Part III Taking Chances for Fun and Profit

1. Part IIITaking Chances for Fun and Profit Chapter 8    Are Your Curves Normal? Probability and Why it Counts

2. 0900 Quiz #3 N=26 • 2|1389 • 3|01112333335669 • 4|00012334 • X-bar=34.62; Median=13th and 14th dp=33 Mode=33; • S=6.03;

3. 1030 Quiz #3 N=33 • 2|0355678899 • 3|033334668899 • 4|00111223455 • X-bar=34.73; Median=33+1/2=17th dp=36; • Mode=33; s= 7.02;

4. Frequency distribution: 900 quiz scores

5. What you will learn in Chapter 7 • Understanding probability is basic to understanding statistics • Characteristics of the “normal” curve • i.e. the bell-shaped curve • All about z scores • Computing them • Interpreting them

6. Why Probability? • Basis for the normal curve • Provides basis for understanding probability of a possible outcome • Basis for determining the degree of confidence that an outcome is “true” • Example: • Are changes in student scores due to a particular intervention that took place or by chance along?

7. The Normal Curve (a.k.a. the Bell-Shaped Curve) • Visual representation of a distribution of scores • Three characteristics… • Mean, median, and mode are equal to one another • Perfectly symmetrical about the mean • Tails are asymptotic (get closer to horizontal axis but never touch)

8. The Normal Curve

9. Hey, That’s Not Normal! • In general, many events occur right in the middle of a distribution with few on each end.

10. More Normal Curve 101

11. More Normal Curve 101 • For all normal distributions… • almost 100% of scores will fit between -3 and +3 standard deviations from the mean. • So…distributions can be compared • Between different points on the X-axis, a certain percentage of cases will occur.

12. What’s Under the Curve?

13. The z Score • A standard score that is the result of dividing the amount that a raw score differs from the mean of the distribution by the standard deviation. • What about those symbols?

14. ThezScore • Scores below the mean are negative (left of the mean) and those above are positive (right of the mean) • A z score is the number of standard deviations from the mean • z scores across different distributions are comparable

15. What z Scores Represent • The areas of the curve that are covered by different z scores also represent the probability of a certain score occurring. • So try this one… • In a distribution with a mean of 50 and a standard deviation of 10, what is the probability that one score will be 70 or above?

16. Why Use Z scores? • Percentages can be used to compare different scores, but don’t convey as much information • Z scores also called standardized scores, making scores from different distributions comparable; Ex: You get two different scores in two different subjects(e.g Statistics 28 and English 76). They are not yet comparable, so lets turn them into percentages( e.g 28/35=80% and 76/100, 76%). Relatively you did better in statistics.

17. Percentages Verse Z scores • How do you compare to others? From percentages alone, you have no way of knowing. Say µ on English exam was =70 with ó of 8 pts, your 76 gives you a z-score of .75, three-fourths of one stand deviation above the mean; Mean on statistics test is 21, with ó of 5 pts; your score of 28 gives a z score of 1.40 standard deviations above mean; Although English and statistics scores were similar, comparing z scores shows you did much better in statistics

18. Using z scores to find percentiles • Prof Oh So Wise, scores 142 on an evaluation. What is Wise’s percentile ranking? Assume profs’ scores are normally distributed with µof 100 and ó of 25. X-µ 142-100 z= 1.68 ó 25 Area under curve ‘Small Part’ = .0465, equals those who scored above the prof; 1–.0465= 95.35th percentile. Oh so wise is in top 5% of all professors. Not bad at all. Never use from ‘mean to z’ to find percentile!! We’re only concerned with scores above or below a certain rank

19. Starting with An Area Under Curve and Finding Z and then X… • Using the previous parameters of µ of 100 and ó of 25, what score would place a professor in top 10% of this distribution? After some algebra, we have X=µ+z (ó) • 100(µ) + 1.28(z)(25)(ó)=132 (X). A score of 132 would place a professor in top 10 %; • What scores place a professor in most extreme 5% of all instructors?

20. What does ‘most extreme’ mean? • It is not just one end of the distribution, but both ends, or 2.5% at either end; • X= µ + z(ó)= 100+ 1.96(25)= 149 • 100 +-1.96(25)=51; 51 and 149 place a professor at the most extreme 5 % of the distribution;

21. The Difference between z scores

22. What z Scores Really Represent • Knowing the probability that a z score will occur can help you determine how extreme a z score you can expect before determining that a factor other than chance produced the outcome • Keep in mind… z scores are typically reserved for populations. 

23. Hypothesis Testing & z Scores • Any event can have a probability associated with it. • Probability values help determine how “unlikely” the even might be • The key --- less than 5% chance of occurring and you have a significant result

24. Some rules regarding normal distribution • Percentiles – if raw score is below the mean use’ small part’ to find percentile ; if raw score is above the mean, use’ big part’ to find percentile; check to see that you’re right by constructing a frequency distribution and identifying cumulative percentage • If raw scores are on opposite sides of the mean, add the areas/percentages. If raw scores are on same side of mean, subtract areas/percentages

25. Using the Computer • Calculating z Scores

26. Glossary Terms to Know • Probability • Normal curve • Asymptotic • Standard Scores • z scores