Chapter 4
Investigating the Difference in Scores
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Chapter Objectives
After completing this chapter, you should be able to
1. Define and give examples of dependent variable,
independent variable, null hypothesis, two-tailed test,
one-tailed test, degrees of freedom, level of
significance, standard error of the mean, and standard
error of the difference between means.
2. Define Type 1 and Type II errors.
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Chapter Objectives
3. Use and interpret the t-test for independent groups and
the t-test for dependent groups.
4. Use and interpret analysis of variance for independent
groups and analysis of variance for repeated measures.
5. Use Tukey’s honestly significant difference test
(HSD).
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Dependent and Independent Variables
Dependent variable
• Response to treatment (independent variable)
• Variable that is observed for changes as result of
treatment
Independent variable
• Treatment in a study
• Controlled by researcher
Study to determine if change in blood pressure
(dependent variable) after reduction in salt intake
(independent variable)
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Testing for Significance Difference
Between Two Means
Examples: Fitness programs, methods of instruction,
flexibility programs
Hypothesis: a prediction about the difference between two
or more variables
Null hypothesis: predicts there will be no statistical
difference between the means of groups (H0: X1 = X2)
Alternative hypothesis: predicts there will be a difference
in means (H1: X1 = X2)
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Testing for Significance Difference
Between Two Means
Alternative hypothesis used for a two-tailed test;
difference in means can be in either direction.
One-tailed test used when mean difference can occur
only in one direction.
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Degrees of Freedom (df)
df - concept used in all statistical tests; calculated by
subtracting 1 from N (N - 1).
Determined by sample size.
Indicate the number of scores in a distribution that are
free to vary.
When using a t-test (two groups), the degrees of freedom
equal (N1 - 1) + (N2 - 1) = N1 + N2 - 2
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Level of Significance
Probability of rejecting a null hypothesis when it is true.
Two most common levels of significance are .01 and .05.
.01 = If you reject the null hypothesis, there is 1 chance in
100 that you are rejecting the null hypothesis when it is
actually true.
.05 = If you reject the null hypothesis, there are 5 chances
in 100 that you are in error.
Type I error - reject a hypothesis when it is true
Type II error - accept the hypothesis when it is not true
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Level of Significance
Using the .01 level of significance, rather than the .05
level, reduces the risk of making a Type I error but
increases the probability of making a Type II error.
Level of significance below .05 rarely uses.
Best way of reducing the probability of making a Type II
error is to increase the sample size.
Level of significance and df are used together to
determine the value that a statistical test must yield to
reject the null hypothesis. Use values in Appendix B to
determine if you reject or accept the null hypothesis.
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Standard Error of the Mean (SEM)
Distribution of sample means
If a large number of equal-size samples were randomly
drawn from the same population and formed into a
distribution, we would have a sampling distribution of
means.
If means are in close agreement, value of SEM is small;
more confident that any one mean is near the value of the
population mean.
Not practical to calculate the SEM from a sampling
distribution of means; use formula to estimate of SEM.
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Standard Error of the Mean (SEM)
SEM = s
N
See Table 4.1
Group running three days/week: X = 80.9 and SEM = 0.38
SEM is added to and subtracted from X (80.9  0.38)
If this study were repeated a large number of times,
68.26% of the time the means for the group running three
times/week would be in the range of 80.52 to 81.28.
For group running five days/week, 68.26% of the time the
means would be in the range of 90.51 to 92.09 (91.3 
0.79)
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Standard Error of the Mean (SEM)
Range interpreted as the limits of the 68% confidence
intervals for mean.
95% confidence intervals - add and subtract two SEMs
99% confidence intervals – add and subtract three
SEMs
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Standard Error of the Difference
Between Means
Estimate of the size difference to be expected between two
sample means randomly drawn from the same population.
To determine standard error of the difference between
means, it is necessary to square the SEM of each group,
add the results, and find the square root of the sum.
Sx-x = SEM12 + SEM22
Table 4.1: Difference between the means is 10.4 and
Sx-x = 0.87.
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t-Test for Independent Groups
Used to determine the significance of the difference
between two independent means.
Independent means - one group, or variable, does not
influence the other group (variable); groups are not
related.
