Statistics II

1. SCALES OF MEASUREMENT
INFERENTIAL STATISTICS
Desmond Ayim-Aboagye, Ph.D.

2. SCALES OF MEASUREMENT
• Behavioural scientists rely on four scales of
measurement in their research. These scales
are usually called
• 1. Nominal,
• 2. Ordinal,
• 3. Interval and
• 4. Ratio

3. INFERENTIAL STATISTICS
• Inferential statistics permit generalizations to
be made about populations based on sample
data drawn from them.
• It infers population characteristics.

4. STIMULI
• STIMULI , singular STIMULUS
• These are aspects of the outside world that
affect behaviour or conscious experience.

5. NOMINAL SCALE
• NOMINAL SCALES
• A nominal scale uses numbers to identify qualitative differences
among measurement. The measurements made by a nominal
scale are names, labels, or categories, and no quantitative
distinctions can be drawn among them.
•
• Nominal information can be coded: Gender/Sex, Ethnicity, Race,
Religion, etc.
• 1 = Male
• 2 = Female
•
• Or 1 = Presbyterian, 2 = SDA, 3 = Pentecostals, 4 = Baptist
•
• Nominal scales name things

6. ORDINAL SCALES
• An ordinal scale ranks or orders observations based on
whether they are greater than or less than one another.
Ordinal scales do not provide information about how close
or distant observations are from one another.
• Ordinal scales ranks or order things.
Top 5 films John ranking David ranking
1 Citizen 2 4
2 Casablanca 4 1
3 The Godfather 1 2
4 Gone with the wind 3 5
5 Lawrence of Arabia 5 3

7. INTERVAL SCALES
• An Interval scale is quantitative, contains measurably
equal distances between observations, but lacks a true
zero.
•
• The Fahrenheit scale found on most thermometers is an
example of an interval scale. Temperatures can go and fall
below zero scale.
•
• Thermometer is an example.
•
• Interval scales are quantitative measures that lack a true
zero

8. RATIO SCALES
• A ratio scale ranks observations, contains equal and
meaningful intervals, and has a true zero point.
•
• Weight is a ratio scale, so that a 2-ton object is to a 1-
ton object as a 4-ton object is to a 2-ton object (i.e.,
each is twice the weight of the other).
• Height is also a ratio scale. For example, 6 ft person is
twice as tall as a 3-ft person.
•
• Ratio scales are quantitative measures that have a true
zero.

9. THE FOUR BASIC STATISTICAL
SYMBOLS
THE FOUR BASIC SYMBOLS
SYMBOLS MEANING
X,Y
N
𝑘=0
𝑛
Variable X, Variable Y
The Total number of Observations
To sum or the sum of

10. SYMBOLS
• X and Y are variables that take on the values
of some set of observations or data
• N signifies the total number of observations
in a set of data.

11. RESEARCH PROCESS IN PSYCHOLOGY:
EXPERIMENTATION
• STEP 1. Collect observations or rely on existing theory
• STEP 2. Develop a testable hypothesis or hypotheses
• STEP 3. Conduct experiments to test the hypothesis
and to eliminate alternative hypothesis/es
• STEP 4. Analyze the data and interpret the results of
the experiment (s)
• STEP 5. Begin the loop again: Go back on step 1
•
• Figure 2. The Research Loop of Experimentation.
Adapted from Dunn (1999)

12. Experimentation
• An experiment introduces intentional change into some situation
so that reactions to it can be systematically observed, measured,
and recorded.
•
• Experimental group (film with aggressive behaviour is shown)
• Control group (film with romantic undertone)
• Independent variable---- Dependent variable
•
• It is the experimental group that has to be manipulated in order to
get the desired result or reaction.
•
• A replication study, which is usually an experiment, is performed
to repeat or duplicate some scientific result.

13. POPULATION AND SAMPLE
• RANDOM ASSIGNMENT VRS RANDOM SAMPLING
• RANDOM ASSIGNMENT
• It involves assigning participants randomly to the
conditions or groups in an experiment.
• Random assignment to condition is used to equalize
groups at the outset of a study.

