Chapter 2 Statistical Concepts Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright

Chapter 2Chapter 2Statistical ConceptsStatistical Concepts

Robert J. Drummond and Karyn Dayle JonesAssessment Procedures for Counselors and Helping Professionals, 6th editionCopyright ©2006 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458All rights reserved.

Statistical ConceptsStatistical Concepts

Counselors need to know certain statistical Counselors need to know certain statistical concepts to evaluate, use, and interpret tests.concepts to evaluate, use, and interpret tests.


Four Scales of MeasurementFour Scales of Measurement

NominalNominal OrdinalOrdinal IntervalInterval RatioRatio


Nominal DataNominal Data

Names or categoriesNames or categories Classification dataClassification data No ordering No ordering ExamplesExamples: marital status, gender: marital status, gender


Ordinal DataOrdinal Data

Ranked or ordered objects Ranked or ordered objects Ordered but differences between values are Ordered but differences between values are

not important not important ExampleExample: Likert scales: Likert scales

1 = poor, 2 = below average, 3 = average, 4 = above average, 5 = 1 = poor, 2 = below average, 3 = average, 4 = above average, 5 = excellentexcellent

The instructor is prepared for class. The instructor is prepared for class. 1 2 3 4 51 2 3 4 5The instructor's style of lecture is informative. The instructor's style of lecture is informative. 1 2 3 4 5 1 2 3 4 5


Interval DataInterval Data

Ordered, constant scaleOrdered, constant scale No natural zero No natural zero ExamplesExamples: temperature, dates: temperature, dates


Ratio DataRatio Data

Ordered, constant scaleOrdered, constant scale Natural zeroNatural zero ExamplesExamples: height, weight, age, length : height, weight, age, length


VariablesVariables

A variable is any measured characteristic or A variable is any measured characteristic or attribute that differs for different subjects. attribute that differs for different subjects.

For example, if the weights of 30 subjects were For example, if the weights of 30 subjects were measured, then weight would be a variable. measured, then weight would be a variable.


Independent and Dependent Independent and Dependent VariablesVariables

Independent variables: variables that are Independent variables: variables that are manipulated by the experimenter.manipulated by the experimenter.

Dependent variables: variables that are Dependent variables: variables that are measured from the subjects. measured from the subjects.


Independent VariableIndependent Variable

In an experimental study, a variable that In an experimental study, a variable that determines (or causes) the score of a determines (or causes) the score of a test/measurement (the dependent variable) is test/measurement (the dependent variable) is called an independent variable. called an independent variable.

For instance, an experiment might explore the For instance, an experiment might explore the extent to which the extent to which the ability to walkability to walk is affected by is affected by the the number of beersnumber of beers a college student drinks. a college student drinks. In this case, the independent variable would be In this case, the independent variable would be the number of beers the student drinks. the number of beers the student drinks.


Dependent VariableDependent Variable A variable whose score depends on (or is A variable whose score depends on (or is

determined or caused by) another variable is determined or caused by) another variable is called the dependent variable.called the dependent variable.

The independent variable affects the dependent The independent variable affects the dependent variable.variable.

The dependent variable is measured.The dependent variable is measured.

In the earlier example, the dependent variable In the earlier example, the dependent variable would be the measurement of the student’s would be the measurement of the student’s ability to walk. ability to walk.


Quantitative and Qualitative VariablesQuantitative and Qualitative Variables

Quantitative variables are measured on an Quantitative variables are measured on an ordinal,ordinal, interval,interval, or or ratioratio scale. scale.

Qualitative variables are measured on a Qualitative variables are measured on a nominalnominal scale. scale.

If five-year-olds were asked to name their favorite If five-year-olds were asked to name their favorite color, then the variable would be qualitative. color, then the variable would be qualitative.

If the time it took them to respond were measured, If the time it took them to respond were measured, then the variable would be quantitative. then the variable would be quantitative.


Here are some test scores from a math class: Here are some test scores from a math class:

65 91 85 76 85 87 79 93 82 75 65 91 85 76 85 87 79 93 82 75 100100 70 88 78 83 59 70 88 78 83 59 87 69 89 54 74 89 83 80 94 67 77 92 82 70 94 84 87 69 89 54 74 89 83 80 94 67 77 92 82 70 94 84 96 98 96 98 4646 70 90 96 88 72 70 90 96 88 72

It's hard to get a feel for this data in this format because it is It's hard to get a feel for this data in this format because it is unorganized. unorganized.

What scores look likeWhat scores look like


Frequency DistributionFrequency Distribution

Summarizes a set of numbers. Summarizes a set of numbers. Summarizes test data by arranging Summarizes test data by arranging

scores in order of magnitude and scores in order of magnitude and indicating how often each score was indicating how often each score was obtained.obtained.


To construct a frequency To construct a frequency distributiondistribution

First, identify the lowest and highest values in First, identify the lowest and highest values in the list. the list.