Assumptions:
1. Initially the two sample groups come from the same
population.
2. The population is normally distributed.
3. The two groups are representative samples; that is, they
have approximately equal variances.
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t-Test for Independent Groups
Example: Determine if running 5 days/week would develop
cardiorespiratory endurance better than running 3
days/week (see table 4.1).
Steps for use of t-test for independent groups
Null hypothesis: X1 = X2
Randomly select individuals for each group and running
program.
1. Calculate the mean and standard deviation for each
group.
X1 = 80.9
X2 = 91.3
s1 = 1.20
s2 = 2.50
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t-Test for Independent Groups
2. Calculate the SEM for each group.
SEM1 = s1 = 1.2 = 0.38
SEM2 = 0.79
N1
10
3. Calculate the standard error of difference between
groups.
sx-x = SEM12 + SEM22 = (0.38)2 + (0.79)2 = 0.87
4. Calculate the t-ratio by substituting the values in the
formula
t = X1 - X2 = 80.9 - 91.3 = -11.95
sx-x
0.87
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t-Test for Independent Groups
5. Determine the degrees of freedom (df).
df = 10 + 10 – 2 = 18
6. Refer to t-values in appendix B. If the computed t-ratio
is equal to or greater than the critical value in appendix
B, reject the null hypothesis. If the ratio is less than the
critical value, accept the null hypothesis.
t = -11.95 (ignore negative sign)
table value (appendix B); 18 df, .05 level = 2.101, .01
level = 2.878
7. Reject the null hypothesis; significant difference in
means at .01 level; running 5 days/week develops
cardiorespiratory endurance better than running 3
days/week.
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t-Test for Dependent Groups
Use to determine significant difference between two groups
when two groups are not independent but are related to each
other.
Assumptions:
1. The paired differences are random sample from a normal
population.
2. The equal variances assumption is unnecessary, since you
will be working with one group.
Example: Determine if participation in a basketball class
will improve the scores of ninth-grade girls on a speed spot
shooting test. (See table 4.2).
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t-Test for Dependent Groups
Steps for use of t-test for dependent groups
Null hypothesis: There will be no difference in means of speed spot
shooting pretest and posttest (X1 = X2).
Administer test first day of class and again at the conclusion of the
basketball unit.
1. List the pairs of scores so that you can subtract one from the
other.
2. Label a column D and determine the difference for each
pair of scores.
3. Label a column D2, square each D, and sum D2 (D2).
D2 = 50
4. Calculate the mean difference (D).
D = 18; D = 1.8
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t-Test for Dependent Groups
5. Calculate the standard deviation of the difference.
sD = D2 - N(D2) = 50 - 10(1.82)
= 1.33
N
10
6. Calculate the standard error of the difference (sD). The sD is
equivalent to the SEM and is obtained in a similar manner.
Formula is:
sD = sD = 1.33 = .42
N
10
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t-Test for Dependent Groups
7. Calculate the t-ratio by substituting the values in the
formula
t = D = 1.8 = 4.29
sD .42
8. Determine the degrees of freedom. df = 10 - 1 = 9.
9. Refer to the t-values in appendix B. With 9 df, the t-ratio
of 4.29 is greater than the t-value of 2.262 needed for
significance at the .05 level of significance and greater
than the t-value of 3.250 needed at 01 level.
10. Reject the null hypothesis at the .01 level of significance.
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Testing for Significant Difference
Among Three or More Means
Analysis of variance (ANOVA) is used to test for significant
difference among three or more means.
Special Terms and Symbols
N = number of scores
n = the number of scores in a group
k = the number of independent groups or the number of trials
(measures) performed on the same subjects (repeated
measures)
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ANOVA – Special Terms and Symbols
r = number of rows or subjects
grand X = sum of all the scores in all groups
grand X2 = sum of the square of all scores in all groups
total sum of squares (SST) = the sum of the squared deviations of
every score from the grand X; represents the variability of each score
in all groups from the grand X
sum of squares within groups (SSW) = the sum of the squared
deviations of each score from its group X; also referred to as sum of
squares for error
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ANOVA – Special Terms and Symbols
sum of squares between groups (SSB) = the sum of the
squared deviations of each group X from the grand X;
also referred to as treatment sum of squares
mean square for the sum of squares within groups
(MSW) = variance within the groups
mean square for the sum of squares between groups
(MSB) = variance between groups
F distribution = table values required to reject null
hypothesis
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ANOVA – Special Terms and Symbols
degrees of freedom within groups = (N - k);
total number of measures or scores (N) minus number of
groups (k); degrees of freedom for denominator in F
distribution
degrees of freedom between groups = (k - 1); number of
groups (k) minus 1; degrees of freedom for numerator in F
distribution
degrees of freedom between groups = k - 1
degrees of freedom within groups = N - k
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ANOVA for Independent Groups
Used when the sample groups are not related to each other.
