Ø Set of mutually exclusive categories. Ø Classify or categorize subject. Ø No meaningful order to categorization.

Transcription

1 Statistical Tools in Evaluation HPS 41 Fall 213 Dr. Joe G. Schmalfeldt Types of Scores Continuous Scores scores with a potentially infinite number of values. Discrete Scores scores limited to a specific number of values. Levels of Measurement Scales of Measurement Nominal Ordinal Interval Nominal Ø Set of mutually exclusive categories. Ø Classify or categorize subject. Ø No meaningful order to categorization. Ratio Ordinal Scales of Measurement Ø Order to scores so that one can be classified as higher or lower. Ø No common unit of measurement between numbers. Ø Numbers cannot be averaged or used in any way except to indicate better than. Interval Scales of Measurement Ø Have meaningful order and common unit of measurement between scores. Ø Arbitrary zero point.

2 Scales of Measurement Scales of Measurement Ratio Nominal Ø Common unit of measurement and absolute zero point. Ordinal Ø A score of zero indicates lack of value. Interval Ratio Organizing and Graphing Frequency Distribution Helps organize and interpret data. Simple frequency distribution listing of a distribution of scores in order. Easy to construct using a data analysis program (e.g., SPSS). Valid Valid Cumulative Frequency Percent Percent Percent Total Frequency Polygon Histogram Graphing For Frequency Polygon or Histogram: Similar scores are grouped together in an interval. Midpoint of interval is plotted on -axis. Frequency is plotted on Y-axis.

3 SPSS Sample Frequency Polygon SPSS Sample Histogram Count Frequency Std. Dev = Mean = 7.7 N = Skewness Ø An asymmetrical distribution. Normal Curve - no skewness. Positive Skew - tail of curve on right, few high scores. Measurement - process of obtaining test scores. Statistics - methodology for analyzing the scores to enhance interpretation. Negative Skew - tail of curve on left, few low scores. In this course, we use statistics: To describe a set of scores. To standardize scores. To estimate validity and reliability. Descriptive Statistics Central Tendency (how data cluster around the center) Variability (how data spread around the center)

4 Mode Median Most frequently occurring score. 5th percentile Middle score Need to order scores in a frequency distribution Found from cumulative percent column Mean Calculate the Mean, Median, and Mode for Three Distributions Mean = Σ N Mean: Median: Mode: Calculate the Mean, Median, and Mode for Three Distributions Mean: 5 Median: 5 Mode: 5 Calculate the Mean, Median, and Mode for Three Distributions Mean: 5 5 Median: 5 5 Mode: 5 5

5 Calculate the Mean, Median, and Mode for Three Distributions Mean: Median: Mode: So these three distributions are all the same, right? No What makes them different? Measure of Variability Range = High score - Low score Range: Variability A second type of descriptive statistic. Describes spread or heterogeneity of scores. Measures of Variability Range Range Standard Deviation Variance Range = high score - low score. Unstable because it depends on only two scores.

6 Standard Deviation (s) Standard Deviation (s) Average deviation of each score from the mean. Minimum value of s =. Larger s, more heterogeneous the group. σ = standard deviation of population s = standard deviation of sample Definitional Formula: s = Σ( - ) 2 (n - 1) Calculate the Standard Deviation Standard Deviation s = Σ( - ) 2 (N - 1) ( - ) ( - ) ( - ) ( - ) Σ =2 Σ(-)= Σ(-) 2 =26 = 5 s = 26 (4-1) = 8.67 = 2.94 Standard Deviation Calculational Formula: s = [Σ 2 - (Σ ) 2 / n] (n - 1) Σ=2 Σ 2 =126 Standard Deviation 2 s = [126-((2) 2 /4)] (4-1) s = [126-1] s = s = 2.94 Σ=2 Σ 2 =126

