1. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e © 2010 Pearson Education, Inc. All rights reserved. Basic Statistical Concepts Sang Min Lee

2. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-2 Learning Goals 1. The basic principles of statistical analysis. 2. The differences between various scales of measurement. 3. Concepts of reliability and validity. 4. Basic Descriptive statistics. 5. Variability. 6. Measures of central tendency.

3. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-3 Measurement In statistics, the level of measurement of a variable is a classification intended to describe the nature of information contained within numbers assigned to objects, and therefore within the variable.

4. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-4 Four Levels of Measurement 1. Nominal 2. Ordinal 3. Interval 4. Ratio

5. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-5 Nominal ExamplesKey Characteristics Mathematical Operations Appropriate Statistics Nominal Gender Ethnicity Marital Status Religion IdentityNone Chi Square (  2) Percent of Total Mode

6. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-6 Ordinal ExamplesKey Characteristics Mathematical Operations Appropriate Statistics Ordinal Top 10 rating for music The rank order of anything Identity Magnitude RankingMedian Interquartile range

7. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-7 Interval ExamplesKey Characteristics Mathematical Operations Appropriate Statistics Interval Temperature IQ Curved Grades Identity Magnitude Equal interval Addition Subtraction Mean Standard Deviation Correlational &Regression ANOVA Factor Analysis

8. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-8 Ratio ExamplesKey Characteristics Mathematical Operations Appropriate Statistics Ratio Age Weight Annual Income Identity Magnitude Equal Interval True Zero Addition Subtraction Multiplication Division All possible statistics

9. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-9 Reliability and Validity Two concepts important for defining and measuring bias and distortion. Reliability refers to the extent to which assessments are consistent. Validity is the extent to which a test measures the construct it is intended to measure.

10. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-10 Four Reliability Indices 1. Test-Retest Reliability: Reliability coefficient is obtained by administering the same test twice and correlating the scores. 2. Parallel Form Reliability: reliability coefficient is obtained by administering similar, but not identical tests and correlating the scores.

11. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-11 Four Reliability Indices 3. Split-Half Reliability: reliability score is obtained by diving the test into halves, correlating the scores on each half. 4. Internal Consistency: Reliability score is obtained by correlating the individual items of a test to each other.

12. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-12 Reliability Limitations Memory Effect: When using a test-retest approach, responses may be overly consistent because participants remember some of the questions asked previously. Practice Effect: With a parallel forms approach, respondents have taken one version of a test then take a similar form at a later date which may impact results.

13. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-13 Reliability Concepts Internal Consistency: focuses on the degree to which the individual items are correlated with each other. Homogeneous items measure a single construct and correlate highly with each other. Kuder-Richardson Formula 20 (KR-20) and Cronbach’s Alpha are measures of homogeneity.

14. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-14 Three Major Approaches to Validity Content: the extent to which the measurement adequately samples the content domain. Construct: The extent to which the test is an accurate measure of a particular construct or variable. Criterion: the extent to which a test is related to some external criterion of the construct being measured.

15. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-15 Construct Validity One way of assessing construct validity is to demonstrate the evidence of convergent and Discriminant validity. Convergent validity refers to the degree to which scores correlated highly with other tests designed to assess the same construct. Discriminant validity refers to the degree scores do not correlate highly with other tests that are not designed to assess the same trait.

16. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-16 Criterion Validity Two types of Criterion Validity: 1. Concurrent validity: yields scores at the time of administration, and 2. Predictive validity: yields scores at a later time from administration.

17. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-17 Descriptive Statistics Descriptive Statistics are used to describe the basic features of the data in a study including when the data are or what they show. The Frequency Distribution is a summary of the frequency of individual values or ranges of values for a variable.

18. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-18 Frequency Distribution Years of Experience (X) Frequency (f) PercentageCumulative Percentage 13-15110.0%100.00% 10-1200.0%90.0% 7-9110.0%90.0% 4-6550.0%80.0% 1-3330.0% N=10100.00%

19. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-19 Histogram Histogram: also known as a bar graph, uses bars to indicate the frequencies for each value of X (which are placed along the horizontal x-axis). The height of the bars indicates the frequencies (which are on the vertical Y-axis).

20. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-20 Frequency Polygon Plots data on the x-axis and frequency on the y- axis. More effective when comparing two or more distributions within the same figure.

21. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-21 Types of Distribution For quantitative variables, the pattern of scores in the distribution can be determined by visually examining the frequency polygon. In large data sets if a pattern exists it typically fits one of three patterns: Symmetrical Skewed Multimodal

22. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-22 Symmetrical Distribution Positively Skewed Normal Distribution Negatively Skewed Normal Distribution

23. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-23 Measures of Central Tendency There are several common methods to estimate measure of central tendency. Intended to describe the most average scores in the distribution. Most common measures are the mean, the median, and the mode.

24. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-24 Mean Most commonly used method of describing central tendency. Arithmetic average of all scores. Add up all of the values and divide by the number of values. Represented by the following equation: Add equation 19.4

25. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-25 Median The midpoint of the ordered list of values. List the values in rank order, then find the point below which one-half of the scores lie. Represented by this mathematical equation: Add equation 19.3

26. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-26 Mode The easiest measure to understand since it is determined by inspection rather than computation. Reports the most frequent score in the variable. Useful when studying nominal variables. Not often a useful indicator of central tendency in a distribution.

27. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-27 Variability Refers to the extent to which the scores in a distribution differ from each other. A distribution lacking variability is referred to as homogenous. Distribution with much variability is referred to as heterogeneous. Three frequently used measures of variability are the range, variance, and standard deviation.

28. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-28 Range The simplest measure of variability. Take the highest score and subtract the lowest score. A quick measure of variability but excludes a information from the other scores.

29. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-29 Variance Represents how close the scores in the distribution are to the mean. Variance is the average of the squared deviations from the mean. Represented by the following equation: Add equation 19.6

30. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-30 Standard Deviation The standard deviation is the square root of the variance. Indicates the average difference between individual scores and the group mean. Add equation 19.9

31. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-31 Standard Normal Distribution & Standard Scores Z Scores: if the mean and standard deviation are known, individual scores can be represented relative to the entire set of scores in the distribution through standardization. When you standardize a raw score to a Z score, it provides information about how far a person is from the mean. One standard deviation above the mean has a Z- score of 1, one below = -1, and at the mean = 0.

32. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-32 Standard Normal Distribution and Z Scores

33. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-33 T Scores Standardized scores which are widely used to report performance on standardized tests and inventories. A score of 50 represents the mean. A difference of 10 from the mean indicates a difference of one standard deviation. T= 50 + 10z

34. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-34 Standard Normal Distribution and T-Scores

35. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-35 Stanines Relating stanines to the normal curve, each stanine score represents a wide band of raw scores and percentile ranks. A normal distribution is divided into 9 intervals, each of which has a width of one half of a standard deviation excluding the first and last intervals. Useful in comparing performance across content areas.

36. Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e Sheperis/Daniels/Young © 2010 Pearson Education, Inc. All rights reserved. 15-36 Standard Normal Distribution and Stanines