Psychology 10

1. Psychology 10 Analysis of Psychological Data February 24, 2014

2. Katie Coburn • Jack is in Washington DC reviewing grants; back on Wednesday.

3. The Plan for Today • Introducing Z scores. • Random variables and probability distributions. • What is probability, anyway? • The law of large numbers. • The frequentist approach to probability.

4. Z scores • Sometimes a special linear transformation of the form Z = (X – M) / s will be particularly useful. • Is that really a linear transformation? • Z = -M / s + (1 / s) X . • a = - M / s b = 1 / s • Yes, that’s linear.

5. Apply the rules for change under linear transformation • Z = -M / s + (1 / s) X . • MZ = -M / s + (1 / s) M = 0. • sZ = (1 / s) s = 1. • So the Z transformation of any variable will have a mean of zero and a standard deviation of one. • This is sometimes called “putting the variable in standard form.”

6. Characteristics of Z scores • The sign of the Z score tells us whether the score is above or below the mean of the distribution. • The magnitude of the Z score tells us how far above or below, in standard deviation units. • For example, a Z score of -0.4 represents an individual score that is four tenths of a standard deviation below the mean.

7. Comparisons using Z scores • Because the Z score is scale free, it can help us compare variables that are reported in different metrics. • Tonight’s in-class exercise illustrates that process.

8. Activity • 40 people took exam 1. Your score was 80. • S X = 3,160, and S X2 = 252,136. • 35 people took exam 2. Your score was 68. • S X = 2,240, and S X2 = 146,760. • Which of your exam scores represents better performance relative to the rest of the class?

9. More about Z scores • Recall that a Z score “standardizes” any variable; that is, it changes the scale so that the mean is zero and the standard deviation is one. • The Z score does not change any other aspect of the shape of a distribution. • Z = (X – M) / s. (Or, for a population, Z = (X – m) / s ).

10. Things we know about Z scores • The sign tells us the position of an original score compared to the mean of the distribution: • Positive Z score: above the mean; • Negative Z score: below the mean. • The magnitude tells us how many standard deviations away from the mean the original score was.

11. Uses for Z scores • We just saw that Z scores are useful when we need to compare variables that are measured differently. • In our class exercise, we saw that a score of 68 on one test was actually better than a score of 80 on another.

12. Uses for Z scores • Another use of Z scores is for changing a variable to a new metric with some other mean and standard deviation. • Procedure: • First, standardize the variable (calculate a Z score). • Then multiply by the desired standard deviation and add the desired mean.

13. An example of metric change • In our exercise, the first test had M = 79 and s = 8. • Suppose we want to change our observed score of 80 to a metric in which the mean is 50 and the standard deviation is 10. • Standardize: (80 – 79) / 8 = 0.125 • Change to target metric: 0.125  10 + 50 = 51.25.

14. Metric change • The other test had a mean of 64 and standard deviation of 10. • Standardize our score of 68: Z = (68 – 64) / 10 = 0.4. • Change to the new metric: 0.4  10 + 50 = 54. • Again, we can see that the second result is better: 54 > 51.25.

15. Metric change (cont.) • This sort of change of scale is very common in educational testing. • Most large standardized tests are developed with mean = 0 and standard deviation = 1. • Parents don’t like negative achievement scores.

16. One more thing about Z scores • As stated before, the Z score doesn’t change anything other than the scale of the variable. • Unfortunately, the word “normalizing” has sometimes been applied as a synonym for “standardizing.”

17. More about Z scores (cont.) • This has created the misunderstanding that calculating a Z score somehow changes the distribution of the variable into a normal distribution. • That belief is a serious error!

18. Random variables • Think about students at Merced High School. • We can imagine with a high degree of confidence that all of them are either male or female. • Let’s denote males as “0” and females as “1”. • What proportion of each would we expect to observe if we were to visit the school?

19. Random variables (cont.)

20. Random variables (cont.) • Think about what we just did: • Specified some values; • Described the relative frequencies of those values. • That sounds like a distribution. • But we never actually observed anything!

21. Random variables (cont.) • Recall that a variable, for us, is numbers that convey information about a well defined entity. • A random variable is numbers that could convey information about a well defined entity; but those numbers exist only in our imagination. • As soon as we actually observe a random variable, it is no longer a random variable.

22. Probability distributions

23. Probability distributions

24. Probability distributions

25. Probability distributions • A probability distribution is the set of values that a random variable could take on, if we were to observe it… • …together with the long-run relative frequencies or probabilities of those values.

26. What is probability, anyway? • Problems with the book’s approach. • Book: probability of A = (number of outcomes classified as A) / (total number of possible outcomes). • Works OK for analytical approach to things like playing card draws. • Doesn’t work so well for things like a coin toss.

27. Defining probability • Why doesn’t it work so well for things like a coin toss? • Coins are not necessarily “fair.” I could have a trick coin that comes up heads 70% of the time. • That coin would still have two possible outcomes, of which “heads” is one.