This document provides an introduction to applied statistics and statistical methods. It discusses objectives such as going beyond the mean and tests of differences like t-tests and ANOVA. It also covers descriptive statistics such as measures of central tendency and dispersion, inferential statistics like z-scores and confidence intervals, and tests of differences including independent and paired t-tests and nonparametric alternatives.

1. Introduction to applied statistics &
applied statistical methods
Prof. Dr. Chang Zhu1

2. objectives
• Going beyond the mean
• Tests of differences
t-tests
Wilcoxon and Mann Whitney tests

3. homogeneity of variance test
the Levene’s test
Analyse > Compare Means > One-way ANOVA
p > .05: equality of variance
p <.05 : no equality of variance
Test of Homogeneity of Variances
Levene
Statistic Df1 Df2 Sig.
3.766 1 56 .057

4. descriptive statistic
• measures of central tendency
(mean, median, mode)
• measures of spread or dispersion
(range, variance, standard deviation)
show the characteristics of the sample,
and the population as well

5. descriptive statistic
• How does the mean (M) and the standard
deviation (SD) tell us about the
population?

6. descriptive statistic
standard deviation between the sample means:
standard error (SE)
Central limit theorem (Field, 2009): The
sampling distribution (the frequency
distribution of sample means) has a
standard deviation calculated as:
s: standard deviation of the sample
n: the sample size

7. descriptive statistic
• In a normal distribution with a mean of 0 (zero) and
a standard deviation of 1, we can calculate the
probability of a score occurring.
by looking at the probability table
(SPSS will do this for us)

8. z-scores
• We can convert a raw score into a z-score (a
score in a normal distribution with a mean of 0,
SD = 1):
Important:
95% of z-scores lie between -1.96 and 1.96
or 95% of the scores will be within the limit +/- 2
SD from the mean in a normal distribution.

9. z-scores
Convert a raw score into z-score, given that:
raw score: = 7
mean: = 4
std. deviation = 3
z = 2 a person who gave a score of 7 is quite above than the average,
having 2 standard deviations from the mean.

10. the confidence interval
Based on the SE, we can calculate the boundaries
within with the population mean will fall.
• upper bound: mean + 1.96*SE
• lower bound: mean – 1.96*SE
confidence interval: a range of scores the
population mean will fall within

11. questions
• standard error (SE)
• z-score
• confidence interval (CI)

12. the confidence interval by error bars
The two groups differ in their means and confidence
intervals, hence they are likely to come from different
populations.

13. Gender N Mean Variance Std. Deviation
Stress at the
start of the
week
Female 16 14.81 28.16 5.307
Male 16 18.94 63.26 7.954
?
tests of differences

14. independent samples test
SPSS output
Analyse>Compare Means > Independent-Samples Test
t-test for Equality of Means
t df
Sig.
(2-tailed)
Mean
Difference
Std. Error
Difference
95% Confidence
Interval of the
Difference
Lower Upper
-1.726 30 .095 -4.125 2.390 -9.007 .757
-1.726 26.146 .096 -4.125 2.390 -9.037 .787
t = mean difference/Std. Error difference

15. independent samples t-tests
1. Do male and female participants have
different levels of computer use?
2. Does age group 1 have a higher level of
computer use than age group 2?
What do we need to find out?
How many groups involved in the study?
15

16. paired samples t-tests
• Are the mean exam score in September
different to their exam score in November?
• Are the deep learning approach of students
reported at the beginning of the study
significantly different to the deep learning
approach of students reported at the end of
the study?
What do we need to find out?
How many groups involved in the study?
16

17. test of differences
The t-test assesses whether the means of two
groups are statistically different from each other.
Assumptions:
• normal distribution
• homogeneity of variance (equal variances)
• data measured at interval level (scale)
violated? non-parametric equivalent tests

18. nonparametric tests of differences
A group of students were asked to rank the extent to which they fear of
statistics from 1 (scared) to 7 (not scared at all) at 2 different times
time 1 1 3 5 2 7 3 3 4 6
time 2 5 2 5 2 6 1 7 6 2
difference in ranks -4 1 0 0 1 2 -4 -2 4
ties positive negative

19. parametric tests nonparametric tests objectives
the single sample t-test whether the observed
mean is different from a
set value
the independent t-test the Wilcoxon rank-sum
test and the Mann
Whitney test
comparing means from
two independent groups of
individuals
the paired t-test the Wilcoxon signed
rank test
comparing the means of
two sets of observations
from the same individuals
or from pairs of individuals
tests of differences

