Transcript data prep and descriptive stats

Data Preparation
Steps in Data Preparation
Editing
Coding
Entering Data
Data Tabulation
Reviewing Tabulations
Statistically adjusting the data (e.g.
weighting)
Editing
 Carefully checking survey data for
 Completeness (no omissions)
 Non-ambiguous (e.g. two boxes checked instead of
one)
 Right informant (e.g. under age, when all supposed
to be over 18)
 Consistency
 e.g. charging something when the person does not
own a charge card
 Accuracy (e.g. no numbers out of range)
 Most important purpose is to eliminate or at
least reduce the number of errors in the raw
data.
Solutions
1. Ideally re-interview respondent
2. Eliminate all unacceptable surveys (case wise
deletion) (if sample is large and few unacceptable)
3. In calculations only the cases with complete
responses are considered (pair wise deletion)
(means that some statistics will be based on
different sample sizes)
4. Code illegible or missing answers into a a “no valid
response” category
5. substitute a neutral value - typically the mean
response to the variable, therefore the mean
remains unchanged
Coding
• The process of systematically and consistently
assigning each response a numerical score.
• The key to a good coding system is for the coding
categories to be mutually exclusive and the entire
system to be collectively exhaustive.
• To be mutually exclusive, every response must fit
into only one category.
• To be collectively exhaustive, all possible responses
must fit into one of the categories.
• Exhaustive means that you have covered the entire
range of the variable with your measurement.
Coding
• Coding Missing Numbers: When respondents fail
to complete portions of the survey.
– Whatever the reason for incomplete surveys, you
must indicate that there was no response provided
by the respondent.
– For single digit responses code as “9”, 2 digit code
as “99”
Coding Open-Ended Questions: When open-ended
questions are used, you must create categories.
– All responses must fit into a category
– similar responses should fall into the same
category.
e.g. Who services your car? ______________
Possible categories: self, garage, husband, wife,
friend, relative etc.
• To make it collectively exhaustive add an “other” or
“none of the above” category
–Only a few i.e. < 10% should fit into this category
Precoded Questionnaires: Sometimes you can place
codes on the actual questionnaire, which simplifies
data entry.
This…
Are you:
Male
Female
How satisfied are you with our product?
___Very Satisfied
___Somewhat Satisfied
___Somewhat Dissatisfied
___Very Dissatisfied
___No opinion
Becomes this…
Are you: (1) Male
(2) Female
How satisfied are you with our product?
_1__Very Satisfied
_2__Somewhat Satisfied
_3__Somewhat Dissatisfied
_4__Very Dissatisfied
_5__No opinion
1. Are you solely responsible for taking care of your
automotive service needs ___ Yes ___ No
2. If No who performs the simple maintenance ___________
3. If scheduled maintenance is done on your automobile,
how do you keep track of what has been done
•Not tracked
•auto dealer records
•mental recollection
•other
4. How often is your automobile serviced?
•Once per month
•Once every three months
•Once every six months
•Once per year
•Other _______________
Code Book
Col.
No
Question
No.
Question Des.
Range of permissible values
1-3
ID #
N/A
001-200 (this also means the surveys
themselves should be numbered)
4
1
Responsible for
Maintenance
0= No. 1=yes, 9= blank
5
2
perform simple
maintenance
0=husband, 1=boyfriend, 2=father, 3=mother,
4=relative, 5=friend, 6=other, 9=blank
5
3
How maintenance
tracked
0=not tracked, 1=auto dealer records,
2=personal records, 3=mental recollection,
4=other, 9=blank
6
4
How often
maintenance
performed
1=Once per month, 2=3 month, 3=6 months ,
4=year , 5=other, 9= blank
In questions that permit multiple responses, each possible
response option should be assigned a separate column
6. Which magazines do you read, choose all that apply.
•
Time
•
National Geographic
•
Readers Digest
•
Chatelaine
•
MacLean's
Col. No
Question No.