The low value here is 46, and the high is 100. The low value here is 46, and the high is 100. A set of categories that would work here is 41-A set of categories that would work here is 41-

50, 51-60, 61-70, 71-80, 81-90, and 91-100. 50, 51-60, 61-70, 71-80, 81-90, and 91-100.


Frequency Distribution--here's a Frequency Distribution--here's a finished productfinished product

ScoresScores FrequencyFrequency

41-5041-50 11

51-6051-60 22

61-7061-70 66

71-8071-80 88

81-9081-90 1414

91-10091-100 99


ScoresScores TallyTally

41-5041-50 //

51-6051-60 ////

61-7061-70 ////////////

71-8071-80 ////////////////

81-9081-90 ////////////////////////////

91-10091-100 //////////////////


HistogramHistogram Bar graph used to show the distribution of test dataBar graph used to show the distribution of test data


Frequency PolygonFrequency Polygon


Measures of Central TendencyMeasures of Central Tendency

Tell us the typical or average performance of a Tell us the typical or average performance of a group of test scoresgroup of test scores

Mean: the average of test scores = Σ X/NMean: the average of test scores = Σ X/N Median: the middle score, divides the Median: the middle score, divides the

distribution in halfdistribution in half Mode: the score that appears most frequently Mode: the score that appears most frequently

in a set of scoresin a set of scores

In the sample of test scores, what is the mean, median In the sample of test scores, what is the mean, median and mode?and mode?


Measures of VariabilityMeasures of Variability

The spread of scoresThe spread of scores RangeRange: a measure of the spread of scores. : a measure of the spread of scores.

The high score The high score −− the low score + 1 the low score + 1

exampleexample: 1,2,2,6,6,6,6,7,7,8,8,8,15,18,18,18,20: 1,2,2,6,6,6,6,7,7,8,8,8,15,18,18,18,20

20 20 −− 1 + 1 equals 20 1 + 1 equals 20

VarianceVariance: tells us how widely spread, or scattered, the : tells us how widely spread, or scattered, the scores are from the meanscores are from the mean

In the sample of test scores, what is the range? In the sample of test scores, what is the range?


Measures of Variability, Measures of Variability, continuedcontinued

Standard Deviation Standard Deviation Tells how tightly a set of scores is clustered around Tells how tightly a set of scores is clustered around

the average of those same values. It's a measure of the average of those same values. It's a measure of dispersal, or variation, in a group of numbers. dispersal, or variation, in a group of numbers.

The square root of the variance The square root of the variance The most widely used statistic of variabilityThe most widely used statistic of variability Higher standard deviation—the higher the volatility Higher standard deviation—the higher the volatility

of test scores of test scores Lower standard deviation—the more stable the test Lower standard deviation—the more stable the test

scores scores


Normal DistributionNormal Distribution

Symmetric with scores more concentrated in the Symmetric with scores more concentrated in the middle than in the tailsmiddle than in the tails

Described as bell shaped Described as bell shaped Symmetrical curveSymmetrical curve A A meanmean of 0 and a of 0 and a standard deviationstandard deviation of 1 of 1 Mean = median = modeMean = median = mode


Normal DistributionNormal Distribution


Skewness—asymmetrical Skewness—asymmetrical curvecurve

Positively skewed Positively skewed distribution—the majority distribution—the majority of test scores are at of test scores are at lower levelslower levels

Negatively skewed Negatively skewed distribution—the majority distribution—the majority of test scores are at of test scores are at higher levelshigher levels


Measures of RelationshipMeasures of Relationship

Show us how two variables are related to one Show us how two variables are related to one another. another.

““CorrelationCorrelation”” For example, a measure of relationship can tell us For example, a measure of relationship can tell us

how the how the GRE GRE score is related to score is related to successsuccess in graduate in graduate schoolschool..


Positive CorrelationPositive Correlation


Negative CorrelationNegative Correlation


Pearson Product Moment CorrelationPearson Product Moment Correlation

A measure of relationship between two quantitative A measure of relationship between two quantitative variables (variables (rr))

The The correlation coefficientcorrelation coefficient ranges from ranges from −−1.00 to 1.00 to +1.00+1.00

A correlation of 0 indicates no relationshipA correlation of 0 indicates no relationship A coefficient of +1.00 indicates a perfect positive A coefficient of +1.00 indicates a perfect positive

relationshiprelationship


RegressionRegression

A statistical tool for predictionA statistical tool for prediction For example, if we know the correlation between an For example, if we know the correlation between an

IQ test and a reading test, we can predict the IQ test and a reading test, we can predict the individual’s reading score from her IQ test.individual’s reading score from her IQ test.


Factor AnalysisFactor Analysis

A statistical technique used to identify how many A statistical technique used to identify how many factors are needed to explain or account for the factors are needed to explain or account for the intercorrelation of a set of variables.intercorrelation of a set of variables.

ExampleExample: Used to reduce the number of adjectives : Used to reduce the number of adjectives defining the construct “personality” to a small group defining the construct “personality” to a small group of representative but uncorrelated terms.of representative but uncorrelated terms.


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Chapter 2 Statistical Concepts Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright