distribution: Measures of Location
X = X i/ni=1
nX
© 2007 Prentice Hall 16-22
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.
Statistics for Frequency Distribution: Measures of Location
© 2007 Prentice Hall 16-23
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.
CV = sx/X
Statistics for Frequency Distribution: Measures of Variability
© 2007 Prentice Hall 16-24
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.
Statistics for Frequency Distribution: Measures of Shape
© 2007 Prentice Hall 16-25
Skewness of a DistributionFig. 15.2
Skewed Distribution
Symmetric Distribution
Mean Media
n Mode
(a)Mean Median
Mode (b)
© 2007 Prentice Hall 16-26
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.
© 2007 Prentice Hall 16-27
Gender and Internet UsageTable 15.3
Gender
RowInternet Usage Male Female Total
Light (1) 5 10 15
Heavy (2) 10 5 15
Column Total 15 15
© 2007 Prentice Hall 16-28
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.
© 2007 Prentice Hall 16-29
Internet Usage by GenderTable 15.4
Gender Internet Usage Male Female Light 33.3% 66.7% Heavy 66.7% 33.3% Column total 100% 100%
© 2007 Prentice Hall 16-30
Gender by Internet UsageTable 15.5
Internet Usage Gender Light Heavy Total Male 33.3% 66.7% 100.0% Female 66.7% 33.3% 100.0%
© 2007 Prentice Hall 16-31
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
Some Association between the Two
Variables
No Association between the Two
Variables
Introduce a Third Variable
Introduce a Third Variable
Original Two Variables
© 2007 Prentice Hall 16-32
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.
3 Variables Cross-Tab:Refine an Initial Relationship
© 2007 Prentice Hall 16-33
Purchase of Fashion Clothing by Marital StatusTable 15.6
Purchase of Fashion
Current Marital Status
Clothing Married Unmarried
High 31% 52%
Low 69% 48%
Column 100% 100%
Number of respondents
700 300
© 2007 Prentice Hall 16-34
Purchase of Fashion Clothing by Marital Status and GenderTable 15.7
Purchase of Fashion Clothing
Sex Male
Female
Married Not Married
Married Not Married
High 35% 40% 25% 60%
Low 65% 60% 75% 40%
Column totals
100% 100% 100% 100%
Number of cases
400 120 300 180
© 2007 Prentice Hall 16-35
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.
3 Variables Cross-Tab:Initial Relationship was Spurious
© 2007 Prentice Hall 16-36
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%
100%
750
© 2007 Prentice Hall 16-37
Ownership of Expensive Automobiles by Education Level and Income Levels
Table 15.9
Own Expensive Automobile
College Degree
No College Degree
College Degree
No College Degree
Yes 20% 20% 40% 40%
No 80% 80% 60% 60%
Column totals 100% 100% 100% 100%
Number of respondents
100 700 150 50
Low Income High Income
Income
© 2007 Prentice Hall 16-38
Table 15.10 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 15.10.
3 Variables Cross-Tab:Reveal Suppressed Association
© 2007 Prentice Hall 16-39
Desire to Travel Abroad by AgeTable 15.10
Desire to Travel Abroad Age
Less than 45 45 or More
Yes 50% 50%
No 50% 50%
Column totals 100% 100%
Number of respondents 500 500
© 2007 Prentice Hall 16-40
Desire to Travel Abroad by Age and Gender
Table 15.11
© 2007 Prentice Hall 16-41
Consider the cross-tabulation of family size and the tendency to eat out frequently in fast-food restaurants as shown in Table 15.12. No association is observed.
When income was introduced as a third variable in the analysis, Table 15.13 was obtained. Again, no association was observed.
Three Variables Cross-TabulationsNo Change in Initial Relationship
© 2007 Prentice Hall 16-42
Eating Frequently in Fast-Food Restaurants by Family Size
Table 15.12
© 2007 Prentice Hall 16-43
Eating Frequently in Fast Food-Restaurantsby Family Size and Income
Table 15.13
© 2007 Prentice Hall 16-44
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
Statistics Associated with Cross-Tab: Chi-Square
© 2007 Prentice Hall 16-45
Statistics Associated with Cross-Tab: Chi-Square
compares the of 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
2
n
nnf cr
e
© 2007 Prentice Hall 16-46
From Table 3 in the Statistical Appendix, the probability of exceeding a chi-square value of 3.841 is 0.05.
The calculated chi-square is 3.333. 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.
Statistics for Cross-Tab: Chi-Square