Download descriptive statistics

Descriptive Statistics
Univariate Statistics
Chi Square
ANOVA
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
Inferential Statistics
Two main methods:
1. estimation
 the sample statistic is used to estimate a
population parameter
 a confidence interval about the estimate is
constructed.
2. hypothesis testing
 a null hypothesis is put forward
 Analysis of the data is then used to
determine whether to reject it.
Inferential statistics generally require that
sampling be random
TYPES OF DATA
• Nominal : gender, type of customer
(loyalty), flavor/color liked, etc.
• Ordinal/Ranking :type of user, preferred
brand, brand awareness, etc.
• Interval: Attitudinal or satisfaction scales.
Are you satisfied with your education at U of L?
3 4 5 Satisfied
Dissatisfied 1 2
• Ratio: Income, price willing to pay, age, etc.
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
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?
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
Bar Graph
90
80
70
60
East
West
North
50
40
30
20
10
0
1st Qtr
2nd Qtr
3rd Qtr
4th Qtr
Measures of Central Location or
Tendency
• Mean: average value
• Mode: the most frequent category
• Median: the middle observation of the data
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 m = SX/N (population)
X
¯ = Sxi/n (sample)
where m/X
¯ is the population/sample mean
and N/n is the number of scores.
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
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 (m) and the
standard deviation (s).
Normal Distribution
s
-
m
a
b
Area between a and b = P(a=X =b)

Normal Distributions with
different Mean
-
m1
0
m
2

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
Measures of Dispersion or
Variability
• Minimum, Maximum, and Range
• Variance
• Standard Deviation
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.
s2 = 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
s2 = S(xx
)
¯ i
X= 4
s 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
s2 = S(xx
)
¯ i
x
X
¯ = 4.6
s2=0.26
S = 0.52
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
s2 = S(xx
)
¯ i
5
X=
¯ 3
s2=4
S=2
1
x
6.
5
x
5
2/n
S =  S(xx
)
¯ i
Normal Distributions
with different SD
s2
-
s1
s3
m

