Basic concepts of measurement

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  • 1. Basic Concepts in Measurement R. K. PRADHAN
  • 2. 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.
  • 3. QuantitativeImplicit in these statements is the notionthat some people are more shy, forexample, than othersThis kind of statement is inherentlyquantitative.Quantitative: It is subject to numericalqualification.If it can be numerically qualified, it canbe measured.
  • 4. Measurement• The process of assigning numbers to objects in such away that specific properties of the objects are faithfullyrepresented by specific properties of the numbers.• Psychological tests do not attempt to measure the totalperson, but only a specific set of attributes.
  • 5. 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.
  • 6. Individual Differences• The cornerstone of psychological measurement - thatthere are real, relatively stable differences betweenpeople.• This means that people differ in measurable ways intheir behavior and that the differences persist over asufficiently long time.•Researchers are interested in assigning individualsnumbers that will reflect their differences.• Psychological tests are designed to measure specificattributes, not the whole person.•These differences may be large or small.
  • 7. Types of Measurement Scales1. Nominal2. Ordinal3. Interval4. Ratio
  • 8. Types of Measurement ScalesNominal Scales - there must be distinct classes but these classeshave no quantitative properties. Therefore, no comparison can be madein terms of one being category being higher than the otherFor example - there are two classes for the variable gender -- males andfemales. There are no quantitative properties for this variable or theseclasses and, therefore, gender is a nominal variable.Other Examples: country of origin biological sex (male or female) animal or non-animal married vs. single
  • 9. Nominal ScaleSometimes numbers are used to designatecategory membershipExample:Country of Origin1 = United States 3 = Canada2 = Mexico 4 = OtherHowever, in this case, it is important to keep inmind that the numbers do not have intrinsic
  • 10. Types of Measurement ScalesOrdinal Scales - there are distinct classes but theseclasses have a natural ordering or ranking. Thedifferences can be ordered on the basis of magnitude.For example - final position of horses in athoroughbred race is an ordinal variable. The horsesfinish first, second, third, fourth, and so on. Thedifference between first and second is not necessarilyequivalent to the difference between second and third,or between third and fourth.
  • 11. Ordinal Scales Does not assume that the intervals between numbers are equalExample: 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
  • 12. 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. Itcaptures the properties of nominal and ordinal scales -- used by mostpsychological tests.Designates an equal-interval ordering - The distance between, forexample, a 1 and a 2 is the same as the distance between a 4 and a 5Example - celsius temperature is an interval variable. It is meaningful tosay 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 14degrees celsius. Notice, however, that 0 degrees celsius does not have anatural meaning. That is, 0 degrees celsius does not mean the absence ofheat!
  • 13. Types of Measurement Scales (cont.)Ratio Scales - captures the properties of the other types ofscales, but also contains a true zero, which represents theabsence of the quality being measured..For example - heart beats per minute has a very natural zeropoint. Zero means no heart beats. Weight (in grams) is also aratio variable. Again, the zero value is meaningful, zerograms 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
  • 14. 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 scaleshave the fewest measurement properties and ratio having themost properties including the properties of all the scalesbeneath it on the hierarchy.• The goal is to be able to identify the type of measurementscale, and to understand proper use and interpretation of thescale.
  • 15. Evaluating Psychological TestsThe 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 aperson takes the test two alternate forms of the test or when anindividual takes the same test on two or more differentoccasions.Validity - has to do with the ability to measure what it issupposed to measure and the extent to which it predictsoutcomes.
  • 16. Why Statistics?Statistics are important because they give us a method foranswering questions about meaning of those numbers.Three statistical concepts are central to psychologicalmeasurement:Variability - measure of the extent to which test scores differ.Correlation - relationship between scoresPrediction - forecast relationships
  • 17. Variability• Variability is the foundation of psychological testing• Variability refers to the spread of the scores within adistribution.•Tests depends on variability across individuals --- if therewas no variability then we could not make decisions aboutpeople.• The greater the amount of variability there is amongindividuals, the more accurately we can make thedistinctions among them.
