Download Comparing 2 Groups: between

Comparing Means
One-sample t-test
Independent-sample t-test
Paired-sample t-test
One-way ANOVA
By Qin Xiaoqing
Normal distribution and sampling
distribution




Normal distribution is made up of individual
scores.
Sampling distribution is composed of group
means.
With more groups means, a curve of their
distribution would be symmetric. This symmetric
curve is not called a normal distribution but rather
a sampling distribution of means
A sampling distribution is the distribution of a
sample statistic that would be obtained if all
possible samples of the same size were drawn
from a given population.
Means and standard deviation for
sampling distribution


When computing a mean, the individual
differences are averaged out. The high and low
scores disappear and only a central measure
(mean) was left.
Since sampling distribution is made of means, it
is much more compact then scores in a single
distribution. Therefore, the standard deviation
will be smaller in the distribution of means.
Population mean and standard
error



When taking the average of a group of scores,
we call the central balance point the mean.
When taking the average of a group of means,
we call it the population mean.
The standard deviation of the sampling
distribution is called the “standard error” and,
when based on sample statistics, it estimates
random sampling error.
As the size of the sample increases, sampling
error or the standard error will decrease.
Sample size and distribution


The size of the groups will also influence the
sampling distribution of means. The larger the N
for each group, the more the means will
resemble each other.
The more scores there are in each group, the
greater the chance of a normal distribution. With
a normal distribution, the mean becomes quite
precise as a measure of central tendency. The
means of large classes should be very similar to
each other.
Features of sampling distribution



For 30 or more samples, sampling distribution of
means is normally distributed.
Its mean is equal to the mean of the population.
Its standard deviation, called standard error of
means, is equal to the standard deviation of the
population divided by the square root of the
sample size.
From sample to population
Sample
Statistic
Population
Estimates
Parameter
Comparison between the mean of
sample and that of population
x μ
Zx 
Sx

Where, x is the mean of sample, μis the
population mean, and S is the standard
deviation of means.
x
Sx
Sx 
N

Where Sx is the standard deviation of the
sample, N is the size of the sample.
One-sample t-test

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H0=There is no effect of group
on the dependent variable.
If the mean for the class was 80,
the published μfor the test was
65, SD for the class was 30, the
class size was 36.
T value = 3.0
Df = N-1 = 36-1 =35
The t critical value for df 30=
2.042 (p<.05), 2.750 (p<.01),
t obs
x μ

Sx
80  65
t obs 
Sx
80  65
t obs 
30  36
Exercise for one-sample test
14 students, mean of reading achievement
for the class is 80 and SD is 5, the
published mean is 85. It is believed that all
the students were better than average.
 Research hypothesis?
 Significant level? 1- or 2-tailed?
 Dependent variable? Independent variable?
 t-value? Critical value? df?

Key to the exercise
80  85
t obs 
 3.73
5  14



t critical value (p.05) = 2.16
Df = 13
Reject H0
Comparing two sample means

To find an individual score in a normal distribution,
we use a z-score formula:
difference between score and mean
Z
standard deviation

To place a sample mean in a distribution of means,
we used a t-test formula:
difference between 2 sample means
t
standard error of difference s between means
xe  xc
t obs 
S xe  xc


Example for independent-sample ttest
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Research hypothesis: no
effect of group on
listening comprehension
P=.05, 2-tailed
df = (n1-1)+(n2-1)=36
The t critical value for df
30= 2.042 (p<.05), 2.750
(p<.01),
Group
n
mean
SD
Con.
19
11.7
4.0
Exper.
19
10.5
4.7
Calculation of t-value and
conclusion
xe  xc
t obs 