How does the Normal Distribution
help to make decisions?
Suppose you are about to introduce new
“Guacamole Doritos” to the market.
• Need to determine:
– Desired flavor intensity (How hot it should be)
– Package size offered
– Introduction price
• What do you do in order to answer your
questions?
ASK THE CONSUMER
• How?
TAKE A SAMPLE
• How can you be sure that what you
conclude on the sample would be true for
the whole population?
Suppose you conducted a research study
• Took a random sample of n=100 subjects
• They tasted the new "Guacamole Doritos”
• They rated the flavor of the chip on the
following scale:
1
Too
Mild
2
3
4
5
Perfect
Flavor
6
7
Too
Hot
Results show : x1 = 2.3 and S1= 1.5
• Can you conclude that on average the target
population thought the flavor was mild?
• Suppose you take a series of random
samples of n=100 subjects:
x2 = 3.7 and S2 = 2
x3 = 4.3 and S3 = 0.5
x4 = 2.8 and S4 = .97
..
.
x50 = 3.7 and S50 = 2
The Sampling Distribution
The means of all the samples will have their own
distribution called the sampling distribution of the
means
It is a normal distribution
The sampling distribution of a proportions is a
binomial that approximates a normal distribution in
large samples (30+)
The mean of the sampling distribution of the mean =
X = (ΣXi)/n
It equals the population parameter
Sampling Distribution
The standard deviation of the sampling distribution is
called the sampling error of the mean (or proportion).
= s = s / n
X
The formula for the proportion is
sp= π(1-π)/n
Often the population standard deviation s is
unknown and has to be estimated from the sample
S = s  Σ(Xi-X)/n-1
Population distribution of the Doritos’ flavor (X)
s
X
m
Sample distribution of the x Doritos’ flavor
x
1 2
3 4
5
6
7
• What relationship does the Population
Distribution have to the Sample Distribution?
The Central Limit Theorem
Let x1, x2….. xn denote a random sample selected from
a population having mean m and variance s2. Let X
denote the sample mean. If n is large, the X has
approximately a Normal Distribution with mean m and
variance s2/n.
• The Central Limit Theorem does not mean that
the sample mean = population mean.
• It means that you can attach a probability to that
value and decide.
Interpretation
• The process of making pertinent inferences
and drawing conclusions
• concerning the meaning and implications of
a research investigation
• You do not need to know the population
distribution in order to take decisions.
• In order to draw conclusions n must be “big
enough.”
• How big?, it DEPENDS
Univariate Statistics
• Test of statistical significance
• Hypothesis testing one variable at a time
• Hypothesis
• Unproven proposition
• Supposition that tentatively explains certain
facts or phenomena
• Assumption about nature of the world
What is a Hypothesis Test?
• It is used when we want to make inferences
about a population.
• Generally we have a particular theory, or
hypothesis, about certain events like:
– The average age of our regular customers
– The average money spent per week on fast food
restaurants
– The percentage of unsatisfied customers of our
store.
Basic Concepts
• The hypothesis the researcher wants to test is called
the alternative hypothesis H1.
• The opposite of the alternative hypothesis us 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
General Procedure for
Hypothesis Test
1.
2.
3.
4.
5.
Formulate H1 and H0
Select appropriate test
Choose level of significance
Calculate the test statistic
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.
7 Reject or do not reject H0
8 Draw a conclusion
1. Formulate H1and H0
• Null hypothesis represents status quo.
• Alternative hypothesis represents the
desired result.
• Example: One-Sample t-test
– The manager of Pepperoni Pizza has developed
a new baking method with lower costs and
wishes to test it with some customers. He asked
customers to rate the difference between both
pizzas on a scale from -10 (old style) to +10
(new style)
1. Formulate H1and H0
• As a manager you would like to observe a
difference between both pizzas
• Since the new baking method is cheaper,
you would like the preference to be for it.
– Null Hypothesis
H0 m=0
– Alternative
H1 m0
Two tail
test
or
H1 m >0
One tail
test
2. Select Appropriate Test
• The selection of a proper Test depends on:
– Scale of the data
• categorical
• interval
– the statistic you seek to compare
• proportions
• 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
3. Choose Level of Significance
• Whenever we draw inferences about a population, there is
a risk that an incorrect conclusion will be reached
• The significance level states the probability of incorrectly
rejecting H0. This error is commonly known as Type I
error, and we denote the significance level as .
• Significance Level selected is typically .05 or .01
– In our example the Type I error would be rejecting the null
hypothesis that the pizzas are equal, when they really are
perceived equal by the customers of the entire population.
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, the Type II error would be not
rejecting the null hypothesis that the pizzas are
equal, when they are perceived to be different
by the customers of the entire population.
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
4. Calculate the Test Statistic
Example
• If we are testing whether the consumer
perceives a difference between the pizzas
– We would need a statistic for the mean
– We know that X N(m, s2/n)
Perceived difference between the pizzas (X) for a given
population of size N with mean m and variance estimated
from the sample s2/n
• If we suppose Ho true, then m=0 and
X N(0, s2/n)
• If we standardized X, we would get
X- 0  N(0, 1)
Z = s/n
• Since we do not know the population value
of s, we would have to estimate it with the
SD of the sample.
• But…..X no longer has a Normal
distribution, now X has a T distribution with
n-1 degrees of freedom.
t = X- 0  T(n-1)
s/n
-
0

• X= perceived difference between the pizzas
• m = real population mean, that equals zero if H0 is
true.
• x = 3.5, observed sample mean
• SD= 2.1, observed sample standard deviation
• n=40
3.5 - 0
• =.01
 T (39)
t=
2.1/40
t =10.54
T=.005(39)=2.074
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.
• For example:
– In reference to the null hypothesis, if H0
hypothesized that there would be no difference
between the pizzas, a sample mean value of 2.5
would be high, but even more extreme would
be a value of 3.5.
– If the p-value is 0.03, it would mean that if we
take 100 samples we would observe only three
samples with an extreme value of 3.5.
– It would be concluded that we have enough
evidence to reject H0.
6. Compare with the level of significance,  and determine if
the critical value falls in the rejection region
Do not Reject H0
1-
Reject H0
Reject H0
/2
/2
-2.074
0
2.074
10.54
7 & 8. Reject or do not reject H0 and draw a conclusion
Since the statistic t falls in the rejection area we reject Ho
and conclude that the perceived difference between the
pizzas is different from zero.
Hypothesis Test for Two
Independent Samples
•Test for mean difference:
– Null Hypothesis
H0 m1= m2
– Alternative
H1 m1 m2
•Under H0 m1- m2 = 0. So, the test concludes whether
there is a difference between the parameters or not.
– e.g. high income consumers spend more on sports activities
than low income consumers
– The proportion of brand-loyal users in segment 1 is different
from that in segment 2
•Can be used for examining differences between
means and proportions
Test for Means Difference
• Suppose X measures the preference for a
mouthwash flavor (cool mint) on a scale from
1-dislike to 5-like
• We want to know if the flavor preference is
different between the type of user (heavy or
light) H0: mH= mL
H1: mH mL
• It would be the same to test if the difference
is zero or not. H0: mH-mL= 0
t=
(XH-XL)- (mH-mL)
S X -X
H
L
 T(nH-nL -2)
• So, if we reject H0 we can conclude that the
means of the independent samples are
different.
Test for Variance Difference
• Tests if the variance ratio is equal to 1
H0: sH/ sL= 1
H1: sH/ sL  1
• So, if we reject H0 we can conclude that the
variances of the independent samples are
different.
Test for Variance Difference
• The test statistic has an F Distribution:
2
SH
f=
2
 F (nH-1)(nL -1)
SL
F