14. Conveni
ence
Sample
Experimenta
l Group
Control
Group
Random Assignment (Experimental or Control Group)

15. Populat
ion
Sample
Random Selection Allows Generalization
Back to the population
Random Selection Process
RANDOM SAMPLING

16. SAMPLING PROCEDURE WITH
RANDOMIZATION
• SIMPLE RANDOM SAMPLE
• Each member of a population has the same
chance as every other member of being
selected for inclusion in a sample.

17. RANDOM ASSIGNMENT
• RANDOM ASSIGNMENT
• Each member of a group is assigned at
random to one condition within an
experiment or study.

18. SYTEMATIC RANDOM ASSIGNMENT
• SYTEMATIC RANDOM ASSIGNMENT
• Each member of a group is assigned at
random to one condition within an
experiment or study.

19. SAMPLING
• SAMPLING
• A list of population's members is created and
then every nth member is included in the
sample.

20. STRATIFIED RANDOM SAMPLING
• STRATIFIED RANDOM SAMPLING
• A population is divided into subgroups
(strata) and then a random sample of
individuals is drawn from each subgroup.

21. CLUSTER SAMPLING
• CLUSTER SAMPLING
• Small units within a population (clusters) are
identified at random and then each person in
a unit is included in the sample.

22. SAMPLING PROCEDURES WITHOUT
RANDOMIZATION
• CONVENIENCE or HAPHAZARD SAMPLE
• A researcher recruits people who are accessible, available and willing to
take part in a piece of research. Individuals in the sample are meant to
be more or less representative of the larger population.
• QUOTA SAMPLE
• It involves sampling a specified number of participants from a special
interest group or groups within the larger population.
• CENSUS
• A sample that includes each and every member (or score, unit, or
observation) within a population.

23. SAMPLING ERROR
• SAMPLING ERROR
• Is the difference existing between a sample
statistic and its corresponding population
parameter.

24. ADVANTAGE OF LARGER SAMPLE
LARGER SAMPLES
Larger samples characterize a population
more accurately than smaller samples.
LARGER SAMPLES
Larger samples exhibit smaller amounts of
sampling error than do smaller samples.

25. CENTRAL LIMIT THEOREM
• The Central Limit Theorem proposes that as the size
of any sample, N, becomes infinitely large in size, the
shape of the sampling distribution of the mean
approaches normality—that is, it takes on the
appearance of the familiar bell-shaped curve – with a
mean equal to μ, the population's mean, and the
standard deviation equal to σ/ 𝑵, which is known as
the standard error of the mean. As N increases in size,
the standard error of the mean or 𝝈χ will decrease in
magnitude, indicating that the sample will be close in
value to the actual population μ. Thus, it will also be
true that μ χ ≅ 𝛍 and that 𝝈χ ≅ 𝛔/ 𝑵.

26. CLT Two Characteristics
• Despite a parent population's shape, mean, or standard deviation,
the central limit theorem can be used to describe any distribution
of sample means. Thus, a population does not have to be normally
distributed in order for the central limit theorem to be true. The
central limit theorem, then, can be applied to any population as
long as samples can be randomly drawn and be of a reasonable
fixed size.
• As N increases in size, the shape of the distribution of sample
means quickly approaches normality. When an N = 30 observations
or greater, a sampling distribution will take on the familiar,
symmetric bell-shaped curve. Interestingly, if a population is
normally distributed to begin with, then even small fixed size N
samples (i.e., < 30) will create a normally shaped sampling
distribution of means.

27. OPERATIONAL DEFINITION IN
BEHAVIOURAL RESEARCH
• An experimenter manipulates independent variable,
whereas dependent variables are measured by
experimenter,
•
• An independent variable is the variable that is
manipulated by a researcher. In experimental
research, it must have two or more levels.
•
• A dependent variable or measure is the outcome
variable, the one that is assessed to determine if the
experimental treatment had any effect.

28. Types of dependent measures
• A behavioural measure is the one that can be seen or observed
directly.
• Self-report measures are almost as common as behavioural
measures, and they are just what you would expect: people's
verbal reactions to questions that allow respondents to give
detailed responses.
• "How did you feel about the story you just read?" Open-ended
• "How old are you?" Close-ended
• Physiological measures are markers of much more private,
internal psychological states. Examples: Pupil dilation, blood
pressure, heart rate, galvanic skin response, an indicator of
electrodermal activity.
• Behavioroid measures. For example participants are asked to
volunteer to perform some activity in the future—devoting time
to community service or visiting patients in a nursing home.