S xe  xc


xe  xc


S xe  xc 
4.7 2
19
Se2 Sc2

Ne Nc

4.02
19
 1.42
 1.2
t obs 
 .845
1.42
Conclusion: The 2 groups came from a homogeneous group.
Assumptions underlying t-test
1.
2.
3.
4.
5.
There are only 2 levels of one independent
variable to compare.
Each subject is assigned to one and only one
group.
The data are truly continuous.
The mean and SD are the most appropriate
measures to describe the data.
The distribution in the respective population
from which the samples were drawn is normal,
and variances are equivalent.
Paired-sample t-test
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df=npairs-1
S D : Standard error of differences between the 2
means
n=the number of pairs
D=difference between pairs
x1  x 2
t
SD
SD
SD 

n
 n D
D 2  1
n 1
2
 n

No
pretest
posttest
D
D2
1
21
33
12
144

2
17
17
0
0

3
22
30
8
64
4
13
23
10
100

5
33
36
3
9

6
20
25
5
25
7
19
21
2
4
8
14
19
5
25
9
20
19
-1
1
10
31
35
4
16
∑X=21
0
∑D2=38
∑Y=258 ∑D=48
8
Mean=
21
Mean=
258
df=10-1=9
n= 10
t-value=-3.63
t critical value=2.262
Conclusion: H0 rejected
388  (1  10)( 48) 2
SD 
 4.18
10  1
4.18
SD 
 1.323
10
X1  X2 21  25.8
t obs 

 3.63
SD
1.323
Interpretation of results from
paired-sample t-test
Group
n
Mean
SD
t value
df
p
Pretest
38
17.3
5.1
-2.81
37
.008
Post
38
19.4
4.9
1.
t critical value for df 30 =2.042
2.
Conclusion: H0 rejected
One-way ANOVA
The one-way ANOVA is used to compare
the means of more than 2 groups on one
variables in order to see whether the
differences are caused by chance or by
treatment effect.
 H0: mean1= mean2 = mean3…

F value
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Error variability = within-group variance = S2within
Error variability + treatment effect = between–group
variance =S2between
If S2between = S2within, then we know our treatments
are all similar.
If S2between ＞ S2within, we can say there is treatment
difference.
Fobs= S2between / S2within
When F value=1, no effect for treatment, when F
value > 1, effect for treatment
Example for one-way ANOVA
Natural
Trans
Silent
Sit/Fun
Drama
1
16
15
14
14
10
2
14
13
13
10
8
3
10
12
15
9
10
4
13
13
17
11
9
5
12
13
11
11
12
6
20
20
11
12
5
7
20
19
12
10
8
8
23
22
10
13
8
9
19
19
13
9
7
10
18
17
12
8
9
∑X
165
163
128
107
86
∑X2
2879
2771
1678
1177
772
N=50; ∑X=649; ∑X2=9277; (∑X)2=421201
5 steps for calculation of F value
1.
2.
3.
4.
5.
Square each score and add to determine the sum
of squares total (SST)
Find the sum of squares between groups (SSB)
SSB=[(∑X1)2÷ n1 …+(∑Xk)2÷nk] - (∑X)2÷N
Find the sum of squares within each group (SSW).
SSW=SST-SSB
Average each of SSW and SSB to make them
sensitive to their respective degrees of freedom.
The result is the variance values.
S2B=SSB÷(K-1); S2W=SSW÷(N-K)
Calculate the F ratio: S2B÷S2W
Results of F ration calculation
1.
2.
3.
4.
5.
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N=50; ∑X=649; ∑X2=9277; (∑X)2=421201
SST= ∑X2-(∑X)2/50 = 9277-421201/50 = 852.98
SSB=(2722.5+2656.9+1634.4+1144.9+739.6) -8424.2
=478.28
SSW= 852.98-478.28=374.7
S2B=SSB÷(K-1)=119.57; S2W=SSW÷(N-K)=8.33
F ratio: S2B÷S2W = 14.35
df in numerator S2B: K-1=4; df in denominator S2W: N-K
=45.
Check the intersection of 4 and 45 on the F distribution
chart: Critical value for 44 df is 2.58
Conclusion: H0 is rejected at the .05 level.
SPSS results
Source of
variance
ANOVA
score
Between Groups
Within Groups
Total
Sum of
Squares
478.280
374.700
852.980
df
4
45
49
Mean Square
119.570
8.327
F
14.360
Sig.
.000
Assumptions underlying ANOVA
Type of variables
 Independent observations
 Normal distribution
 Equal variances of scores in each group
 A minimum of at least 5 observations per
cell.

Group differences: multiple
comparison
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2 ways to precisely locate differences among
means: planning the comparison ahead of time
(a priori) and post hoc comparison of means
(post hoc).
In the first case, there are preplanned
comparisons and hypotheses (directional) to test.
In the second, exploratory comparisons are
made and the analyses will be 2-tailed tests.