0
f
SPSS Output
Independent Samples Test
Levene's Test for
Equality of Variances
F
Domestic gross Equal variances
assumed
Equal variances
not assumed
Foreign gross
Equal variances
assumed
Equal variances
not assumed
Income
Equal variances
assumed
Equal variances
not assumed
1.655
2.202
2.591
Sig.
.203
.143
.112
t-test for Equality of Means
t
df
Sig. (2-tailed)
Mean
Difference
Std. Error
Difference
95% Confidence
Interval of the
Difference
Lower
Upper
-1.383
64
.171
-10.08
7.283
-24.627
4.474
-1.777
54.679
.081
-10.08
5.670
-21.441
1.288
-1.585
64
.118
-16.52
10.426
-37.349
4.307
-1.856
43.600
.070
-16.52
8.903
-34.468
1.427
-2.429
64
.018
-3.42
1.409
-6.239
-.608
-2.885
45.151
.006
-3.42
1.187
-5.814
-1.034
Test for Proportion Difference
• Suppose X measures the number of
individuals that preferred the mouthwash
flavor cool mint.
• We want to know if the proportion of people
who preferred cool mint is different between
heavy and light users.
H0: H= L
H1: H L
• It would be the same to test if the difference
is zero or not. H0: H- L= 0
• The sample proportion would be equal to
number
of
favorable
cases
p=
total number of cases
 N(p, pq/n) where q=1-p
• How is the difference between two
proportions distributed?
• The difference of two independent sample
proportions is distributed as:
1- 2  N(p1-p2, p1q1/n1+ p2q2/n2)
• Therefore, under H0: H- L= 0, the test
statistic is as follows:
z=
p1-p2-0
p1q1/n1+ p2q2/n2
 N
Example
• Suppose we are the brand manager for Tylenol, and
a recent TV ad tells the consumers that Advil is
more effective (quicker) treating headaches than
Tylenol.
• An independent random sample of 400 people with
a headache is given Advil, and 260 people report
they feel better within an hour.
• Another independent sample of 400 people is taken
and 252 people that took Tylenol reported feeling
better. Is the TV ad correct?
Tylenol vs Advil
• We would need to test if the difference is
zero or not.
H0: A - T = 0;
H1: A - T  0
pA = 260/400= 0.65
pT = 252/400= 0.63
.65 - .63 -0
= 0.66
z=
(.65)(.35)/400+ (.63)(.37)/400
Tylenol vs Advil
 = 0.10 N(0,1) = 1.64
 -1
/2
-
/2
-1.64
0
0.66
1.64

Test for Means Difference on
Paired Samples
•What is a paired sample?
–When observations from two populations occur
in pairs or are related then they are not
independent
–When you want to measure brand recall before
and after an ad campaign.
–When employing a consumer panel, and
comparing whether they increased their
consumption of a certain product from one period
to another.
Test for Means Difference on
Paired Samples
• Since both samples are not independent we
employ the differences as a random sample
di=x1i-x2i
i=1,2,…,n
• Now we can test this variable to compare it
to against any other value.
SPSS Output
Paired Samples Statistics
Pair
1
RATING1
RATING2
Mean
2.10
-1.33
N
40
40
Std. Deviation
4.717
3.083
Std. Error
Mean
.746
.488
Paired Samples Test
Paired Differences
Pair 1
RATING1 - RATING2
Mean
3.43
Std. Deviation
5.857
Std. Error
Mean
.926
95% Confidence
Interval of the
Difference
Lower
Upper
1.55
5.30
t
3.699
df
39
Sig. (2-tailed)
.001
Cross Tabulation
and Chi Square Test
for Independence
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.
Cross tabulation
GROUPINC * Gender Crosstabulation
GROUPINC
income <= 5
5<Income<= 10
income >10
Total
Count
% within GROUPINC
% within Gender
% of Total
Count
% within GROUPINC
% within Gender
% of Total
Count
% within GROUPINC
% within Gender
% of Total
Count
% within GROUPINC
% within Gender
% of Total
Gender
Female
Male
10
9
52.6%
47.4%
55.6%
18.8%
15.2%
13.6%
5
25
16.7%
83.3%
27.8%
52.1%
7.6%
37.9%
3
14
17.6%
82.4%
16.7%
29.2%
4.5%
21.2%
18
48
27.3%
72.7%
100.0%
100.0%
27.3%
72.7%
Total
19
100.0%
28.8%
28.8%
30
100.0%
45.5%
45.5%
17
100.0%
25.8%
25.8%
66
100.0%
100.0%
100.0%
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
Chi-square Test for Independence
• The Chi-square test for independence
determines whether two variables are
associated or not.
H0: Two variables are independent
H1: Two variables are not independent
Chi-square test results are unstable if cell count is lower than 5
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)
Awareness of Tire
Manufacturer’s Brand
Men
Women
Total
Aware
50/39
10/21
60
Unaware
15/21
65
25/14
35
40
100
Chi-Square Test: Differences Among
Groups Example
X
2
( 50  39 ) 2
(10  21) 2