Assumptions
1. The samples are randomly drawn from a normally
distributed population.
2. The variances of the samples are approximately equal.
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ANOVA for Independent Groups
Example: Determine if there is a difference in three
programs designed to improve trunk extension.
(See table 4.3).
Null hypothesis: X1 = X2 = X3
Instructor randomly assigns subjects to one of three
groups; each group participates in a different trunk
extension program; all subjects are given the same trunk
extension test.
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Steps for Calculation of ANOVA for
Independent Groups
1. Sum each group of scores to get X1, X2, and X3. Add
the group sums to get a grand X (X1 + X2 + X3 =
grand X).
171 + 154 + 152 = 2904
2. Square each score and sum the squared scores of each
group (X2).
Add the X2 for each group to get a grand X2
(X12 + X22 + X32 = grand X2).
3665 + 2972 + 2904 = 9541
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Steps for Calculation of ANOVA for
Independent Groups
3. Calculate the correction factor (C). C is necessary
because the raw scores are used rather than
deviations of the scores from the mean.
C = (grand X)2
C = (477)2 = 9480.38
total N
24
4. Calculate the total sum of squares (SST).
SST = grand X2 - C
SST = 9541 - 9480.38 = 60.62
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Steps for Calculation of ANOVA for
Independent Groups
5. Calculate the sum of squares between groups (SSB).
SSB = (X1)2 + (X2)2 + (X3)2 - C
n1
n2
n3
SSB = (171)2 + (154)2 + (152)2 - 9480.38 = 27.25
8
8
8
6. Calculate the sum of squares within groups (SSW)
SSW = SST - SSB
SSW = 60.62 - 27.25 = 33.37
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Steps for Calculation of ANOVA for
Independent Groups
7. Calculate the mean square for the sum of squares between
groups.
MSB = SSB
k-1
MSB = 27.25 = 13.63
3–1
8. Calculate the mean square for the sum of squares within
groups.
MSW = SSW
N-k
MSW = 33.37 = 1.59
24 - 3
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Steps for Calculation of ANOVA for
Independent Groups
9. Calculate the F-ratio (see table 4.4).
F = MSB
MSW
F = 13.63 = 8.57
1.59
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Steps for Calculation of ANOVA for
Independent Groups
10. Refer to the F-distribution in appendix C to
determine if F is significant.
Numerator degrees of freedom = (k - 1) = ((3 - 1) = 2
Denominator degrees of freedom = (N - k) = (24 - 3) =
21
.05 table value = 3.47
.01 table value = 5.78
F-ratio of 8.57 greater than 5.78; significant difference
in three means at the .01 level.
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Post Hoc Test
If significant difference in means, must determine which means
are significantly different from each other. Test used for this
purpose is called post hoc test.
Tukey’s honestly significant difference test (HSD)
HSD = q(, k, N-k) MSW (used if n’s in each group are
n
equal)
HSD = q(, k, N-k)
MSW 1 + 1 (used if n’s in any two 2
2
n 1 n2
groups are unequal)
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Post Hoc Test
Since n’s are equal in table 4.3, use first formula.
Use appendix D to fine q.
q = about 4.64 (k = 3, N - k = 21); since table jumps from 20
to 24 for the denominator, will use 3,20.
HSD = 4.64 1.59 = 4.64 0.19875 = 2.069
8
2.069 represents the minimum raw-score difference between
any two means that may be declared significant.