7 Variance (s 2 ) Average squared deviation from the mean. Standard deviation squared. Not used for description. Used with higher level statistics like regression analysis or analysis of variance. Percentile Rank Percentage of subjects that scored below a given score. Read from cumulative percent column in a simple frequency distribution. Percentile ranks are ordinal data. s 2 = Σ( - ) 2 (n - 1) s 2 = [Σ 2 - (Σ) 2 / n] (n - 1) Standard Scores Z - score Change variables to a constant mean and standard deviation. Different units of measurement are converted to the same unit (standardized) and can then be averaged. standard score with a mean = and standard deviation = 1. Z = ( - ) S T -score standard score with a mean = 5 and standard deviation = 1. T = 1(Z) + 5 Provide descriptions of relative performance on one or more tests. Z-scores

8 Example use of Z-scores Don t know Student A Subject Raw Score Math 3 English 7 Science 12 On which test did Student A perform best? Example use of Z-scores Still Don t Know Student A Subject Raw Score Mean Math 3 25 English 7 65 Science On which test did Student A perform best? Example use of Z-scores Now we know Student A Subject Raw Score Mean SD Math English Science On which test did Student A perform best?

9 Example use of Z-scores Student A Subject Raw Score Mean SD Z-score Math English Science Why use standard scores? To combine different units of measurement. To assign different weights to each score. On which test did Student A perform best? Math The test with the highest standard score. Characteristics of Normal Curve Characteristics of Normal Curve Symmetric Asymptotic Unimodal Area Characteristics of Normal Curve- IQ Tests Score Distribution- Skewedness

10 Y Y Using the Normal Curve to Determine Meaningful Test Score = mean + Z (standard deviation) If mean = 5 and SD = 1, what is score above which 1% of scores would fall? = (1) Z = 1.28 comes from normal curve for 9 th percentile. = 628 Determining Relationships between Scores Graphing Correlation Graphing Each subject must have a score on two variables; an and a Y score. Coordinates of and Y are plotted. Ø Coordinate - paired and Y score for a subject. scores are placed on horizontal axis. Ø abscissa Y scores are placed on vertical axis. Ø ordinate Regression Line Line of Best Fit Ø Straight line drawn through the data points. Ø Represents the trend in the data. Characteristics of Correlations Direction of r Direction Magnitude (size) + Positive (+) or Negative (-)

11 Positive Relationship Negative Relationship When high scores on one measure are associated with high scores on the other measure. When high scores on one measure are associated with low scores on the other measure. The closer the data points fall to the line of best fit, the higher the relationship. 5 Examine sample graphs on following slides. Percent Fat r = BMI Mile Run 15 1 VO2 max 4 r =.49 3 r = Percent Fat Percent Fat 5

12 1 r = r =? PACER 1 4 PACER Percent Fat PACER 1 1 r =. 1 5 PACER 2 4 VO2 max 4 3 r =? PACER Height (in.) r =? VO2 max r = -.24 Mod. Pull-up Height (in.) Percent Fat 5

13 Mod. Pull-up Y r = -.42 Correlation (r) Mathematical technique to determine the relationship between two sets of scores Percent Fat Pearson Product-moment Correlation (r) Estimates the linear relationship between variables. Magnitude (strength) of r How close r is to +1. or -1.. Higher absolute value of r, the stronger the correlation. r = perfect positive correlation. r = perfect negative correlation. Factors that influence magnitude of r: Linearity Ø If the relationship between two variables is curvilinear, Pearson r will underestimate the true relationship. 1 Factors that influence magnitude of r: Reliability Ø Low reliability on one or both variables will decrease the correlation. r =

14 Y Y Factors that influence magnitude of r: Effect of Restricted Range of Scores on r: Range of Scores 9 9 Ø A restricted range of scores on one or both variables will decrease the correlation. 7 7 Ø r will be smaller for a homogeneous group than 5 5 for a heterogeneous group A high r does not necessarily indicate a cause-and-effect relationship. Causal t-tests Additional Statistics Ø used to compare two means. Ø is one mean significantly higher than another mean? Ø this is sometimes used to demonstrate known groups evidence of validity.