1. Describing a Score’s Position within a Distribution Lesson 5

2. Science & Probability n Learn about populations by studying samples l Introduction of error n Drawing conclusions l Cannot make states with certainty l Probability statements n Use of normal distribution l Can calculate probability of a result l Natural variables ≈ normal ~

3. Probability: Definitions n Probability(P) of an event (A) l Assuming each outcome equally likely P(A) = # outcomes Classified as A total # possible outcomes P(drawing ♥ ) = P(7 of ♥ ) = P(15 of ♥ ) = P( ♥ or ♦ or ♣ or ♠ ) = ~

4. Standard Normal Distribution n AKA Unit Normal Distribution n Parameters  = 0,  = 1 n z scores l Or standard scores Distance & direction from  in units of  ~

5. Standard Normal Distribution 120-2 f Z scores (  )

6. Other Standardized Distributions n Many natural variables ≈ normal n Standardized distributions l Have defined or set parameters IQ:  = 100,  = 15 ACT:  = 18,  = 6 SAT:  = 500,  = 100 ~

7. 11513010085 70 f 120-2z scores IQ IQ Scores  = 100  = 15

8. The Normal Distribution & Probability n Area under curve = frequency l Area under curve represents all data n Proportion (p) including all scores = 1 l p for any area under curve can be calculated l Proportion = probability that a score(s) is in distribution l Table A.1, pg 797 ~

9. 11513010085 70 f IQ Total area under curve = 1.0 0.5 Probability of obtaining IQ score below the median? Percentile rank of 70? Greater than 115? Use z scores.

10. Using z scores n AKA standard scores distance from mean in units of  n Uses l Determining probabilities l Percentile rank or scores l Compare scores from different distributions n *Technically must use parameters l text uses sample statistics: ~

11. z Score Equation z = X -  

12. Using z scores n Distance and direction relative to mean l Standard Normal Distribution  = 0,  = 1 n Answer questions by 1 st finding z score l What proportion of population have IQ scores greater than 115? l What is the percentile rank for IQ score of 70? l What percentage of people have IQ scores between 70 and 115?

13. 11513010085 70 f IQ z score for 115? IQ Score z score for 70?

14. Handy Numbers n Standard Normal Distribution l z scores l Proportions of distribution u i.e., area under curve, table A.1 n 3 handy proportions l Same for all normal distributions l Between z = 0 and ±1 l Between z = 1 and 2 (also -1 & -2) l Beyond z = ±2 (area in tails) ~

15. Areas Under Normal Curves 120-2 f.34.14.02.34.14.02 Z scores (  )

16. 24301812 6 f 120-2z scores ACT ACT Scores  = 18  = 6 What % of students scored b/n 18 and 24? % greater than 30? % less than 30?

17. Comparing Scores from Different Distributions n How to compare ACT to SAT? l Use z scores 1. Raw ACT score  z score 2. Use z score to compute Raw SAT score

18. Areas Under Normal Curves 120-2 f standard deviations.34.14.02.34.14.02

19. Percentile Rank & Percentile n Percentile rank l % of scores ≤ a particular score (X i ) l 84 th percentile: 84% of IQ scores ≤ 115 n Percentile l Raw score (X i ) associated with a particular percentile rank l IQ score of 100 is the 50 th percentile n Use z scores & table to determine ~

20. 11513010085 70 f IQ Scores.34.14.02.34.14.02 120-2z scores 2 d 16 th 50 th 84 th 98 th percentile rank IQ