Question Des.
Range of permissible values
15
6
Time
0 =read, 1= not read
16
6
Readers Dig.
0 =read, 1= not read
17
6
MacLean's
0 =read, 1= not read
18
6
National Geo.
0 =read, 1= not read
19
6
Chatelaine
0 =read, 1= not read
For rank order questions, separate columns are also needed
7. Please rank the following brands of toothpaste in order
of preference (1-5) with 1 being the most important
•
Crest
•
Aquafresh
•
Pepsodent
•
Colgate
•
Arm & Hammer
Col.# Q.
No.
Question Des.
Range of permissible values
20
7
Crest rank
0 =blank, 1 = most important, 2 =2nd most
important, 3 =third, 4=fourth, 5= fifth
21
7
Colgate rank
0 =blank, 1 = most important, 2 =2nd most
important, 3 =third, 4=fourth, 5= fifth
22
7
Acquafresh rank 0 =blank, 1 = most important, 2 =2nd most
important, 3 =third, 4=fourth, 5= fifth
23
7
A & H rank
0 =blank, 1 = most important, 2 =2nd most
important, 3 =third, 4=fourth, 5= fifth
25
7
Pepsodent rank
0 =blank, 1 = most important, 2 =2nd most
important, 3 =third, 4=fourth, 5= fifth
Preparing the Data for Analysis
Variable Re-specification
• Existing data modified to create new variables
• Large number of variables collapsed into fewer
variables
• E.g. If 10 reasons for purchasing a car are given they
might be collapsed into four categories e.g.
performance, price, appearance, and service
• Creates variables that are consistent with research
questions
Entering Data
• Problems can occur during data entry, such as
transposing numbers and inputting an infeasible
code(e.g out of range)
– E.g. Score on range of 1-5 then 0, 6, 7, and 8 are
unacceptable or out of range (might be due to
transcription error)
• Always check the data-entry work.
Descriptive Statistics
Five types of statistical analysis
Descriptive
What are the characteristics of
the respondents?
Inferential
What are the characteristics of
the population?
Differences
Are two or more groups the same
or different?
Associative
Are two or more variables related
in a systematic way?
Predictive
Can we predict one variable if we know
one or more other variables?
Descriptive Statistics
 Summarization of a collection of data in a
clear and understandable way
 the most basic form of statistics
 lays the foundation for all statistical
knowledge
Measures of central tendency (mean, median, mode)
Measures of dispersion (range, standard deviation,
and coefficient of variation)
Measures of shape (skewness and kurtosis)
The tradeoff in descriptive statistics
• If you use fewer statistics to describe the distribution of a
variable, you lose information but gain clarity.
• When should one use fewer statistics?
– When dropping the number of statistics would leave more
information per remaining statistic.
– When the information you drop is unimportant to one’s
research question.
Type of
Measurement
Type of
descriptive analysis
Two
categories
Nominal
More than
two categories
Frequency table
Proportion (percentage)
Frequency table
Category proportions
(percentages)
Mode
Type of
Measurement
Type of
descriptive analysis
Ordinal
Rank order
Median
Interval
Arithmetic mean
Ratio
means
Data Tabulation
• Tabulation: The organized arrangement of data in a
table format that is easy to read and understand.
– Tabulate the data to count the number of responses to each
question.
• Simple Tabulation: tabulating of results of only one
variable informs you how often each response was
given.
• Frequency Distribution: A distribution of data that
summarizes the number of times a certain value of a
variable occurs expressed in terms of percentages.
Frequency Tables
The arrangement of statistical data in a row-andcolumn format that exhibits the count of
responses or observations for each category
assigned to a variable
• How many of certain brand users can be called loyal?
• What percentage of the market are heavy users and
light users?
• How many consumers are aware of a new product?
• What brand is the “Top of Mind” of the market?
More on relative frequency distributions
• Rules for relative frequency distributions:
–
–
–
–
–
Make sure each observation is in one and only one category.
