Basic concepts of measurement

Basic Concepts in
Measurement

R. K. PRADHAN

Can Psychological Properties Be
Measured?
A common complaint: Psychological
variables can’t be measured.

We regularly make judgments about
who is shy and who isn’t; who is
attractive and who isn’t; who is smart
and who is not.

Quantitative
Implicit in these statements is the notion
that some people are more shy, for
example, than others
This kind of statement is inherently
quantitative.
Quantitative: It is subject to numerical
qualification.
If it can be numerically qualified, it can
be measured.

Measurement
• The process of assigning numbers to objects in such a
way that specific properties of the objects are faithfully
represented by specific properties of the numbers.

• Psychological tests do not attempt to measure the total
person, but only a specific set of attributes.

Measurement (cont.)
•Measurement is used to capture some “construct”
- For example, if research is needed on the construct of
“depression”, it is likely that some systematic
measurement tool will be needed to assess depression.

Individual Differences
• The cornerstone of psychological measurement - that
there are real, relatively stable differences between
people.

• This means that people differ in measurable ways in
their behavior and that the differences persist over a
sufficiently long time.

•Researchers are interested in assigning individuals
numbers that will reflect their differences.

• Psychological tests are designed to measure specific
attributes, not the whole person.

•These differences may be large or small.

Types of Measurement Scales
1. Nominal

2. Ordinal

3. Interval

4. Ratio

Nominal Scales - there must be distinct classes but these classes
have no quantitative properties. Therefore, no comparison can be made
in terms of one being category being higher than the other

For example - there are two classes for the variable gender -- males and
females. There are no quantitative properties for this variable or these
classes and, therefore, gender is a nominal variable.

Other Examples:
country of origin
biological sex (male or female)
animal or non-animal
married vs. single

Nominal Scale
Sometimes numbers are used to designate
category membership

Example:
Country of Origin
1 = United States 3 = Canada
2 = Mexico 4 = Other

However, in this case, it is important to keep in
mind that the numbers do not have intrinsic

Ordinal Scales - there are distinct classes but these
classes have a natural ordering or ranking. The
differences can be ordered on the basis of magnitude.

For example - final position of horses in a
thoroughbred race is an ordinal variable. The horses
finish first, second, third, fourth, and so on. The
difference between first and second is not necessarily
equivalent to the difference between second and third,
or between third and fourth.

Ordinal Scales
Does not assume that the intervals between numbers are equal

Example:
finishing place in a race (first place, second place)

1st place 2nd place 3rd place 4th place

1 hour 2 hours 3 hours 4 hours 5 hours 6 hours 7 hours 8 hours

Types of Measurement Scales (cont.)
Interval Scales - it is possible to compare differences in magnitude,
but importantly the zero point does not have a natural meaning. It
captures the properties of nominal and ordinal scales -- used by most
psychological tests.

Designates an equal-interval ordering - The distance between, for
example, a 1 and a 2 is the same as the distance between a 4 and a 5

Example - celsius temperature is an interval variable. It is meaningful to
say that 25 degrees celsius is 3 degrees hotter than 22 degrees celsius,
and that 17 degrees celsius is the same amount hotter (3 degrees) than 14
degrees celsius. Notice, however, that 0 degrees celsius does not have a
natural meaning. That is, 0 degrees celsius does not mean the absence of
heat!

Ratio Scales - captures the properties of the other types of
scales, but also contains a true zero, which represents the
absence of the quality being measured..

For example - heart beats per minute has a very natural zero
point. Zero means no heart beats. Weight (in grams) is also a
ratio variable. Again, the zero value is meaningful, zero
grams means the absence of weight.

Example:
the number of intimate relationships a person has had
0 quite literally means none
a person who has had 4 relationships has had twice as
many as someone who has had 2


• Each of these scales have different properties (i.e.,
difference, magnitude, equal intervals, or a true zero point)
and allows for different interpretations

• The scales are listed in hierarchical order. Nominal scales
have the fewest measurement properties and ratio having the
most properties including the properties of all the scales
beneath it on the hierarchy.

