Chapter 2 Statistical Concepts Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Statistical Concepts  Counselors need to know certain statistical concepts to evaluate, use, and interpret tests. Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Four Scales of Measurement  Nominal  Ordinal  Interval  Ratio Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Nominal Data  Names or categories  Classification data  No ordering  Examples: marital status, gender Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Ordinal Data  Ranked or ordered objects  Ordered but differences between values are not important  Example: Likert scales 1 = poor, 2 = below average, 3 = average, 4 = above average, 5 = excellent The instructor is prepared for class The instructor's style of lecture is informative Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Interval Data  Ordered, constant scale  No natural zero  Examples: temperature, dates Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Ratio Data  Ordered, constant scale  Natural zero  Examples: height, weight, age, length Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Variables  A variable is any measured characteristic or attribute that differs for different subjects. For example, if the weights of 30 subjects were measured, then weight would be a variable. Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Independent and Dependent Variables  Independent variables: variables that are manipulated by the experimenter.  Dependent variables: variables that are measured from the subjects. Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Independent Variable  In an experimental study, a variable that determines (or causes) the score of a test/measurement (the dependent variable) is called an independent variable. For instance, an experiment might explore the extent to which the ability to walk is affected by the number of beers a college student drinks. In this case, the independent variable would be the number of beers the student drinks. Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Dependent Variable  A variable whose score depends on (or is determined or caused by) another variable is called the dependent variable.  The independent variable affects the dependent variable.  The dependent variable is measured. In the earlier example, the dependent variable would be the measurement of the student’s ability to walk. Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Quantitative and Qualitative Variables  Quantitative variables are measured on an ordinal, interval, or ratio scale. ordinal,interval,ratio ordinal,interval,ratio  Qualitative variables are measured on a nominal scale. nominal  If five-year-olds were asked to name their favorite color, then the variable would be qualitative.  If the time it took them to respond were measured, then the variable would be quantitative. Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Here are some test scores from a math class: It's hard to get a feel for this data in this format because it is unorganized. What scores look like Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Frequency Distribution  Summarizes a set of numbers.  Summarizes test data by arranging scores in order of magnitude and indicating how often each score was obtained. Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

To construct a frequency distribution  First, identify the lowest and highest values in the list.  The low value here is 46, and the high is 100.  A set of categories that would work here is , 51-60, 61-70, 71-80, 81-90, and Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Frequency Distribution--here's a finished product ScoresFrequency Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

ScoresTally / // // ////// ////// //////// //////// ////////////// ////////////// ///////// ///////// Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Histogram  Bar graph used to show the distribution of test data Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Frequency Polygon Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Measures of Central Tendency Tell us the typical or average performance of a group of test scores  Mean: the average of test scores = Σ X/N  Median: the middle score, divides the distribution in half  Mode: the score that appears most frequently in a set of scores In the sample of test scores, what is the mean, median and mode? Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Measures of Variability The spread of scores  Range: a measure of the spread of scores. The high score − the low score + 1 example: 1,2,2,6,6,6,6,7,7,8,8,8,15,18,18,18,20 20 − equals 20  Variance: tells us how widely spread, or scattered, the scores are from the mean In the sample of test scores, what is the range? Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Measures of Variability, continued  Standard Deviation  Tells how tightly a set of scores is clustered around the average of those same values. It's a measure of dispersal, or variation, in a group of numbers.  The square root of the variance  The most widely used statistic of variability  Higher standard deviation—the higher the volatility of test scores  Lower standard deviation—the more stable the test scores Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Normal Distribution  Symmetric with scores more concentrated in the middle than in the tails  Described as bell shaped  Symmetrical curve  A mean of 0 and a standard deviation of 1  Mean = median = mode Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Normal Distribution Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Skewness — asymmetrical curve  Positively skewed distribution—the majority of test scores are at lower levels  Negatively skewed distribution—the majority of test scores are at higher levels Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Measures of Relationship  Show us how two variables are related to one another.  “Correlation”  For example, a measure of relationship can tell us how the GRE score is related to success in graduate school. Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Positive Correlation Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Negative Correlation Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Pearson Product Moment Correlation  A measure of relationship between two quantitative variables (r)  The correlation coefficient ranges from −1.00 to  A correlation of 0 indicates no relationship  A coefficient of indicates a perfect positive relationship Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Regression  A statistical tool for prediction  For example, if we know the correlation between an IQ test and a reading test, we can predict the individual’s reading score from her IQ test. Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Factor Analysis  A statistical technique used to identify how many factors are needed to explain or account for the intercorrelation of a set of variables.  Example: Used to reduce the number of adjectives defining the construct “personality” to a small group of representative but uncorrelated terms. Robert J. Drummond and Karyn Dayle Jones Assessment Procedures for Counselors and Helping Professionals, 6 th edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.