20. Mean (SD) Mean
difference
t Sig.
score or
measurement at
time point 1
score or
measurement at
time point 2
Paired sample t-test results
reporting the results

21. reporting the results
(APA styles)

22. PRACTICE

23. what tests to use?
A TV company have started a reality TV show where 32 members of the public are
left to fend for themselves on a desert island. They have asked a psychologist to
monitor the psychological well-being of the contestants and he records a number of
indices of mental health. He is initially interested in the amount of stress
experienced by the contestants during their first week on the island and
hypothesises that:
(1)the females will report higher levels of stress than the males at the start as well
as at the end of the week (H1)
(2)the level of stress experienced by all the participants is increased by the end of
the week of the reality TV show (H2)
The data is named TVshow.sav
H1: Analyze > Compare Means > Independent-Samples Test
H2: Analyze > Compare Means > Paired-Samples T-Test

24. what tests to use?
We want to know if people who intend to get a Ph.D. or Psychology Doctor (PhD
holder) in psychology are more likely to rely on a calendar or day-planner to
remember what they are supposed to be doing (i.e., are people who might become
professors more absent minded than other people).
The ordinal variable planner measures the extent to which a person relies on a
calendar/day planner, ranging from 1 (strongly agree) to 5 strongly disagree).
The data file is named planner_use.sav.
Analyse > Nonparametric Tests > Legacy Dialogs >
2 Independent-Samples

25. Practice 1
- use the data file TVshow.sav
• Did females experience a higher level of stress at the
start of the week than males?
• Did females experience a higher level of stress at the
end of the week than males?
Which test should you use?
independent t-test
Practical guidelines page 2
Analyze > Compare Means > Independent-Samples Test

26. Practice 1
Independent Samples Test
Levene's Test for
Equality of
Variances t-test for Equality of Means
F Sig. t df
Sig.
(2-tailed)
Mean
Difference
Std. Error
Difference
95% Confidence
Interval of the
Difference
Lower Upper
Stress at the start of the
week
Equal variances assumed 3.211 .083 -1.726 30 .095 -4.125 2.390 -9.007 .757
Equal variances not
assumed
-1.726 26.146 .096 -4.125 2.390 -9.037 .787
Stress at the end of the
week
Equal variances assumed .038 .847 .522 30 .606 2.188 4.193 -6.376 10.751
Equal variances not
assumed
.522 29.673 .606 2.188 4.193 -6.380 10.755
Analyse>Compare Means > Independent-Samples Test

27. Practice 1
Conclusion?
At the start of the week, the male participants experienced a
higher level of stress (M= 18.94, SE = 1.99) than the females
(M=14.81, SE = 1.32). This difference was significant t(30) = -
1.73, p < .05. Therefore, hypothesis 1 is not supported
because the psychologist assumed that the females
experienced a higher level of stress than males.
(Practical guidelines page 3)

28. Practice 2
- use the data file planner_use.sav
• Do people who intend to do a PhD degree or PhD
holders are more likely to use a calendar or day
planner?
Which test to use?
independent t-test (nonparametric)
Practical guidelines page 6
Analyse > Nonparametric Tests > Legacy Dialogs >
2 Independent-Samples

29. Practice 2
Ranks
Intend To Get PhD or PsyD N Mean Rank Sum of Ranks
I rely on a calendar / day-planner
to remember what I am supposed
to do.
Intend to do a PhD 11 27.32 300.50
PhD holder 35 22.30 780.50
Total 46
Analyse > Nonparametric Tests > Legacy Dialogs >
2 Independent-Samples

30. Practice 2
Analyse > Nonparametric Tests > Legacy Dialogs >
2 Independent-Samples
Test Statisticsb
I rely on a calendar / day-planner to remember what I am
supposed to do.
Mann-Whitney U 150.500
Wilcoxon W 780.500
Z -1.169
Asymp. Sig. (2-tailed) .242
Exact Sig. [2*(1-tailed
Sig.)]
.284a
Exact Sig. (2-tailed) .252
Exact Sig. (1-tailed) .127
Point Probability .006
a. Not corrected for ties.
b. Grouping Variable: Intend To Get PhD or PsyD

31. Practice 2
Conclusion?
People who intend to do a PhD do not differ significantly from
PhD degree holders with regard to the use of day planner to
remember what they are supposed to be doing , U = 150.50, z =
-1.169, p > .05, ns.
(Practical guidelines page 7)

32. Assignment 3
•
Lecture 3_practical guidelines_assignment
(p. 10)