  • 18. VariabilityThere 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 202. Interquartile Range - difference between the 75th and 25thpercentile.3. Variance - the degree of spread within the distribution (thelarger the spread, the larger the variance). It is the sum of thesquared differences from the mean of each score, divided bythe number of scores4. Standard Deviation - a measure of how the average scoredeviates or spreads away from the mean.
  • 19. 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.
  • 20. Example – using individual data valuesQuestion: 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
  • 21. Example – using individual data valuesQuestion: 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 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
  • 22. 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 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 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
  • 23. 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 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 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
  • 24. Normal Distribution Curve• Many human variables fall on a normal or close to normal curveincluding IQ, height, weight, lifespan, and shoe size.• Theoretically, the normal curve is bell shaped with the highestpoint at its center. The curve is perfectly symmetrical, with noskewness (i.e., where symmetry is absent). If you fold it in half at themean, both sides are exactly the same.•From the center, the curve tapers on both sides approaching the Xaxis. However, it never touches the X axis. In theory, thedistribution of the normal curve ranges from negative infinity topositive infinity.•Because of this, we can estimate how many people will compare onspecific variables. This is done by knowing the mean and standarddeviation.
  • 25. Scatter Plots• An easy way to examine the data given is by scatter plot. When we plot thepoints from the given set of data onto a rectangular coordinate system, we have ascatter plot.• Is often employed to identify potential associations between two variables, whereone may be considered to be an explanatory variable (such as years of education)and another may be considered a response variable
  • 26. Relational/Correlational ResearchRelational Research …• Attempts to determine how two or more variables are related toeach other.•Is used in situations where a researcher is interested indetermining whether the values of one variable increase (ordecrease) as values of another variable increase. Correlation doesNOT imply causation!•For example, a researcher might be wondering whether there is arelationship between number of hours studied and exam grades.The interest is in whether exam grades increase as number of studyhours increase.
  • 27. 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 thevariables.•A positive correlation means that high scores on one variable tend to go withhigh scores on the other variable, and that low scores on one variable tend to gowith low scores on the other variable.•A negative correlation means that high scores on one variable tend to go withlow scores on the other variable.•The further the value of r is away from 0 and the closer to +1 or -1, thestronger the relationship between the variables.
  • 28. Coefficients of Determination•By squaring the correlation coefficient, you get the amount of varianceaccounted for between the two data sets. This is called the coefficient ofdetermination.• A correlation of .90 would represent 81% of the variance between the two setsof data (.90 X .90 = .81)• A perfect correlation of 1.00 represents 100% of the variance. If you knowone 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 suggestingthat other factors are involved (.30 X .30 = .09)
  • 29. Factor AnalysisIs a statistical technique used to analyze patterns ofcorrelations among different measures.The principal goal of factor analysis is to reduce thenumbers of dimensions needed to describe data derivedfrom a large number of data.It is accomplished by a series of mathematical calculations,designed to extract patterns of intercorrelations among aset of variables.
  • 30. Prediction/Linear Regression• Linear regression attempts to model the relationship betweentwo variables by fitting a linear equation to observed data. Onevariable is considered to be an explanatory variable, and theother is considered to be a dependent variable.Formula : Y = a + bX ---------- Where X is the independentvariable, Y is the dependent variable, a is the intercept and b isthe slope of the line.• Before attempting to fit a linear model to observed data, amodeler should first determine whether or not there is arelationship between the variables of interest
  • 31. Prediction/Linear Regression• The method was first used to examine the relationshipbetween the heights of fathers and sons. The two were related,of course, but the slope is less than 1.0. A tall father tended tohave sons shorter than himself; a short father tended to havesons taller than himself. The height of sons regressed to themean. The term "regression" is now used for many sorts ofcurve fitting.• Linear regression analyzes the relationship between twovariables, X and Y. For each subject (or experimental unit),you know both X and Y and you want to find the best straightline through the data.
  • 32. 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
  • 33. Comparing Reliability and Validity
  • 34. Thank you.