X1 - X2 = 21.38 - 19.25 = 2.13*
X2 - X3 = 19.25 - 19.00 = 0.25
X1 - X3 = 21-38 - 19.00 = 2.38*
(*significant difference)
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Analysis of Variance for Repeated Measures
Used when repeated measures are made on the same
subjects.
Basic assumptions:
1. The samples are randomly selected from a normal
population.
2. The variances for each measurement are approximately
equal.
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Analysis of Variance for Repeated Measures
Example: Determine if loudness of sound is a factor in
the making of foul shots (see table 4.5).
Null hypothesis: X1 = X2 = X3
In this technique, N refers to the total number of
measurements.
10 subjects are measured 3 times each, so N = 30.
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Steps in ANOVA for Repeated Measures
1. Calculate the grand X and grand X2.
X1 + X2 + X3 = grand X
100 + 110 + 94 = 304
X12 + X22 + X32 = grand X2
1016 + 1224 + 890 = 3130
2. Calculate the correction factor.
C = (grand X)2 = (304)2 = 3080.53
N
30
3. Calculate the total sum of squares (SST).
SST = grand X2 - C
SST = 3130 - 3080.53 = 49.47
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Steps in ANOVA for Repeated Measures
4. Calculate the sum of squares between groups (SSB);
this is the sum of squares between trials.
SSB = (X1)2 + (X2)2 + (X3)2 - C
n1
n2
n3
SSB = (100)2 + (110)2 + (94)2 - 3080.53 = 13.07
10
10
10
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Steps in ANOVA for Repeated Measures
5. Calculate the sum of squares to determine the effects of
the test-retest (SSsubjects). Add the scores of the three
trials for each subject, square the sum, and add the
squared sums for each subject. The sum of the squared
sums for each subject is divided by k (the number of
trials), and C is subtracted from the resulting value.
SSsubjects = ( rows)2 – C
k
= (11+12 +10)2 + (9+10+9)2 + . . . (9 +10 +10)2 - 3080.53
3
SSsubjects = 3108.67 – 3080.53 = 28.14
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Steps in ANOVA for Repeated Measures
6. Compute the sum of squares for error (also referred to as
within-group sum of squares). This calculation represents
the sum of the squared deviations of each score from its
group X.
SSE = SST - SSB - Sssubjects
SSE = 49.47 - 13.07 - 28.14 = 8.26
7. Calculate the mean square for the sum of squares between
groups
(trials).
MSB = SSB = 13.07 = 6.54
k–1
3-1
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Steps in ANOVA for Repeated Measures
8. Calculate the mean square for the sum of squares between
subjects.
MSsubjects = Sssubjects
r-1
MSsubjects = 28.14 = 3.13
10 - 1
9. Calculate the mean square for the sum of squares for error.
MSE =
SSE
(k - 1)(r - 1)
MSE =
8.26 = 0.46
(2)(9)
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Steps in ANOVA for Repeated Measures
10. Calculate the F-ratio (see table 4.6).
F-ratio = MSB
MSE
F-ratio = 6.54 = 14.22
0.46
11. Refer to the F-distribution in appendix C. The degrees of
freedom for the numerator are (k-1) = (3-1) = 2. For the
denominator, the degrees of freedom are (k-1)(r-1) =
(3-1)(10-1) = 18. The F-ratio of 14.22 is greater than the
6.01 value needed for significance at .01 level, so we
reject the null hypothesis. Now use Tukey’s HSD test to
determine which means are significantly different.
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Post Hoc Test
Tukey’s HSD test for ANOVA for repeated measures
HSD = q(, k, [k-1][r-1]) MSE
r
q = value obtained from table in appendix D = 4.70
 = significance level = .01
k = number of trials or measures performed on the same
subjects = 3
r = number of rows or number of subjects = 10
(k-1)(r-1) = degrees of freedom for denominator in
appendix D = (3-1)(10-1) = 18
MSE = mean square for the sum of squares for error = 0.46
(table 3.8)
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Post Hoc Test
HSD = q(, k, [k-1][r-1]) MSE
r
HSD = 4.70 0.46 = 1.008
10
X1 - X2 = 10 - 11 = -1
X2 - X3 = 11- 9.4 = 1.6*
X1 - X3 = 10 - 9.4 = 0.6
*significant difference between X2 (medium sound) and
X2 (loud sound)
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