Practical Statistics
Mean Comparisons
There are six statistics that will
answer 90% of all questions!
1.
2.
3.
4.
5.
6.
Descriptive
Chi-square
Z-tests
Comparison of Means
Correlation
Regression
t-test and ANOVA are for the means of
interval and ratio scales
These are very common statistics….
William S. Gosset
1876-1937
Published under the
name: Student
t-test come in three types:
1. A sample mean against a hypothesis.
t-test come in three types:
1. A sample mean against a hypothesis.
2. Two sample means compared to each other.
t-test come in three types:
1. A sample mean against a hypothesis.
2. Two sample means compared to each other.
3. Two means within the same sample.
t-test
The standard error for means is:
SE 
0
n
 x
t-test
Hence for one mean compared to a hypothesis:
t
x  H0
0
n
Each t value comes with a certain degree
of freedom df = n - 1
t-test
IQ has a mean of 100 and a standard deviation of
15. Suppose a group of immigrants came into
London. A random sample of 400 of these
Immigrants found an average IQ of 98.
Does this group have an IQ below the
population average?
t-test
The test statistic looks like this:
98  100  2
t

 2.67
15
0.75
400
There are n – 1 = 399 degrees of freedom.
The results are printed out by a computer or looked
up on a t-test table.
The critical value for
399 degrees of
freedom is about 1.97.
Of course, we could look this
up on the internet….
http://www.danielsoper.com/statcalc/calc08.aspx
For the IQ test: t(399) = 2.67, p = 0.00395
t-test
Since the test was “one-tailed,” the critical value
of t would be -1.65.
Therefore, t(399) = -2.67 would indicate
that the immigrants IQ is below normal.
t-test come in three types:
1. A sample mean against a hypothesis.
2. Two sample means compared to each other.
3. Two means within the same sample.
t-test
The standard error of the difference
between two means looks like this:
SE 
  12  22 

   1  2

 n1 n2 
t-test
Therefore the test statistic would look like this:
t
( x1  x2 )  ( 1  2 )
 12
n1

 22
n2
With degrees of freedom = n(1) + n(2) - 2
t-test
Usually this is simplified by looking at the difference
between two samples; so that:
t
( x1  x2 )
1
2 1
sp (  )
n1 n2
Where:
(n1  1) S  (n2  1) S
S 
n1  n2  2
2
p
2
1
2
2
Suppose that a new product was test marketed in
the United States and in Japan. The company
hypothesizes that customers in both countries
would consume the product at the same rate.
A sample of 500 in the U.S. used an average of 200 kilograms
a year (sd = 20), while a sample of 400 in Japan used an
average of 180 kilograms a year (sd = 25).
Test the hypothesize…..
H0  1  2  0
The test would start be computing:
(500  1)20  (400  1)25
2
S 
2
p
= 500
500  400  2
2
(200  180)
t
 0.89
500
The results would be written as:
(t(898) = 0.89, ns),
and the conclusion is
that there is no difference in the
consumption rate between the U.S. and
Japanese customers.
(200  180)
t
 0.89
500
But this is wrong!
Can you see why?
It is caused by a common mistake of
confusing the sampling distribution
with a the sample distribution.
(200 180)
t
1
1
(500)(

)
500 400
The results are written as:
(t(898) = 13.33, p < .0001),
and the conclusion is that there is a large
difference in the consumption rate between
the U.S. and Japanese customers.
t-test come in three types:
1. A sample mean against a hypothesis.
2. Two sample means compared to each other.
3. Two means within the same sample.
t-test come in three types:
3. Two means within the same sample.
This t-test is used with correlated samples and/or
when the same person or object is measured
twice in the same sample.
Student
T1
T2
d
Tom
Jan
Jason
Halley
Bill
89
88
87
90
75
90
91
86
90
79
1
3
-1
0
4
The measurement of interest is d.
H0 : Average of d = 0
That is… the average difference
between test 1 and test 2 is zero.
t-test
The sampling error for this t-test is:
SE 
2
D
S
 SD
n
Were d = score(2) – score(1)
t-test
The t-test is:
D
t
SD
The degrees of freedom = n - 1
Examples can be found at these sites:
http://en.wikipedia.org/wiki/T-test
http://canhelpyou.com/statistics/tTestDependentSamples.html
Suppose there are more than two groups
that need to be compared.
The t-test cannot be utilized for two reason.
1. The number of pairs becomes large.
Suppose there are more than two groups
that need to be compared.
The t-test cannot be utilized for two reason.
1. The number of pairs becomes large.
2. The probability of t is no longer accurate.
Hence a new statistic
is needed:
The F-test
Or
Analysis of Variance (ANOVA)
R.A. Fisher
1880-1962
The F-test
Compares the means of two or more groups
by comparing the variance between groups
with the variance that exists within groups.
F is the ratio of variance:
2
1
2
2
S
S
http://controls.engin.umich.edu/wiki/index.php/Factor_analysis_and_ANOVA
The F-test
http://www.statsoft.com/textbook/distribution-tables/
The F-test
The probability distribution is dependent upon
the degrees of freedom between and the
degrees of freedom within.
The F-test
Typical output looks like this:
In SPSS ANOVA looks like this:
Descriptives
overall
N
1.00
2.00
3.00
4.00
Total
64
24
38
6
132
Mean
8.5781
8.2917
9.1053
8.0000
8.6515
Std. Deviation
1.83272
1.60106
.92384
1.09545
1.56773
Std. Error
.22909
.32682
.14987
2
.44721S1
2
.13645S 2
95% Confidence Interval for
Mean
Lower Bound
Upper Bound
8.1203
9.0359
7.6156
8.9677
8.8016
9.4089
6.8504
9.1496
8.3816
8.9215
Minimum
1.00
3.00
7.00
7.00
1.00
Maximum
10.00
10.00
10.00
10.00
10.00
ANOVA
overall
Between Groups
Within Groups
Total
Sum of
Squares
13.823
308.147
321.970
df
3
128
131
Mean Square
4.608
2.407
F
1.914
Sig.
.131
Service Encounter
The average age of Iowans over 18 is
approximately 47. Is the sample a cross-section
of this population by age?
A sample mean against a hypothesis.
Service Encounter
Is the measure of personality different between
men and women?
Two sample means compared to each other.
Service Encounter
Is the measure of personality different between
men and women?
Two sample means compared to each other.
Service Encounter
Is the measure of personality different between
men and women?
Service Encounter
Do respondents like themselves better than the
service provider?
Two means within the same sample.
Service Encounter
Do respondents like themselves better than the
service provider?
Two means within the same sample.
Service Encounter
Is the measure of personality different between
shopping times?
Service Encounter
Is personality difference by perception of
service encounter?
More than two sample means compared to
each other.
Service Encounter
Is personality difference by perception of
service encounter?
More than two sample means compared to
each other.