Use categories of equal width.
Choose an appealing number of categories.
Provide labels
Double-check your graph.
• Definitions:
– A histogram is a relative frequency distribution of a
quantitative variable
– A bar graph is a relative frequency distribution of a qualitative
variable
WebSurveyor Bar Chart
How did you find your last job?
643 Netw orking
213 print ad
179 Online recruitment site
112 Placement firm
18 Temporary agency
1.5 %
Temporary agency
9.6 %
Placement firm
15.4 %
Online recruitment site
18.3 %
print ad
55.2 %
Netw orking
0
100
200
300
400
500
600
700
How many times per week do you use mouthwash ?
1__ 2__ 3__ 4__ 5__ 6__ 7__
112223333344444445555566677
1
2
2
3
3
5
4
7
5
5
7
6
5
4
3
2
1
6
3
7
2
0
1
2
3
4
5
6
7
Normal Distribution

-

a
b

Normal Distributions
 Curve is basically bell shaped
from -  to 
 symmetric with scores
concentrated in the middle (i.e. on
the mean) than in the tails.
Mean, medium and mode
coincide
They differ in how spread out
they are.
 The area under each curve is 1.
The height of a normal
distribution can be specified
mathematically in terms of two
parameters: the mean () and the
standard deviation ().
Skewed Distributions
Occur when one tail of the distribution is longer than the other.
Positive Skew Distributions
 have a long tail in the positive direction.
 sometimes called "skewed to the right"
 more common than distributions with negative skews
E.g. distribution of income. Most people make under $40,000 a
year, but some make quite a bit more with a small number making
many millions of dollars per year
 The positive tail therefore extends out quite a long way
Negative Skew Distributions
have a long tail in the negative direction.
called "skewed to the left."
negative tail stops at zero
• Kurtosis: how peaked a distribution is. A
zero indicates normal distribution, positive
numbers indicate a peak, negative numbers
indicate a flatter distribution)
Peaked
distribution
Flat distribution
Thanks, Scott!
Summary statistics
–central tendency
–Dispersion or variability
A quantitative measure of the degree to
which scores in a distribution are spread
out or are clustered together;
Descriptive Analysis: Measures of
Central Tendency
• Mode: the number that occurs most often
in a string (nominal data)
• Median: half of the responses fall above
this point, half fall below this point
(ordinal data)
• Mean: the average (interval/ratio data)
Mode
 the most frequent category
users
25%
non-users 75%
Advantages:
• meaning is obvious
• the only measure of central tendency that can be used
with nominal data.
Disadvantages
• many distributions have more than one mode, i.e. are
"multimodal
• greatly subject to sample fluctuations
• therefore not recommended to be used as the only
measure of central tendency.
Median
the middle observation of the data
number times per week consumers use mouthwash
112223333344444445555566677
Frequency
distribution of
Mouthwash
use per week
Light user
Mode
Median
Mean
Heavy user
The Mean (average value)
sum of all the scores divided by the number of scores.
 a good measure of central tendency for roughly
symmetric distributions
 can be misleading in skewed distributions since it can be
greatly influenced by extreme scores in which case other
statistics such as the median may be more informative
 formula  = SX/N (population)
X
¯ = Sxi/n (sample)
where /X
¯ is the population/sample mean
and N/n is the number of scores.
Normal Distributions with
different Means
-
1
0

2

Measures of Dispersion or
Variability
• Minimum, Maximum, and Range (Highest
value minus the lowest value)
• Variance
• Standard Deviation (A measure’s distance
from the mean)
Distribution of Final Course Grades in MGMT 3220Y
25
Frequency
20
- 1 SD
15
+ 1 SD
10
5
RANGE
0
Frequency
F
D
C
B
A
3
10
20
23
12
Grade
Variance
• The difference between an observed value and
the mean is called the deviation from the mean
• The variance is the mean squared deviation
from the mean
• i.e. you subtract each value from the mean,
square each result and then take the average.
