1. Reasoning in Psychology Using Statistics Psychology 138 2015

2. Reasoning in Psychology Using Statistics Annoucement Quiz 3 is posted, due Friday, Feb. 20 at 11:59 pm –Covers Tables and graphs Measures of center Measures of variability –You may want to have a calculator handy Exam 2 is two weeks from today (Wed. Mar. 4 th )

3. Reasoning in Psychology Using Statistics Outline for 2 classes Transformations: z-scores Normal Distribution Using Unit Normal Table –Combines 2 topics Today

4. Reasoning in Psychology Using Statistics Location Where is Bone student center? –Reference point –Direction –Distance –CVA Rotunda –North (and 10 o West) –Approx. 1625 ft. 1625 ft.

5. Reasoning in Psychology Using Statistics Locating a score Where is a score within distribution? –Negative or positive sign on deviation score –Obvious choice is mean –Reference point –Direction –Distance Subtract mean from score (deviation score). –Value of deviation score μ

6. Reasoning in Psychology Using Statistics Locating a score μ X 1 = 162 X 2 = 57 X 1 - 100 = +62 X 2 - 100 = -43 Reference point Direction

7. Reasoning in Psychology Using Statistics Locating a score  X 1 = 162 X 2 = 57 X 1 - 100 = +62 X 2 - 100 = -43 Below Above Direction

8. Reasoning in Psychology Using Statistics Locating a score  X 1 = 162 X 2 = 57 X 1 - 100 = +62 X 2 - 100 = -43 Distance

9. Reasoning in Psychology Using Statistics Transforming a score Deviation score is valuable, BUT measured in units of measurement of score AND lacks information about average deviation SO, convert raw score (X) to standard score (z). Raw score Population mean Population standard deviation Direction and Distance

10. Reasoning in Psychology Using Statistics Transforming scores μ If X 1 = 162, If X 2 = 57, X 1 - 100 = +1.24 50 X 2 - 100 = -0.86 50 Direction. Sign of z-score (+ or -): whether score is above or below mean Distance. Value of z-score: distance from mean in standard deviation units z-score: standardized location of X value within distribution z =

11. Reasoning in Psychology Using Statistics Transforming scores  If X 1 = 26, If X 2 = 16, X 1 - 20 = +1.2 5 X 2 - 20 = -0.8 5 Direction. Sign of z-score (+ or -): whether score is above or below mean Distance. Value of z-score: distance from mean in standard deviation units z-score: standardized location of X value within distribution z = μ = 20 σ = 5

12. Reasoning in Psychology Using Statistics Transforming distributions –Called a standardized distribution Has known properties (e.g., mean & stdev) Used to make dissimilar distributions comparable –Comparing your height and weight –Combining GPA and GRE scores –z-distribution One of most common standardized distributions Can transform all observations to z-scores if know distribution mean & standard deviation Can transform all of scores in distribution

13. Reasoning in Psychology Using Statistics Properties of z-score distribution Shape: Mean: Standard Deviation:

14. Reasoning in Psychology Using Statistics Properties of z-score distribution Shape: Same as original distribution of raw scores. Every score stays in same position relative to every other score. μZμZ μ 15050 transformation originalz-score Note: this is true for other shaped distributions too: –e.g., skewed, mulitmodal, etc.

15. Reasoning in Psychology Using Statistics Properties of z-score distribution Shape: Same as original distribution of raw scores. Every score stays in same position relative to every other score Mean μZμZ μ transformation 15050 = 0 X mean = 100 = 0 o If X = μ, z = ? o Mean z always = 0

16. Reasoning in Psychology Using Statistics Properties of z-score distribution Shape: Same as original distribution of raw scores. Every score stays in same position relative to every other score Mean: always = 0 Standard Deviation:

17. Reasoning in Psychology Using Statistics Properties of z-score distribution Shape: Same as original distribution of raw scores. Every score stays in same position relative to every other score Mean: always = 0 Standard Deviation: μμ transformation 15050 = +1 X +1std = 150 +1 For z, z is in standard deviation units

18. Reasoning in Psychology Using Statistics Properties of z-score distribution Shape: Same as original distribution of raw scores. Every score stays in same position relative to every other score Mean: always = 0 Standard Deviation: μμ transformation 15050 = +1 = -1 X -1std = 50 +1 For z, X +1std = 150

19. Reasoning in Psychology Using Statistics Properties of z-score distribution Shape: Same as original distribution of raw scores. Every score stays in same position relative to every other score Mean: always = 0 Standard Deviation: always = 1, so it defines units of z- score

20. Reasoning in Psychology Using Statistics μ 15050 μ +1 From z to raw score: If know z-score and mean & standard deviation of original distribution, can find raw score (X) –have 3 values, solve for 1 unknown transformation  z = -0.60  X = (-0.60)( 50) + 100 X = 70  X = (z)( σ) + μ  (z)( σ) = (X - μ) = -30 +100

21. Reasoning in Psychology Using Statistics Another student got 420. What is her z-score? A student got 580 on the SAT. What is her z-score? Example 1 SAT examples Population parameters of SAT: μ= 500, σ= 100

22. Reasoning in Psychology Using Statistics Student said she got 1.5 SD above mean on SAT. What is her raw score? SAT examples Population parameters of SAT: μ= 500, σ= 100 X = z σ + μ = (1.5)(100) + 500 = 150 + 500 = 650 Standardized tests often convert scores to:  = 500, σ = 100 (SAT, GRE)  = 50,  = 10 (Big 5 personality traits) Example 2

23. Reasoning in Psychology Using Statistics Suppose you got 630 on SAT & 26 on ACT. Which score should you report on your application? Example 3 SAT examples SAT:  = 500,  = 100 ACT:  = 21,  = 3 z-score of 1.67 (ACT) is higher than z-score of 1.3 (SAT), so report your ACT score.

24. Reasoning in Psychology Using Statistics Example 4 Example with other tests On Aptitude test A, a student scores 58, which is.5 SD below the mean. What would his predicted score be on other aptitude tests (B & C) that are highly correlated with the first one? Test B: μ = 20, σ = 5 X B 20? How much: 1? 2.5? 5? 10? Test C: μ = 100, σ = 20 X C 100? How much: 20? 10? If X A = -.5 SD, then z A = -.5 X B = z B σ + μ X C = z C σ + μ = (-.5)(20) + 100 = -10 + 100 = 90 Find out later that this is true only if perfectly correlated; if less so, then X B and X C closer to mean. = (-.5)(5) + 20= -2.5 + 20 = 17.5

25. Reasoning in Psychology Using Statistics Formula Summary Population Sample Mean Standard Deviation Z-score

26. Reasoning in Psychology Using Statistics Wrap up In lab –Using SPSS to convert raw scores into z-scores; copy formulas with absolute reference Questions?