Measurement Statistics

Basic Concepts in Measurement

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Can Psychological Properties Be Can Psychological Properties Be Measured?Measured?

A common complaint: A common complaint: Psychological Psychological variables can’t be measuredvariables can’t be measured..

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

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QuantitativeQuantitative

Implicit in these statements is the Implicit in these statements is the notion that some people are more shy, notion that some people are more shy, for example, than othersfor example, than others

This kind of statement is inherently This kind of statement is inherently quantitativequantitative..

QuantitativeQuantitative: It is subject to : It is subject to numerical qualification.numerical qualification.

If it can be numerically qualified, it can If it can be numerically qualified, it can be measured.be measured.

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MeasurementMeasurement

• 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.

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Measurement (cont.)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.

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MeasurementMeasurement

MeasurementMeasurement--defined as application of --defined as application of rules to assign numbers to objects (or rules to assign numbers to objects (or attributes).attributes).

Measurement rulesMeasurement rules--the procedures used --the procedures used to transform the qualities of attributes into to transform the qualities of attributes into numbers (e.g., type of scale used).numbers (e.g., type of scale used).

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Why bother assigning numbers?Why bother assigning numbers?

quantifying something that is expected to quantifying something that is expected to vary.vary.

individual differencesindividual differences -- premise that -- premise that people will vary (get different scores) on people will vary (get different scores) on the attributethe attribute

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Individual DifferencesIndividual Differences• The cornerstone of psychological measurement - The cornerstone of psychological measurement - that there are real, relatively stable differences that there are real, relatively stable differences between people.between people. • This means that people differ in measurable ways This means that people differ in measurable ways in their behavior and that the differences persist in their behavior and that the differences persist over a sufficiently long time.over a sufficiently long time.

•Researchers are interested in assigning Researchers are interested in assigning individuals numbers that will reflect their individuals numbers that will reflect their differences. differences. • Psychological tests are designed to measure Psychological tests are designed to measure specificspecific attributes, not the whole person. attributes, not the whole person.

•These differences may be large or small. These differences may be large or small.

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Scales of measurementScales of measurement Three important properties:Three important properties:

Magnitude--property of “moreness”. Higher Magnitude--property of “moreness”. Higher score refers to more of something.score refers to more of something.

Equal intervals--is the difference between any Equal intervals--is the difference between any two adjacent numbers referring to the same two adjacent numbers referring to the same amount of difference on the attribute?amount of difference on the attribute?

Absolute zero--does the scale have a zero point Absolute zero--does the scale have a zero point that refers to having none of that attribute?that refers to having none of that attribute?

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Types of Measurement Scales

1. Nominal

2. Ordinal

3. Interval

4. Ratio

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Nominal Scales - there must be distinct classes but these classes have no quantitative properties. Therefore, no comparison can be made in terms of one 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 originbiological sex (male or female)animal or non-animalmarried vs. single

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Nominal ScaleNominal Scale Sometimes numbers are used to designate Sometimes numbers are used to designate

category membershipcategory membership

Example: Example: Country of OriginCountry of Origin1 = United States1 = United States 3 = Canada3 = Canada

2 = Mexico2 = Mexico 4 = Other4 = Other

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

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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.

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Ordinal ScalesOrdinal Scales

Does not assume that the intervals between numbers Does not assume that the intervals between numbers are equalare equal

ExampleExample::

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

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

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

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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!

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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 nonea person who has had 4 relationships has had twice as many as someone who has had 2 1616


• 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.

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Types of scalesTypes of scales

Nominal scales--qualitative, not quantitative Nominal scales--qualitative, not quantitative distinction (no absolute zero, not equal distinction (no absolute zero, not equal intervals, not magnitude)intervals, not magnitude)

Ordinal scales--ranking individuals (magnitude, Ordinal scales--ranking individuals (magnitude, but not equal intervals or absolute zero)but not equal intervals or absolute zero)

Interval scales--scales that have magnitude Interval scales--scales that have magnitude and equal intervals but not absolute zeroand equal intervals but not absolute zero

Ratio scales--have magnitude, equal intervals, Ratio scales--have magnitude, equal intervals, and absolute zero (so can compute ratios)and absolute zero (so can compute ratios)

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Test Your Knowledge:

A professor is interested in the relationship between the number of times students are absent from class and the letter grade that students receive on the final exam. He records the number of absences for each student, as well as the letter grade (A,B,C,D,F) each student earns on the final exam. In this example, what is the measurement scale for number of absences?

a)Nominal b) Ordinal c) Interval d) Ratio

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In the previous example, what is the measurement scale of letter grade on the final exam?

a) Nominal b) Ordinal c) Interval d) Ratio

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A researcher is interested in studying the effect of room temperature in degrees Fahrenheit on productivity of automobile assembly workers. She controls the temperature of the three manufacturing facilities, such that employees in one facility work in a room temperature of 60 degrees, employees in another facility work in a room temperature of 65 degrees, and the last group works in a room temperature of 70 degrees. The productivity of each group is indicated by the number of automobiles produced each day. In this example, what is the measurement scale of room temperature?

a) Nominal b) Ordinal c) Interval d)Ratio

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In the previous example, what is the measurement scale of productivity?