2 = S(x¯ xi)2/n
• Because it is squared it can never be negative
Standard Deviation
• The standard deviation is the square root of
the variance
2/n
S =  S(xx
)
¯ i
• Thus the standard deviation is expressed in
the same units as the variables
• Helps us to understand how clustered or
spread the distribution is around the mean
value.
Measures of Dispersion
Suppose we are testing the new flavor of a fruit punch
Dislike 1
1.
2
3 4 5 Like Data
x
x
2.
3.
x
x
2/n
2 = S(xx
)
¯ i
X= 4
 2= 1
S=1
5
3
x
6.
5
3
x
4.
5.
3
5
2/n
S =  S(xx
)
¯ i
Measures of Dispersion
Dislike 1
2
3
4
1.
2.
5 Like Data
x
5
4
x
3.
x
5
4.
x
5
5.
x
5
6.
2/n
2 = S(xx
)
¯ i
x
X
¯ = 4.67
2=0.22
S = 0.47
4
2/n
S =  S(xx
)
¯ i
Measures of Dispersion
Dislike 1
1.
4
5 Like Data
1
x
x
4.
5.
3
x
2.
3.
2
1
x
2/n
2 = S(xx
)
¯ i
5
X=
¯ 3
2=4
S=2
1
x
6.
5
x
5
2/n
S =  S(xx
)
¯ i
Normal Distributions
with different SD
2
-
1
3


Cross Tabulation
• A statistical technique that involves tabulating the
results of two or more variables simultaneously
• informs you how often each response was given
• Shows relationships among and between variables
• frequency distribution for each subgroup compared
to the frequency distribution for the total sample
• must be nominally scaled
Cross-tabulation
• Helps answer questions about whether two
or more variables of interest are linked:
– Is the type of mouthwash user (heavy or
light) related to gender?
– Is the preference for a certain flavor (cherry
or lemon) related to the geographic region
(north, south, east, west)?
– Is income level associated with gender?
• Cross-tabulation determines association not
causality.
Dependent and Independent Variables
• The variable being studied is called the
dependent variable or response variable.
• A variable that influences the dependent
variable is called independent variable.
Cross-tabulation
• Cross-tabulation of two or more variables is
possible if the variables are discrete:
– The frequency of one variable is subdivided by the
other variable categories.
• Generally a cross-tabulation table has:
– Row percentages
– Column percentages
– Total percentages
• Which one is better?
DEPENDS on which variable is considered as
independent.
Contingency Table
• A contingency table shows the conjoint
distribution of two discrete variables
• This distribution represents the probability
of observing a case in each cell
– Probability is calculated as:
Observed
cases
P=
Total cases
Cross tabulation
GROUPINC * Gender Crosstabulation
Gender
Female
GROUPINC
income <= 5
5>Income<= 10
Count
9
19
% within GROUPINC
52.6%
47.4%
100.0%
% within Gender
55.6%
18.8%
28.8%
% of Total
15.2%
13.6%
28.8%
5
25
30
% within GROUPINC
16.7%
83.3%
100.0%
% within Gender
27.8%
52.1%
45.5%
7.6%
37.9%
45.5%
3
14
17
% within GROUPINC
17.6%
82.4%
100.0%
% within Gender
16.7%
29.2%
25.8%
4.5%
21.2%
25.8%
18
48
66
27.3%
72.7%
100.0%
100.0%
100.0%
100.0%
27.3%
72.7%
100.0%
Count
Count
% of Total
Total
Total
10
% of Total
income >10
Male
Count
% within GROUPINC
% within Gender
% of Total
General Procedure for
Hypothesis Test
1. Formulate H0 (null hypothesis) and H1
(alternative hypothesis)
2. Select appropriate test
3. Choose level of significance
4. Calculate the test statistic (SPSS)
5. Determine the probability associated with
the statistic.
•
Determine the critical value of the test
statistic.
