Statistical Tools in Evaluation

1. Statistical Tools in Evaluation Part II

2. Percentiles and the Normal Curve

3. Comparing Different Groups

4. Consider two different distributions (groups of scores) with different Xs and SDs. The same score in both distributions will have different meanings. Class 1 Class 2 X = 32 X = 28 SD = 2 SD = 4 Same score of 34 in both groups is different within each group. Class 1: 34 is 1 SD above X; percentile rank of 84.13% (50 + 34.13) Class 2: 34 is 1.5 SD above the X; percentile rank is 90.925%

5. Distributions

6. Question: • Can scores from different kinds of tests be added to make a total score?

7. Example How do you combine scores on the 1-mile run test and sit-up test into an overall fitness score?

8. Standard Scores • Conversion of raw scores into standard scores • Allows for the combining and comparison of information that is in different measurement units • Examples: time + distance + repetitions, etc.

9. Types of Standard Scores • Z scores – represent the number of standard deviations a raw score is from the mean • Z scale has a X of 0 and a SD of 1 (see normal curve)

10. Calculation: Z = X - X SD *high to low Z = X – X SD *low to high

11. Examples and Practice Problems Class 1 Class 2 X = 32 X = 28 SD = 2 SD = 4 score = 30 score = 30 Z = + .5 Z = -1

12. Z scores can be decimals and neg. values • Z scores represent position on the normal curve

13. Other Types of Standard Scores?

14. Types of Standard Scores • T scores – represent Z scores that have been converted for ease of use • T scale has a X of 50 and a SD of 10 ( No decimals or negative numbers! ) • T scores represent position on the normal curve

15. T = 10 ( X – X ) + 50 SD *high to low T = 10 ( X – X ) + 50 SD *low to high T score calculations:

16. Put another way….. T = 10 Z + 50 (Use to convert Z score to T score)

17. Examples and Practice Problems X = 80 X = 25 X = 65 X = 8 SD = 5 SD = 10 T = 10(65-80) +50 T = 10(8 – 25) +50 5 10 T = 33 T = 20

18. T-score and the Normal Curve

19. Practical Use of T scores • Add values from unrelated measurements • Example: Fitness Test Battery • Curl ups • Pull ups • Flexibility • Mile run • Compare to 50th percentile: T50 x 4 = 200

20. Example • Curl-ups Score = 21 T = 30 • Pull-ups Score = 13 T = 70 • Flexibility Score = 8 cm T = 42 • Mile Run Score = 6:05 T = 58 • Fitness Score = ??? T total = 200 = Average

21. Questions?

22. Relationships • Similarities • Connections • What one factor has to do with another

23. Determining Relationships Between Scores Graphing Technique Scattergram /Scatter plot Graph used to demonstrate the relationship between two groups of variables or scores.

24. Determining Relationships Between Scores Scattergram

25. Correlations • A correlation is a statistical technique used to express the relationship between two sets of scores or variables • Correlation coefficient (r) - a number that is used to represent the strength of the relationship

26. Key points: • r values range from - 1.00 to + 1.00 • The larger the r, the stronger the relationship. • 1.00 is a perfect relationship (either pos. or neg.) • A correlation coefficient near zero indicates no relationship; the variables are not related in any way

27. Key points: (+) correlation means a positive relationship; as one variable increases, so does the other one High Low High Low

28. Key points: (-) correlation means an inverse relationship; as one variable goes up, the other goes down High Low Low High

29. Key points: • (0) Low correlation means no relationship; as one goes up the other goes up or down – not connected High Low Low High

30. Significance of the Correlation Coefficient: r = .80 - 1.00: very strong relationship r = .60 - .79: strong r = .40 - .59: moderate r = < .40: weak • Correlation coefficient meaning should be considered within the context of the study. • Not cause and effect – just strength of connection.

31. Coefficient of determination (r2) The amount of variability in one score that can be accounted for (explained) by the variability in another score. Example 1: leg power and vertical jump height r = .90 r2 = .81 81% of the variability in VJH can be explained by the variability in LP

32. Coefficient of determination (r2) The amount of variability in one score that can be accounted for (explained) by the variability in another score. Example 2: shoe size and type of car you drive r = .30 r2 = .09 9% of the variability in the type of car you drive can be explained by the variability in shoe size

33. Graphing Technique Scattergram Graph used to demonstrate the relationship between two groups of variables or scores Line of best fit (regression line) - A line representing the trend in the data (the plotted points)

34. Example r = .79

35. Example r = - .68

36. Cause and Effect? • Correlations only show strength of relationship!!! • They don’t show cause and effect!!!!

37. Prediction-Regression Analysis • When two variables are correlated, it becomes possible to predict one variable from the measure on the other

38. Prediction-Regression Analysis • Simple prediction – use one variable to predict another Y’ = bX + c also the equation for a line Y’ = [r(sy/sx)](X - X) + Y

39. Prediction-Regression Analysis measured predicted measured

40. Prediction-Regression Analysis • Multiple Prediction – Use several variables to predict one variable. • R – multiple correlation coefficient • R2 – percentage of variance in one score explained by the prediction equation.

41. Prediction-Regression Analysis Example – prediction equation for the one mile walk test: VO2max = 132.853 – (.0769 x W) – (.3877 x A) + (6.315 x G) – (3.2649 x T) – (.1565 x HR)

42. Questions?

43. Probability Probability of an event occurring is the number of possible outcomes that satisfy the conditions of the desired event divided by the total number of possible outcomes (page 54). 1 coin 2 coins P(heads) = 1 P(both heads) = 1 2 4

44. Probability • What is the probability that a score is untrue when many scores are gathered? • What is the chance of making a mistake and saying the score is true when it is not? • 10 out of 100? • 5 out of 100? • 1 out of 100?

45. Probability • How much risk of being wrong do you want to take? • For surgery • For money • 10 out of 100? • 5 out of 100? • 1 out of 100?

46. Probability Probability • The normal curve is used to make probability statements. • The normal curve represents 100 possible outcomes. • Use tabled values to find probability – “alpha”

47. Probability Probability • The normal curve is used to make probability statements. • Alpha level is set ahead of time • p < .10 - probability of being wrong 10 out of 100 times • p < .05 - probability of being wrong 5 out of 100 times • p < .01 - probability of being wrong 1 out of 100 times

48. Probability • One Tailed Test • Extreme or untrue score could only be high or low • Two Tailed Test • Extreme or untrue score could be either high or low

49. Questions?

50. Determining the Difference Between Group Scores Null Hypothesis and Significant Difference • Inferential statistics provides a method to make an educated guess as to whether two groups are really different (due to a treatment effect) • or if a difference is just due to measurement error or random chance