Independent Groups ttest
When the
means of two groups are to be compared (where each group consists of
subjects that are not related) then the Excel twosample ttest
procedure is used to perform these calculations.
(NOTE: If your observations are related across “group” as paired or
repeated measurements, this in an INCORRECT version of the ttest. For
that case, see the tutorial on the Paired ttest.)
Assumptions: Subjects are randomly assigned to one of two groups. The
distribution of the means by group is normal with equal variances. Sample
sizes between groups do not have to be equal.
Test: The hypotheses for the comparison of means from two
independent groups are:
_{
Ho: m1 = m2
(means of the two groups are equal)}
Ha:
m1 ¹
m2
(means are not equal)
The test statistic is a student’s ttest with N‑2 degrees of freedom,
where N is the total number of subjects. A low p‑value indicates evidence
to reject the null hypothesis in favor of the alternative. In other words,
there is evidence that the means are not equal.
For example,
suppose we are interested in comparing SCORES across GROUPS, where there
are two groups. The purpose is to determine if the mean SCORE on a test is
different for the two groups tested (i.e., control and treatment
groups). The example data is shown here:
Group 
Scores 
1 
20 
1 
23 
1 
32 
1 
24 
1 
25 
1 
28 
1 
27.5 
2 
25 
2 
46 
2 
56 
2 
45 
2 
46 
2 
51 
2 
34 
2 
47.5 
In this
example, GROUP contains two values, 1 or 2, indicating which group each
subject was in. The ttest will be performed on the values in the variable
(column) named SCORE.
An
independent group ttest is done in two steps:
Step 1:
Decide if the variances are equal in both groups, which determines the
type of ttest to perform (one that assumes equal variances or one that
doesn’t make that assumption.) A conservative approach suggested in some
texts is to always assume unequal variances. Another approach is to do a
statistical test to determine equality.
Step 2:
Depending on you decision about the equality of variances you either
perform the version of the ttest that assumes equality of variances or
other one that doesn’t make that assumption.
Determine
Equality of Variance
If you take the conservative approach, skip this test and proceed to the
version of the ttest that does not assume equality of variance.
To do a statistical test to determine equality of variance, follow these
instructions. (The test for equality of variances is an Ftest.)
1.
In Excel, select Tools/ Data Analysis / FTest Two Sample for
Variance.
2.
In the FTest Two Sample for Variance dialog box: For the Input
Range for Variable 1, highlight the seven values of Score in group 1
(values from 20 to 27.5). For the input range for Variable 2, highlight
the eight values of Score in group 2 (values from 25 to 47.5). Leave the
other items at their default selections. This dialog box is shown below.
Click OK.
3.
The following results are produced by Excel:
FTest TwoSample
for Variances 






Variable 1 
Variable 2 
Mean 
25.64285714 
43.8125 
Variance 
15.22619048 
96.42410714 
Observations 
7 
8 
df 
6 
7 
F 
0.157908545 

P(F<=f) onetail 
0.019378053 

F Critical
onetail 
0.23771837 

Notice the highlighted probability p=0.01937. This is a onetail pvalue
associated with the test for equality of variance. Generally, if this
value is less than 0.05 you assume that the variances are NOT equal.
a.
If the variances are assumed to NOT be equal, proceed with the
ttest that assumes nonequal variances.
b.
If the variances are assumed to be equal, proceed with the
ttest that assumes equal variances.
Perform the ttest
The process of doing the ttest in Excel is similar for both the equal and
unequal variances case – the main difference is which version you select
from the menu. Suppose you select the unequal version of the twosample
ttest – this is how you proceed:
1.
Select Tools/ Data Analysis/ tTest: Two Sample assuming
Unequal Variances
2.
For the Input Range for Variable 1, highlight the seven values of
Score in group 1 (values from 20 to 27.5). For the input range for
Variable 2, highlight the eight values of Score in group 2 (values from
25 to 47.5). Leave the other items at their default selections. This
dialog box is shown below. Click OK.
3.
The following output is created:
tTest: TwoSample
Assuming Unequal Variances 






Variable 1 
Variable 2 
Mean 
25.64285714 
43.8125 
Variance 
15.22619048 
96.42410714 
Observations 
7 
8 
Hypothesized Mean
Difference 
0 

Df 
9 

t Stat 
4.816944724 

P(T<=t) onetail 
0.000475506 

t Critical
onetail 
1.833112923 

P(T<=t) twotail 
0.000951012 

t Critical
twotail 
2.262157158 

Notice that
the two sample mean values (variance) are 25.64(15.23) and 43.81(96.42).
The two tailed calculated tstatistic is 4.82 and the
highlighted pvalue for this test
is p=0.001. (0.000951012) Since the pvalue is less than 0.05, this
provides evidence to reject the null hypothesis of equal means.
As an example
of how this might be reported in a journal article:
Methods: A
preliminary test for the equality of variances indicates that the
variances of the two groups were significantly different F=.157, p=.02.
Therefore, a twosample ttest was performed that does not assume equal
variances.
Results: The mean score for group 1 (M=25.64 SD= 3.9021, N= 7) was
significantly smaller than the scores for group 2 (M=42.81, SD=9.82, N=
8.) using the twosample ttest for unequal variances, t(9) = 4.82, p
<= 0.001. (Technically, the degrees of freedom for this unequal
variances ttest should be 9.4 instead of 9, but Excel unfortunately
rounds off the DF, so it is reported incorrectly. Years ago, it used to be
conventional to round down if you were constructing a table for a
probability level, but most statistical programs now calculate the correct
pvalue using a fractional DF through interpolation.)
Notice that the variance is reported rather than the standard deviation,
as shown in the Excel results table. You can calculate the standard
deviation using Tools/ Data Analysis / Descriptive statistics.
When the variances are assumed equal, the analysis is similar, select
Tools/ Data Analysis/ tTest: Two Sample assuming Equal Variances
End of
tutorial
See
http://www.stattutorials.com/EXCEL