39
21
2
(15  26 )
( 25  14 ) 2


26
14
 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
X2 with 1 d.f. at .05 critical value = 3.84
Chi-square Test for Independence
• Under H0, the joint distribution is
approximately distributed by the Chisquare distribution (2).
Chi-square
3.84
2
Reject H0 
22.16
Analysis of Variance
(ANOVA)
What is an ANOVA?
• One-way ANOVA stands for Analysis of
Variance
• Purpose:
– Extends the test for mean difference between
two independent samples to multiple samples.
– Employed to analyze the effects of
manipulations (independent variables) on a
random variable (dependent).
Definitions
• Dependent variable: the variable we are
trying to explain, also known as response
variable (Y).
• Independent variable: also known as
explanatory variables (X).
Therefore, we would like to study whether
the independent variable has an effect on
the variability of the dependent variable
Continuous Dependent variable
One or More Independent
Variable
One Independent
Variable
Binary
Categorical
Categorical and
Continuous
Continuous
t Test
ANOVA
ANCOVA
Regression
One Factor
One-Way
ANOVA
More than
one Factor
N-Way
ANOVA
What does ANOVA tests?
H0 m1= m2 = m3 …..= mn
H1 m1 m2  m3 …..  mn
•
The null hypothesis tests whether the mean of all
the independent samples is equal
•
The alternative hypothesis specifies that all the
means are not equal
Comparing Antacids
• Non comparative ad:
– Acid-off provides fast relief
• Explicit Comparative ad:
– Acid-off provides faster relief than Tums
• Non explicit comparative ad
– Acid-off provides the fastest relief
Comparing Antacids
Brand
Attitude
Means
Type of Ad
Non
Comparative
Explicit
Comparative
Non Explicit
Comparative
Comparing Antacids
Brand
Attitude
Means
Type of Ad
Non
Comparative
Explicit
Comparative
Non Explicit
Comparative
Decomposition of the Total
Variation
Within
Category
Variation
SSwithin
Category
Mean
Independent Variable X
Categories
Total Sample
X1
X2
X3
Xc
….
Y1
Y1
Y1
Y1
Y1
….
Y2
Y2
Y2
Y2
Y2
….
Total
Variation
SSy
Yn
Yn
Yn
Yn
Yn
….
Y1
Y2
Y3
Yc
Y
Between Category Variation SSbetween
Grand
Mean
Decomposition of the Total
Variation
• Total Variation:
SSy = S(Yi- Y)2
SSy =SSbetween + SSwithin
SSy =SSx + SSerror
• Between variation:
c
SSx= S n(Yj- Y)2
j
• Within variation:
c n
SSerror= S S(Yij- Yj)2
j
i
Measurement of the Effects
• We would like to know how strong are the
effects of the independent variable (X) on
the dependent variable (Y).
SSy =SSx + SSerror
SSx =SSy – SSerror
 SSy – SSerror SSx
=
SSy
=
SSy
ANOVA Test
• Under H0 m1= m2 = m3 …..= mn, SSx and SSy
have the same source of variability since the
means are equal between categories.
• Therefore the estimate of the population
variance of Y can be based on either sum of
squares:
Sy=
SSx
=
SSerror
(c-1)
(N-c)
MSx
MSerror
ANOVA Test
• The null hypothesis would be tested with
the F distribution
MS
f=
F distribution
x
MSerror
Reject H0 
f(c-1)(Nc)