Overview

1. Overview • Revising z-scores • Interpreting z-scores • Dangers of z-scores • Sample and populations • Types of samples • An Example • Another Example • The THREE Distributions • Relationships between the THREE Distributions • Summary

2. Example 1 • Ben is a 4th grader in an underperforming school • In one case, Ben’s math exam score is 10 points above the mean in his school • BUT, Ben’s exam score is 10 points below the mean for students in his grade in the country • It is useful to interpret Ben’s performance relative to average performance. Ben’s class Ben’s class Ben’s class Ben’s grade across the country Ben’s grade across the country Mean of students across country = 60 Mean of class = 40 Mean of class = 40

3. Example 2 – Dave is a Math Concentrator • Both distributions have the same mean (40), but different standard deviations (10 vs. 20) • In one case, Dave is performing better than almost 95% of the class. In the other, he is performing better than approximately 68% of the class. • Thus, how we evaluate Dave’s performance depends on how much variability there is in the exam scores Statistics Calculus

4. Standard Scores • We CAN express a person’s score with respect to both (a) the mean of the group and (b) the variability of the scores • How far a person is from the mean • Variability • What is the appropriate referent group • Raw score may contain information we lose by just looking at z score

5. Standard (Z) Scores • Once we are comfortable with selecting the referent group (a) the mean of the group and (b) the variability of the scores for that group we calculate • how far a person (i) is from the mean equals the deviation score X - M • variability = SD

6. Example 1 Ben in class: (50 - 40)/10 = 1 (one SD above the mean) Ben in country (50 - 60)/10 = -1 (one SD below the mean) Ben’s class Ben’s grade across the country Mean of students across country = 60 Mean of class = 40

7. Example 2 An example where the means are identical, but the two sets of scores have different spreads Dave’s Stats Z-score (50-40)/5 = 2 Dave’s Calc Z-score (50-40)/20 = .5 Statistics Calculus

8. Why is the Mean of z-scores always equal to 0? M = 50 BUT what is the mean of the deviation scores? Look back in your notes. x 20 30 40 50 60 70 80 z -3 -2 -1 0 1 2 3

9. Why is the SD of z-scores always equal to 1.0? M = 50 SD = 10 if x = 60, x 20 30 40 50 60 70 80 z -3 -2 -1 0 1 2 3

10. What happens to the shape of the distribution of the raw scores once we standardize? The distribution of a set of standardized scores has the same shape as the unstandardized scores • beware of the “normalization” misinterpretation REMEMBER the z-score is BASED on a normal distribution. If you transform non-normal distributions using the z-score you may accidentally lose information (about skewness, kurtosis, bimodality…)

11. The shape is the same(but the scaling or metric is different)

12. Percentile Scores We can use standard scores to find percentile scores: the proportion of people with scores less than or equal to a particular score. Percentile scores are intuitive ways of summarizing a person’s location in a larger set of scores.

13. The area under a normal curve 50% 34% 34% 14% 14% 2% 2%

14. Sample and Population • Population parameters and sample statistics

15. Why a Sample? • We want to learn about a certain population • The population we are interested in is BIG • If we take a sample from that population we can learn things about the population from the sample • Inferential statistics is all about trying to make an inference from a sample to a population

16. Types of Samples • Random samples • Systematic samples • Haphazard samples • Convenience samples • Biased samples

17. Lets look at a deck of cards • Population – 52 • Mean – 340/52 equals about 6.5 • Let’s calculate the variance up at the board • Ok- well that was an EASY population to deal with • Lets take a sample – deal a hand of solitaire on the computer

18. 14 15 4 0 20 9 Population of scores  = 10.00 and  = 6.05 Sample of 5 scores drawn randomly from the population M = 11.6 and SD = 6.78 Add cards to deck and sample again

19. 14 15 4 0 20 9 • Take the mean of each sample and set it aside • 11.6 • 11 • 9.2 • 12.4 • 11.8 • 6.8 • 12 • 10.2 • 13.2 • 9.4 The distribution of these sample means can be used to quantify sampling error

20. Three Important Distributions • Distribution of the population • Distribution of YOUR sample • Distribution of the means of many samples drawn from the population (sampling distribution) • IF you keep this straight – you are GOLDEN! If you keep confusing these – you are in TROUBLE

21. Things to notice about the sampling distribution 1. The average of these sample means is close to the population mean 2. There is variation in the sample means 3. The distribution of sample means is normal- THIS IS A COMPLEX THOUGHT These three facts make up what is called the CENTRAL LIMIT THEOREM