29. Descriptive vrs Operational Definition
• A descriptive definition explains the
relationship among variables in an abstract,
conceptual manner.
• An operational definition renders
hypothetical, often abstract variables into
concrete operations that can be manipulated
or measured empirically.

30. Writing Descriptive and Operational
Definitions (Adapted from Dunn
1999)
• Write a brief description (three or four sentences) of the theory
being used in the research.
• Write a descriptive definition of the independent variables from
this theory. Do the same for any dependent measures featured in
the theory.
• Write an operational definition for the independent variables and
the dependent measures identified in step 2. Be certain to use
concrete terms and familiar concepts in the operational definition.
• If you are using published research, what operational definitions
have been used to examine these or similar independent
variables? Write down these published operational definitions
and then compare them to those generated in step 3.
• Refine and finalize the operational definitions based on step 4.
Write down the final operational definitions.

31. RELIABILITY AND VALIDITY
• RELIABILITY AND VALIDITY
• A hypothetical construct is an image, an idea,
or theory used to organize hypotheses and
data. Hypothetical constructs enable
researchers to speculate about the processes
underlying, even, causing, thought and
behaviour,

32. RELIABILITY
• Reliability refers to measure's stability or
consistency across time. If used on the same
stimulus, a reliable measure gives approximately
the same result each time it is used.
•
• Take the example of Thermometer and
bathroom scale.
•
• Reliability = Stability = Consistency

33. VALIDITY: ITS VARIOUS FORMS
• VALIDITY: ITS VARIOUS FORMS
•
• Validity is the degree to which an observation
or a measurement corresponds to the
construct that was supposed to be observed
or measured.

34. Different Forms of Validity
• Construct validity examines how well a variable's operational definition reflects the actual nature
and meaning of the theoretical variable.
•
• Face validity. To what degree does a measure or variable appear to accurately represent a
construct?
•
• Convergent validity. To what degree is a measure or variable related to other measures or
variables in predictable ways? Is the association positive and moderate to strong?
•
•
• Discriminant validity. To what degree is a measure or variable unrelated to other measures and
variables that it should not be related to? Is the association low on appositive scale, zero, or
negative?
•
• Internal validity. Did the independent variable create a meaningful and verifiable change in the
independent measure?
•
•
• External validity. To what degree do the results apply to other situations, other persons, and
other times? Can the data be generalized beyond the original setting?

35. RESEARCH DESIGN
• A research design is an organized collection of
procedures used by researchers to collect
behavioural data.
•
• A confounded variable is an uncontrolled
variable that unknowingly but systematically
varies with the independent variable, thereby
preventing a clear interpretation of cause and
effect between the independent variable and
the dependent measure.

36. Populat
ion
Control
Group
Experimen
tal Group
Convenience
Non-random
Sample
Random Assignment Random Selection
Independent
Variable introduced
Dependent Measure
Assessment of Between-Group Differences
Figure 3. Basic Two-Group Experimental Illustrated (Adapted from Dunn 2001

37. Required Readings:
1. Dunn, D. S. 2001. Statistics and Data
Analysis for the Behavioural
Sciences. Toronto: McGraw Hill.
1. Babbie, E. 2007. The Practice of
Social Research. Eleventh Edition.
Thomsom: Wadsworth.
1. Creswell, J. W. 2003. Research
Design: Qualitative, Quantitative,
and Mixed Methods. Second
Edition. Thousand Oaks: Sage
Publications.
1. Healey J. F. 2009. Statistics: A Tool
for Social Research. Eighth Ed.
Cengage Learning.
1. Morgan, S. E., Reichert, T., &
Harrison, T. R. (2002). From
numbers to words: Reporting
statistical results for the social
sciences. Boston, MA: Allyn and
Bacon. [ISBN: 9780801332807].
1. Frankfort-Nachmias, C. & Leon-
Guerrero, A. 2006. Social Statistics
for a Diverse Society. 4th Edition.
Thousand Oaks: Pine Forge Press.