General Procedure for
Hypothesis Test
6 a) Compare with the level of significance, 
b) Determine if the critical value falls in the
rejection region. (check tables)
7 Reject or do not reject H0
8 Draw a conclusion
1. Formulate H1and H0
• The hypothesis the researcher wants to test is called
the alternative hypothesis H1.
• The opposite of the alternative hypothesis is the null
hypothesis H0 (the status quo)(no difference between
the sample and the population, or between samples).
• The objective is to DISPROVE the null hypothesis.
• The Significance Level is the Critical probability of
choosing between the null hypothesis and the
alternative hypothesis
2. Select Appropriate Test
• The selection of a proper Test depends on:
– Scale of the data
• nominal
• interval
– the statistic you seek to compare
• Proportions (percentages)
• means
– the sampling distribution of such statistic
• Normal Distribution
• T Distribution
• 2 Distribution
– Number of variables
• Univariate
• Bivariate
• Multivariate
– Type of question to be answered
Example
A tire manufacturer believes that men are more aware of their
brand. To find out, a survey is conducted of 100 customers, 65 of
whom are men and 35 of whom are women.
The question they are asked is: Are you aware of our brand: Yes
or No. 50 of the men were aware and 15 were not whereas 10 of
the women were aware and 25 were not.
Are these differences significant?
Men
Aware
50
Unaware
15
65
Women
10
25
35
Total
60
40
100
1. Formulate H1and H0
We want to know whether brand awareness is
associated with gender. What are the Hypotheses
H0: There is no difference in brand awareness based on gender
H1: There is a difference in brand awareness based on gender
Chi-square test results are unstable if cell count is lower than 5
2. Select Appropriate Test
X2 (Chi Square)
• Used to discover whether 2 or more groups of one variable
(dependent variable) vary significantly from each other with
respect to some other variable (independent variable).
• Are the two variables of interest associated:
– Do men and women differ with respect to product usage
(heavy, medium, or light)
– Is the preference for a certain flavor (cherry or lemon) related
to the geographic region (north, south, east, west)?
H0: Two variables are independent (not associated)
H1: Two variables are not independent (associated)
• Must be nominal level, or, if interval or ratio must be divided into
categories
Awareness of Tire Manufacturer’s Brand
Men
Women
Total
Aware
50/39
10/21
60
Unaware
15/26
65
25/14
35
40
100
Estimated cell
Frequency
E
ij
=
R iC
j
n
Ri = total observed frequency in the ith row
Cj = total observed frequency in the jth column
n = sample size
Eij = estimated cell frequency
3. Choose Level of Significance
• Whenever we draw inferences about a population, there is
a risk that an incorrect conclusion will be reached
• The real question is how strong the evidence in favor of the
alternative hypothesis must be to reject the null
hypothesis.
• The significance level states the probability of rejecting H0
when in fact it is true.
• In the example a Type I error would be committed if we
said that
There is a difference between men and women with respect
to brand awareness when in fact there was no difference
• This error is commonly known as Type I error, The value
of  is called the significance level of the test
• Significance Level selected is typically .05 or .01
• i.e 5% or 1%
•In other words we are willing to accept the risk
that 5% (or 1%) of the time the results we get
indicate that there is a difference between men
and women with respect to brand awareness when
in fact there is no difference
3. Choose Level of Significance
• We commit Type error II when we
incorrectly accept a null hypothesis when it
is false. The probability of committing Type
error II is denoted by .
• In our example we commit a type II error
when we say that.
there is NO difference between men and women
with respect to brand awareness (we accept the
null hypothesis) when in fact there is
Type I and Type II Errors
Null is true
Null is false
Accept null
Reject null
Correctno error
Type I
error
Type II
error
Correctno error
Which is worse?
• Both are serious, but traditionally Type I error has
been considered more serious, that’s why the
objective of hypothesis testing is to reject H0 only
when there is enough evidence that supports it.
