R. G. Bias | School of Information | SZB 562BB | Phone: | i 1 INF 397C Introduction to Research in Library and Information Science Fall, 2003 Day 4

R. G. Bias | School of Information | SZB 562BB | Phone: | i 2 4 things today 1.NEW equation for σ 2.Practice exercises 3.z scores and “area under the curve” 4.Start to look at experimental design (maybe)

R. G. Bias | School of Information | SZB 562BB | Phone: | i 3 NEW equation for σ σ = SQRT(Σ(X - µ) 2 /N) –HARD to calculate when you have a LOT of scores. Gotta do that subtraction with every one! New, “computational” equation –σ = SQRT((Σ(X 2 ) – (ΣX) 2 /N)/N) –Let’s convince ourselves it gives us the same answer.

R. G. Bias | School of Information | SZB 562BB | Phone: | i 4 Practice Questions

R. G. Bias | School of Information | SZB 562BB | Phone: | i 5 z scores – table values z = (X - µ)/σ It is often the case that we want to know “What percentage of the scores are above (or below) a certain other score”? Asked another way, “What is the area under the curve, beyond a certain point”? THIS is why we calculate a z score, and the way we do it is with the z table, on p. 306 of Hinton.

R. G. Bias | School of Information | SZB 562BB | Phone: | i 6 Going into the table You need to remember a few things: –We’re ASSUMING a normal distribution. –The total area under the curve is = 1.00 –Percentage is just a probability x 100. –50% of the curve is above the mean. –z scores can be negative! –z scores are expressed in terms of (WHAT – this is a tough one to remember!) –USUALLY it’ll help you to draw a picture. So, with that, let’s try some exercises.

R. G. Bias | School of Information | SZB 562BB | Phone: | i 7 z table practice 1.What percentage of scores fall above a z score of 1.0? 2.What percentage of scores fall between the mean and one standard deviation above the mean? 3.What percentage of scores fall within two standard deviations of the mean? 4.My z score is.1. How many scores did I “beat”? 5.My z score is.01. How many scores did I “beat”? 6.My score was higher than only 3% of the class. (I suck.) What was my z score. 7.Oooh, get this. My score was higher than only 3% of the class. The mean was 50 and the standard deviation was 10. What was my raw score?

R. G. Bias | School of Information | SZB 562BB | Phone: | i 8 Probability Remember all those decisions we talked about, last week. VERY little of life is certain. It is PROBABILISTIC. (That is, something might happen, or it might not.)

R. G. Bias | School of Information | SZB 562BB | Phone: | i 9 Prob. (cont’d.) Life’s a gamble! Just about every decision is based on a probable outcomes. None of you raised your hands last week when I asked for “statistical wizards.” Yet every one of you does a pretty good job of navigating an uncertain world. –None of you touched a hot stove (on purpose.) –All of you made it to class.

R. G. Bias | School of Information | SZB 562BB | Phone: | i 10 Probabilities Always between one and zero. Something with a probability of “one” will happen. (e.g., Death, Taxes). Something with a probability of “zero” will not happen. (e.g., My becoming a Major League Baseball player). Something that’s unlikely has a small, but still positive, probability. (e.g., probability of someone else having the same birthday as you is 1/365 =.0027, or.27%.)

R. G. Bias | School of Information | SZB 562BB | Phone: | i 11 Just because There are two possible outcomes, doesn’t mean there’s a “50/50 chance” of each happening. When driving to school today, I could have arrived alive, or been killed in a fiery car crash. (Two possible outcomes, as I’ve defined them.) Not equally likely. But the odds of a flipped coin being “heads,”....

R. G. Bias | School of Information | SZB 562BB | Phone: | i 12 Let’s talk about socks

R. G. Bias | School of Information | SZB 562BB | Phone: | i 13 Prob (cont’d.) Probability of something happening is –# of “successes” / # of all events –P(one flip of a coin landing heads) = ½ =.5 –P(one die landing as a “2”) = 1/6 =.167 –P(some score in a distribution of scores is greater than the median) = ½ =.5 –P(some score in a normal distribution of scores is greater than the mean but has a z score of 1 or less is... ? –P(drawing a diamond from a complete deck of cards) = ?

R. G. Bias | School of Information | SZB 562BB | Phone: | i 14 Probabilities – and & or From Runyon: –Addition Rule: The probability of selecting a sample that contains one or more elements is the sum of the individual probabilities for each element less the joint probability. When A and B are mutually exclusive, p(A and B) = 0. p(A or B) = p(A) + p(B) – p(A and B) –Multiplication Rule: The probability of obtaining a specific sequence of independent events is the product of the probability of each event. p(A and B and...) = p(A) x p(B) x...

R. G. Bias | School of Information | SZB 562BB | Phone: | i 15 Prob (II) From Slavin: –Addition Rule: If X and Y are mutually exclusive events, the probability of obtaining either of them is equal to the probability of X plus the probability of Y. –Multiplication Rule: The probability of the simultaneous or successive occurrence of two events is the product of the separate probabilities of each event.

R. G. Bias | School of Information | SZB 562BB | Phone: | i 16 Prob (II) mlhttp:// ml –The product or multiplication rule. "If two chances are mutually exclusive the chances of getting both together, or one immediately after the other, is the product of their respective probabilities.“ –the addition rule. "If two or more chances are mutually exclusive, the probability of making ONE OR OTHER of them is the sum of their separate probabilities."

R. G. Bias | School of Information | SZB 562BB | Phone: | i 17 Let’s try with Venn diagrams

R. G. Bias | School of Information | SZB 562BB | Phone: | i 18 Additional Resources Phil Doty, from the ISchool, has taught this class before. He has welcomed us to use his online video tutorials, available at html html –Frequency Distributions –z scores –Intro to the normal curve –Area under the normal curve –Percentile ranks, z-scores, and area under the normal curve Pretty good discussion of probability:

R. G. Bias | School of Information | SZB 562BB | Phone: | i 19 Think this through. What are the odds (“what are the chances”) (“what is the probability”) of getting two “heads” in a row? Three heads in a row? Six heads in a row?

R. G. Bias | School of Information | SZB 562BB | Phone: | i 20 So then... WHY were the odds in favor of having two people in our class with the same birthday? Think about the problem! What if there were 367 people in the class. –P(2 people with same b’day) = 1.00

R. G. Bias | School of Information | SZB 562BB | Phone: | i 21 Happy B’day to Us But we had 43. Probability that the first person has a birthday: Prob of the second person having the same b’day: 1/365 Prob of the third person having the same b’day as Person 1 and Person 2 is 1/ /365 – the chances of all three of them having the same birthday.

R. G. Bias | School of Information | SZB 562BB | Phone: | i 22 Sooooo... hday.html

R. G. Bias | School of Information | SZB 562BB | Phone: | i 23 hill.com/sites/ /student_view0 /statistics_primer.htmlhttp://highered.mcgraw- hill.com/sites/ /student_view0 /statistics_primer.html Click on Statistics Primer.

R. G. Bias | School of Information | SZB 562BB | Phone: | i 24 More practice problems