• The goal is to be able to identify the type of measurement
scale, and to understand proper use and interpretation of the
scale.

Evaluating Psychological Tests
The evaluation of psychological tests centers on the test’s:
Reliability - has to do with the consistency of the instrument.
A reliable test is one that yields consistent scores when a
person takes the test two alternate forms of the test or when an
individual takes the same test on two or more different
occasions.

Validity - has to do with the ability to measure what it is
supposed to measure and the extent to which it predicts
outcomes.

Why Statistics?
Statistics are important because they give us a method for
answering questions about meaning of those numbers.

Three statistical concepts are central to psychological
measurement:

Variability - measure of the extent to which test scores differ.

Correlation - relationship between scores

Prediction - forecast relationships

Variability
• Variability is the foundation of psychological testing

• Variability refers to the spread of the scores within a
distribution.

•Tests depends on variability across individuals --- if there
was no variability then we could not make decisions about
people.

• The greater the amount of variability there is among
individuals, the more accurately we can make the
distinctions among them.

Variability
There are four major measures of variability:
1. Range - difference between the highest and lowest scores
For Example: If the highest score was 60 & lowest was 40 = range of 20

2. Interquartile Range - difference between the 75th and 25th
percentile.

3. Variance - the degree of spread within the distribution (the
larger the spread, the larger the variance). It is the sum of the
squared differences from the mean of each score, divided by
the number of scores

4. Standard Deviation - a measure of how the average score
deviates or spreads away from the mean.

Standard Deviation

Standard deviation is
a measure of spread
affected by the size of each data value
a commonly calculated and used statistic
equal to
var iance

 typically about 2/3 of data values lie within one
standard deviation of the mean.

Example – using individual data values
Question: Six masses were weighed as 4, 6, 6, 7, 9 and 10 kg
Find the mean, variance and standard deviation of these weights.

Answer: mean x=
∑x =
4 + 6 + 6 + 7 + 9 + 10
=
42
= 7 kg
n 6 6

Variance is the
average square
distance from
the mean

1 2 3 4 5 6 7 8 9 10 weight kg

x

Example – using individual data values

Answer: mean x=
∑x =
4 + 6 + 6 + 7 + 9 + 10
=
42
= 7 kg
n 6 6

Method 1 Variance σ 2 = ∑ ( x−µ )2
n
2 ( 4 − 7 ) 2 + ( 6 − 7 ) 2 + ( 6 − 7 ) 2 + ( 7 − 7 ) 2 + ( 9 − 7 ) 2 + (10 − 7 ) 2
Variance is the σ =
6
average square
distance from
the mean

1 2 3 4 5 6 7 8 9 10 weight kg

x


Answer: mean x=
∑x =
4 + 6 + 6 + 7 + 9 + 10
=
42
= 7 kg
n 6 6

n
2 ( 4 − 7 ) 2 + ( 6 − 7 ) 2 + ( 6 − 7 ) 2 + ( 7 − 7 ) 2 + ( 9 − 7 ) 2 + (10 − 7 ) 2
Variance is the σ =
6
average square
distance from 2 ( −3) 2 + ( −1) 2 + ( −1) 2 + ( 0 ) 2 + ( 2 ) 2 + ( 3) 2 24
σ = = = 4 kg2
the mean 6 6

1 2 3 4 5 6 7 8 9 10 weight kg

x


Answer: mean x=
∑x =
4 + 6 + 6 + 7 + 9 + 10
=
42
= 7 kg
n 6 6

n
2 ( 4 − 7 ) 2 + ( 6 − 7 ) 2 + ( 6 − 7 ) 2 + ( 7 − 7 ) 2 + ( 9 − 7 ) 2 + (10 − 7 ) 2
σ =
6
2 ( −3) 2 + ( −1) 2 + ( −1) 2 + ( 0 ) 2 + ( 2 ) 2 + ( 3) 2 24
σ = = = 4 kg2
6 6
standard deviation σ= var iance = 4 = 2 kg

1 2 3 4 5 6 7 8 9 10 weight kg

x

Normal Distribution Curve
• Many human variables fall on a normal or close to normal curve
including IQ, height, weight, lifespan, and shoe size.