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Select the highest appropriate level of measurement:

Bicycle models:

1= Road2 = Touring3 = Mountain4 = Hybrid5 = Comfort6 = Cruiser


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Educational Level:

1 = Some High school2 =High school Diploma3 = Undergraduate Degree4 = Masters Degree5 = Doctorate Degree


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Number of questions asked during a class lecture


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Select the highest level of measurement:

Categories on a Likert-type scale measuring attitudes:

1 = Strongly Disagree2 = Disagree3 = Neutral4 = Agree5 = Strongly Agree


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Evaluating Psychological TestsEvaluating Psychological TestsThe evaluation of psychological tests centers on the test’s:

ReliabilityReliability - has to do with the consistency of the instrument. A reliable test is one that yields consistent scores when a person takes two alternate forms of the test or when an individual takes the same test on two or more different occasions.

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

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Why Statistics?Why Statistics?

Correlation - relationship between scores.

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

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:

Prediction - forecast relationships .

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Why we need statisticsWhy we need statistics

Statistics for the purposes of description--Statistics for the purposes of description--numbers as summaries.numbers as summaries.

Statistics for making inferences--logical Statistics for making inferences--logical deductions about events that can’t be deductions about events that can’t be observed directly (e.g., opinion polls).observed directly (e.g., opinion polls).

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Three basic statistical conceptsThree basic statistical concepts

Variability--extent to which individuals Variability--extent to which individuals differ on the attributediffer on the attribute

not simply the range of scoresnot simply the range of scores determine how far from the mean determine how far from the mean

each individual’s score is--square each each individual’s score is--square each value--then sum these values and value--then sum these values and divide by the number of peopledivide by the number of people

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VariabilityVariability

There are four major measures of variability:

1. Range - difference between the highest and lowest scoresFor 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.

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Variability (continued)Variability (continued) square root of that value is the square root of that value is the standard standard

deviationdeviation

standard scoresstandard scores ( (zz-scores) -- calculated -scores) -- calculated using the mean (average score) and the using the mean (average score) and the standard deviationstandard deviation

positive values are above the mean, positive values are above the mean, negative values are below the meannegative values are below the mean

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Standard DeviationStandard Deviation

Standard deviation isStandard deviation is a measure of spreada measure of spread affected by the size of each data valueaffected by the size of each data value a commonly calculated and used statistica commonly calculated and used statistic equal to square root of varianceequal to square root of variance

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

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QuestionQuestion:: Six masses were weighed as 4, 6, 6, 7, 9 and 10 kgSix masses were weighed as 4, 6, 6, 7, 9 and 10 kgFind the mean, variance and standard deviation of these weights.Find the mean, variance and standard deviation of these weights.

Example – using individual data valuesExample – using individual data values

AnswerAnswer:: mean mean

x

Variance is the average square distance from

the mean

n

xx

6

1097664

6

42 = 7 = 7

kgkg

1245678910weight kg

3

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Example – using individual data valuesExample – using individual data values

AnswerAnswer:: mean mean n

xx

6

1097664

6

42 = 7 = 7

kgkg

x

Method 1Method 1 Variance Variance n

x

22

)(

6

)710()79()77()76()76()74( 2222222 Variance is the

average square distance from

the mean

1245678910weight kg

3

3535


xx

6

1097664

6

42

x


x

22

)(

6

)710()79()77()76()76()74( 2222222

6

)3()2()0()1()1()3( 2222222

6

24

Variance is the average square distance from

the mean= 4 = 4

kgkg22

= 7 = 7 kgkg

1245678910weight kg

3


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xx

6

1097664

6

42


x

22

)(

6

)710()79()77()76()76()74( 2222222

6

)3()2()0()1()1()3( 2222222

6

24 = 4 = 4

kgkg22

standard deviation standard deviation = =

iancevar 4

= 7 = 7 kgkg


= 2 = 2 kkg g

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VariabilityVariability

• 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.

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Normal Distribution CurveNormal 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. 3939

Relational/Correlational ResearchRelational/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. 4040

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.

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CorrelationCorrelation

used to determine the relationship used to determine the relationship between two variablesbetween two variables

scatterplots involve plotting the scores on scatterplots involve plotting the scores on each of two variables (one along the each of two variables (one along the xx-axis -axis and one along the and one along the yy-axis)-axis)

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Scatter PlotsScatter 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

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PredictionPrediction Correlations used in predictionCorrelations used in prediction

Relation between test score (predictor) Relation between test score (predictor) and the thing to be predicted (criterion) and the thing to be predicted (criterion)

E.g., GREs used to predict likely success E.g., GREs used to predict likely success in graduate schoolin graduate school

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Prediction/Linear RegressionPrediction/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

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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)

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Factor AnalysisFactor 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.

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