Basic Concepts in Measurement Scales and stats 1

Basic Concepts in Measurement (Scales and stats) 1

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. 2

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. 3

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. 4

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. 5

Measurement Measurement--defined as application of rules to assign numbers to objects (or attributes). Measurement rules--the procedures used to transform the qualities of attributes into numbers (e. g. , type of scale used). 6

Why bother assigning numbers? quantifying something that is expected to vary. individual differences -- premise that people will vary (get different scores) on the attribute 7

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. 8

Scales of measurement Three important properties: Magnitude--property of “moreness”. Higher score refers to more of something. Equal intervals--is the difference between any two adjacent numbers referring to the same amount of difference on the attribute? Absolute zero--does the scale have a zero point that refers to having none of that attribute? 9

Types of Measurement Scales 1. Nominal 2. Ordinal 3. Interval 4. Ratio 10

Types of Measurement Scales 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 origin biological sex (male or female) animal or non-animal married vs. single 11

Nominal Scale Sometimes numbers are used to designate category membership Example: Country of Origin 1 = United States 2 = Mexico 3 = Canada 4 = Other However, in this case, it is important to keep in mind that the numbers do not have intrinsic meaning 12

Types of Measurement Scales 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. 13

Ordinal Scales Does not assume that the intervals between numbers are equal Example: finishing place in a race (first place, second place) 1 st place 1 hour 2 hours 2 nd place 3 rd place 3 hours 4 th place 5 hours 6 hours 7 hours 8 hours 14

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! 15

Types of Measurement Scales (cont. ) 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 had twice as many as someone who has had 2 16

Types of Measurement Scales (cont. ) • 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. 17

Types of scales Nominal scales--qualitative, not quantitative distinction (no absolute zero, not equal intervals, not magnitude) Ordinal scales--ranking individuals (magnitude, but not equal intervals or absolute zero) Interval scales--scales that have magnitude and equal intervals but not absolute zero Ratio scales--have magnitude, equal intervals, and absolute zero (so can compute ratios) 18

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 19

In the previous example, what is the measurement scale of letter grade on the final exam? a) Nominal b) Ordinal c) Interval d) Ratio 20

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 21

In the previous example, what is the measurement scale of productivity? a) Nominal b) Ordinal c) Interval d) Ratio 22

Select the highest appropriate level of measurement: Bicycle models: 1= Road 2 = Touring 3 = Mountain 4 = Hybrid 5 = Comfort 6 = Cruiser a) Nominal b) Ordinal c) Interval d) Ratio 23

Select the highest appropriate level of measurement: Educational Level: 1 = Some High school 2 =High school Diploma 3 = Undergraduate Degree 4 = Masters Degree 5 = Doctorate Degree a) Nominal b) Ordinal c) Interval d) Ratio 24

Select the highest appropriate level of measurement: Number of questions asked during a class lecture a) Nominal b) Ordinal c) Interval d) Ratio 25

Select the highest level of measurement: Categories on a Likert-type scale measuring attitudes: 1 = Strongly Disagree 2 = Disagree 3 = Neutral 4 = Agree 5 = Strongly Agree a) Nominal b) Ordinal c) Interval d) Ratio 26

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 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. 27

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. 28

Why we need statistics Statistics for the purposes of description-numbers as summaries. Statistics for making inferences--logical deductions about events that can’t be observed directly (e. g. , opinion polls). 29

Three basic statistical concepts Variability--extent to which individuals differ on the attribute not simply the range of scores determine how far from the mean each individual’s score is--square each value--then sum these values and divide by the number of people 30

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 75 th and 25 th 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 31 deviates or spreads away from the mean.

Variability (continued) square root of that value is the standard deviation standard scores (z-scores) -- calculated using the mean (average score) and the standard deviation positive values are above the mean, negative values are below the mean 32

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 square root of variance l typically about 2/3 of data values lie within one standard deviation of the mean. 33

Example – using individual data values Question: Answer: Six masses were weighed as 4, 6, 6, 7, 9 and 10 kg Find the mean, variance and standard deviation of these weights. = 7 kg mean Variance is the average square distance from the mean 132456789 w 0 e i g h t k g 34

Example – using individual data values Question: Answer: Six masses were weighed as 4, 6, 6, 7, 9 and 10 kg Find the mean, variance and standard deviation of these weights. = 7 kg mean Method 1 Variance is the average square distance from the mean 132456789 w 0 e i g h t k g 35

Question: Answer: Six masses were weighed as 4, 6, 6, 7, 9 and 10 kg Find the mean, variance and standard deviation of these weights. = 7 kg mean Method 1 Variance is the average square distance from the mean = 4 kg 2 132456789 w 0 e i g h t k g 36

Question: Answer: Six masses were weighed as 4, 6, 6, 7, 9 and 10 kg Find the mean, variance and standard deviation of these weights. = 7 kg mean Method 1 Variance = 4 kg 2 standard deviation = =2 kg 37

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. 38

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. 39

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 whethere 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. 40

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. 41

Correlation used to determine the relationship between two variables scatterplots involve plotting the scores on each of two variables (one along the x-axis and one along the y-axis) 42

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 43

Prediction Correlations used in prediction Relation between test score (predictor) and the thing to be predicted (criterion) E. g. , GREs used to predict likely success in graduate school 44

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 + b. X ----- 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 45

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) 46

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. 47