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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 ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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 07458
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 07458
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 07458
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 07458
All rights reserved.
1 2 3 4 5
1 2 3 4 5
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 07458
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 07458
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 07458
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 07458
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 07458
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 07458
All rights reserved.
Quantitative and Qualitative Variables
 Quantitative variables are measured on an
ordinal, interval, or ratio scale.
 Qualitative variables are measured on a
nominal scale.
 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 07458
All rights reserved.
What scores look like
Here are some test scores from a math
class:
65 91 85 76 85 87 79 93 82 75 100 70 88 78
83 59 87 69 89 54 74 89 83 80 94 67 77 92
82 70 94 84 96 98 46 70 90 96 88 72
It's hard to get a feel for this data in this format because
it is unorganized.
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 07458
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 07458
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 4150, 51-60, 61-70, 71-80, 81-90, and 91-100.
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 07458
All rights reserved.
Frequency Distribution--here's a
finished product
Scores
Frequency
41-50
1
51-60
2
61-70
6
71-80
8
81-90
14
91-100
9
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 07458
All rights reserved.
Scores
Tally
41-50
/
51-60
//
61-70
//////
71-80
////////
81-90
//////////////
91-100
/////////
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 07458
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 07458
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 07458
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 07458
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 − 1 + 1 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 07458
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 07458
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 07458
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 07458
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 07458
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 07458
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 07458
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 07458
All rights reserved.
Pearson Product Moment
Correlation
 A measure of relationship between two quantitative
variables (r)
 The correlation coefficient ranges from −1.00 to
+1.00
 A correlation of 0 indicates no relationship
 A coefficient of +1.00 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 07458
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 07458
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 07458
All rights reserved.