• Therefore, we choose  to be as small as possible
without compromising .
• Increasing the sample size for a given α will
decrease β (I.e. accepting the null hypothesis when it is
in fact false)
Awareness of Tire Manufacturer’s Brand
Men
Women
Total
Aware
50/39
10/21
60
Unaware
15/26
65
25/14
35
40
100
Estimated cell
Frequency
E
ij
=
R iC
j
n
Ri = total observed frequency in the ith row
Cj = total observed frequency in the jth column
n = sample size
Eij = estimated cell frequency
Chi-Square Test
R iC j
Estimated cell
E
=
ij
Frequency
n
Ri = total observed frequency in the ith row
Cj = total observed frequency in the jth column
n = sample size
Eij = estimated cell frequency
Chi-Square
statistic
x² =

(Oi  E i )²
Ei
x² = chi-square statistics
Oi = observed frequency in the ith cell
Ei = expected frequency on the ith cell
Degrees of
Freedom
d.f.=(R-1)(C-1)
4. Calculate the Test Statistic
Chi-Square Test: Differences Among Groups
X
+
2
=
( 15
( 50
 39 )
39
 26 )
26
2
+
2
+
( 25
( 10
 21 )
21
 14 )
14
2
2
 2 = 3 .102 + 5 .762 + 4 .654 + 8 .643 =
 2 = 22 .161
d . f . = ( R  1)( C  1)
d . f . = ( 2  1)( 2  1) = 1
Chi-square test results are unstable if cell count is lower than 5
Degrees of Freedom
 the number of values in the final calculation of a statistic that
are free to vary
For example To calculate the standard deviation of a random
sample, we must first calculate the mean of that sample and
then compute the sum of the squared deviations from that mean
While there will be n such squared deviations only (n - 1) of them
are free to assume any value whatsoever.
This is because the final squared deviation from the mean must
include the one value of X such that the sum of all the Xs divided by
n will equal the obtained mean of the sample.
All of the other (n - 1) squared deviations from the mean can,
theoretically, have any values whatsoever..
5. Determine the Probabilityvalue (Critical Value)
•The p-value is the probability of seeing a random
sample at least as extreme as the sample observed given
that the null hypothesis is true.
• given the value of alpha,  we use statistical theory to
determine the rejection region.
• If the sample falls into this region we reject the null
hypothesis; otherwise, we accept it
• Sample evidence that falls into the rejection region is
called statistically significant at the alpha level.
Significance from p-values -continued
• How small is a “small” p-value? This is largely a matter of
semantics but if the
– p-value is less than 0.01, it provides “convincing”
evidence that the alternative hypothesis is true;
– p-value is between 0.01 and 0.05, there is “strong”
evidence in favor of the alternative hypothesis;
– p-value is between 0.05 and 0.10, it is in a “gray area”;
– p-values greater than 0.10 are interpreted as weak or no
evidence in support of the alternative.
5. Determine the Probability-value (Critical Value)
Chi-square Test for Independence
Under H0, the probability distribution is approximately
distributed by the Chi-square distribution (2).
Chi-square
Reject H0
3.84
2

22.16
2 with 1 d.f. at .05 critical value = 3.84
6 a) Compare with the level of significance, 
b) Determine if the critical value falls in the rejection
region. (check tables)
22.16 is greater than 3.84 and falls in the rejection area
In fact it is significant at the .001 level, which means that the
chance that our variables are independent, and we just happened
to pick an outlying sample, is less than 1/1000
Or, in other words, the chance that we have a Type 1 error is less
than .1%
i.e. That there is a .1% chance that we reject the null
hypothesis, that there is no difference between men and
women with respect to brand awareness, and say that there
is, when in fact the null hypothesis is true: there is no
difference.
7 Reject or do not reject H0
Since 22.16 is greater than 3.84 we reject the null
hypothesis
8 Draw a conclusion
Men and women differ with respect to brand
awareness, specifically, men are more brand
aware then women