Reasoning in Psychology Using Statistics

1. Reasoning in PsychologyUsing Statistics Psychology 138 2018

2. Quiz 3 is posted, due Friday, Feb. 23 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. 7th) • Don’t forget about Extra-Credit through SONA Annoucement

3. Transformations: z-scores • Normal Distribution • Using Unit Normal Table • Combines 2 topics Today • Statistical Snowball: • For the rest of the course, new concepts build upon old concepts • So if you feel like you don’t understand something now, ask now, don’t wait. X X X X Outline for next 2 classes

4. CVA Rotunda • North (and 10o West) • Approx. 1625 ft. 1625ft. • Where is Bone student center? • Reference point • Direction • Distance Location

5. Reference point • Direction • Distance μ • Where is a score within distribution? • Obvious choice is mean • Negative or positive sign ondeviation score Subtract mean from score (deviation score). • Value of deviation score Locating a score

6. Reference point Direction μ X1 - 100= +62 X1 = 162 X2 = 57 X2 - 100= -43 Locating a score

7. Below Above μ X1 - 100= +62 X1 = 162 Direction X2 = 57 X2 - 100= -43 Locating a score

8. Distance Distance μ X1 - 100= +62 X1 = 162 X2 = 57 X2 - 100= -43 Locating a score

9. Raw score Population mean Population standard deviation • Direction and Distance • 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). Transforming a score

10. z-score: standardized location of X value within distribution X1 - 100= +1.24 50 X2 - 100= -0.86 μ 50 If X1 = 162, z = • 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 If X2 = 57, z = Transforming scores

11. z-score: standardized location of X value within distribution X1 - 20= +1.2 5 X2 - 20= -0.8 μ 5 μ = 20 σ = 5 If X1 = 26, z = • 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 If X2 = 16, z = Transforming scores

12. Can transform all of scores in distribution Look what happens if sizes are standardized • 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 we know distribution mean & standard deviation Transforming distributions (transforming all the scores)

13. Shape: • Mean: • Standard Deviation: Properties of z-score distribution

14. transformation 50 150 μZ μ • Shape: Same as original distribution of raw scores. Every score stays in same position relative to every other score. original z-score • Note: this is true for other shaped distributions too: • e.g., skewed, mulitmodal, etc. Properties of z-score distribution

15. transformation 50 150 μZ μ Xmean = 100 • Shape: Same as original distribution of raw scores. Every score stays in same position relative to every other score • Mean • If X = μ, z = ? • Meanz always = 0 = 0 = 0 Properties of z-score distribution

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

17. transformation +1 μ μ X+1std = 150 • Shape: Same as original distribution of raw scores. Every score stays in same position relative to every other score • Mean: always = 0 • Standard Deviation: For z, 50 150 = +1 z is in standard deviation units Properties of z-score distribution

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

19. 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 Properties of z-score distribution

20. μ μ transformation 50 150 -1 +1  z = -0.60 • If know z-score and mean & standard deviation of original distribution, can find raw score (X) • have 3 values, solve for 1 unknown  (z)( σ) = (X - μ)  X = (z)( σ) + μ X = 70  X = (-0.60)( 50) + 100 = -30 +100 From z to raw score:

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

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

23. Example 3 Suppose you got 630 on SAT & 26 on ACT. Which score should you report on your application? • SAT: μ = 500, σ = 100 • ACT: μ = 21, σ = 3 z-score of 1.67 (ACT) is higher than z-score of 1.3 (SAT), so report your ACT score. SAT examples

24. Example 4 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 XB < or > 20? How much: 1? 2.5? 5? 10? Test C: μ = 100, σ = 20 XC < or > 100? How much: 20? 10? If XA = -.5 SD, then zA = -.5 XB = zBσ + μ XC = zCσ + μ = (-.5)(20) + 100 = -10 + 100 = 90 Find out later that this is true only if perfectly correlated; if less so, then XB and XC closer to mean. = (-.5)(5) + 20 = -2.5 + 20 = 17.5 Example with other tests

25. X X X X Formula Summary

26. In lab • Using SPSS to convert raw scores into z-scores; copy formulae with absolute reference • Questions? • Applications: Statistics and the Weather • Brian’s Climate Blog • NOAA – Jan 2014 Temperature Anomalies • Brandon Foltz: Understanding z-scores (~22 mins) • StatisticsFun (~4 mins) • Chris Thomas. How to use a z-table (~7 mins) • Dr. Grande. Z-scores in SPSS (~7 mins) Wrap up