• Theoretically, the normal curve is bell shaped with the highest
point at its center. The curve is perfectly symmetrical, with no
skewness (i.e., where symmetry is absent). If you fold it in half at the
mean, both sides are exactly the same.

•From the center, the curve tapers on both sides approaching the X
axis. However, it never touches the X axis. In theory, the
distribution of the normal curve ranges from negative infinity to
positive infinity.

•Because of this, we can estimate how many people will compare on
specific variables. This is done by knowing the mean and standard
deviation.

Scatter Plots
• An easy way to examine the data given is by scatter plot. When we plot the
points from the given set of data onto a rectangular coordinate system, we have a
scatter plot.

• Is often employed to identify potential associations between two variables, where
one may be considered to be an explanatory variable (such as years of education)
and another may be considered a response variable

Relational/Correlational Research
Relational Research …
• Attempts to determine how two or more variables are related to
each other.

•Is used in situations where a researcher is interested in
determining whether the values of one variable increase (or
decrease) as values of another variable increase. Correlation does
NOT imply causation!

•For example, a researcher might be wondering whether there is a
relationship between number of hours studied and exam grades.
The interest is in whether exam grades increase as number of study
hours increase.

Use and Meaning of Correlation Coefficients
• Value can range from -1.00 to +1.00

• An r = 0.00 indicates the absence of a linear relationship.

• An r = +1.00 or an r = - 1.00 indicates a “perfect” relationship between the
variables.

•A positive correlation means that high scores on one variable tend to go with
high scores on the other variable, and that low scores on one variable tend to go
with low scores on the other variable.

•A negative correlation means that high scores on one variable tend to go with
low scores on the other variable.

•The further the value of r is away from 0 and the closer to +1 or -1, the
stronger the relationship between the variables.

Coefficients of Determination

•By squaring the correlation coefficient, you get the amount of variance
accounted for between the two data sets. This is called the coefficient of
determination.

• A correlation of .90 would represent 81% of the variance between the two sets
of data (.90 X .90 = .81)

• A perfect correlation of 1.00 represents 100% of the variance. If you know
one variable, you can predict the other variable 100% of the time
(1.00 X 1.00 = 1.00)

•A correlation of .30 represents only 9% of the variance, strongly suggesting
that other factors are involved (.30 X .30 = .09)

Factor Analysis
Is a statistical technique used to analyze patterns of
correlations among different measures.

The principal goal of factor analysis is to reduce the
numbers of dimensions needed to describe data derived
from a large number of data.

It is accomplished by a series of mathematical calculations,
designed to extract patterns of intercorrelations among a
set of variables.

Prediction/Linear Regression
• Linear regression attempts to model the relationship between
two variables by fitting a linear equation to observed data. One
variable is considered to be an explanatory variable, and the
other is considered to be a dependent variable.
Formula : Y = a + bX ---------- Where X is the independent
variable, Y is the dependent variable, a is the intercept and b is
the slope of the line.

• Before attempting to fit a linear model to observed data, a
modeler should first determine whether or not there is a
relationship between the variables of interest

Prediction/Linear Regression
• The method was first used to examine the relationship
between the heights of fathers and sons. The two were related,
of course, but the slope is less than 1.0. A tall father tended to
have sons shorter than himself; a short father tended to have
sons taller than himself. The height of sons regressed to the
mean. The term "regression" is now used for many sorts of
curve fitting.

• Linear regression analyzes the relationship between two
variables, X and Y. For each subject (or experimental unit),
you know both X and Y and you want to find the best straight
line through the data.

Steps in Test Construction
1. Concept Development
2. Domain Identification
3. Item Construction
4. Item Analysis: Item Difficulty & Item
Discrimination
5. Reliability: Test-Retest, Parallel Form, Slit-half,
Rational Equivalence (Internal-consistency)
6. Validity: Content, Construct, Criterion
7. Norms

Comparing Reliability
and Validity

Basic concepts of measurement

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Basic concepts of measurement