Statistics: Intro Statistics or What’s normal about the normal curve, what’s standard about the standard deviation, and what’s co-relating in a correlation?

Statistics: Intro Overview What’s normal about the normal curve? –The nature of the confusion –One formal answer –An intuitive answer (real-time demo) What’s standard about a standard deviation? –Z-scores What’s co-relating in a correlation?

Statistics: Intro What’s normal about the normal curve(s)? There are a number of ways of mathematically defining and estimating the normal distribution (which defines a class of curves, not one single curve) The main question I want to address today is: what does that math mean? Why are so many things normally distributed? What makes sure that those things stay distributed normally? What stops other things from being normally distributed at all?

Statistics: Intro U: Why do you think height is distributed normally? L: Come again? (sarcastic) U: Why is it that women's height can be graphed using a normal curve? L: That's a strange question. U: Strange? L: No one's ever asked me that before..... (thinking to herself for a while) I guess there are 2 possible theories: Either it's just a fact about the world, some guy collected a lot of height data and noticed that it fell into a normal shape..... U: Or? L: Or maybe it's just a mathematical trick. U: A trick? How could it be a trick? From: Wilensky, U., (1997). What is Normal Anyway? Therapy for Epistemological Anxiety. Educational Studies in Mathematics. Special Issue on Computational Environments in Mathematics Education. Noss R. (Ed.) Volume 33, No. 2. pp

Statistics: Intro L: Well... Maybe some mathematician somewhere just concocted this crazy function, you know, and decided to say that height fit it. U: You mean... L: You know the height data could probably be graphed with lots of different functions and the normal curve was just applied to it by this one guy and now everybody has to use his function. U: So you’re saying that in the one case, it's a fact about the world that height is distributed in a certain way, and in the other case, it's a fact about our descriptions but not about height? L: Yeah. U: Well, if you had to commit to one of these theories, which would it be? L: If I had to choose just one? U: Yeah. L: I don't know. That's really interesting. Which theory do I really believe? I guess I've always been uncertain which to believe and it's been there in the background you know, but I don't know. I guess if I had to choose, if I have to choose one, I believe it's a mathematical trick, a mathematician's game.....What possible reason could there be for height,....for nature, to follow some weird bizarro function?

Statistics: Intro Formal answer 1: The binomial distribution I The chance of an event of probability p happening r times out of n tries: P(r) = n!/(r! (n - r)!) * p r * (1 - p) n-r (Recall: We wondered about this generalization last class.)

Statistics: Intro Formal answer 1: The binomial distribution II Why is it called the binomial distribution? Bi = 2 Nom = thing = the two-thing distribution It can be used wherever: 1.Each trial has two possible outcomes (say, success and failure; or heads and tails) 2.The trials are independent = the outcome of one trial has no influence over the outcome of another trial. 3. The trials are mutually exclusive 4. The events are randomly selected

Statistics: Intro Let’s try it out (Example 6.3 from last class) What are the odds of there being exactly one seven out of two rolls? one way is to roll 7 first, but not second - the odds of this are 1/6 * 5/6 (independent events) = the odds of rolling 7 second are 5/6 * 1/6 (independent events) = since these two outcomes are mutually exclusive, we can add them to get = 0.277

Statistics: Intro The generalization (Example 6.3 from last class) An event of probability p happens r times out of n tries: P(r) = n!/(r! (n - r)!) * p r * (1 - p) n-r p = 1/6; N = 2; r = 1 2!/(1!1!)*1/6 1 *5/6 1 = What are the odds of there being exactly one seven out of two rolls?

Statistics: Intro What does this have to do with the normal distribution?

Statistics: Intro What does this have to do with the normal distribution?

Statistics: Intro Why does this normal distribution happen? [See for the StarLogoT demo used in class. Can you understand: What effect changing the probabilities of each event has? What has to change to skew a normal curve?]

Statistics: Intro The standard deviation Given the non-linear shape of the normal distribution, one has two choices: –A.) Keep the amount of variation in each division constant, but vary the size of the divisions –B.) Keep the size of each division constant, but vary the the amount of variation in each division From:

Statistics: Intro The standard deviation (SD) The SD takes the second approach: it keeps the size of each division constant, but varies the the amount of variation in each division The SD is a measure of average deviation (difference) from the mean It is the square root of the variance, which is the average squared difference from the mean.

Statistics: Intro Z-scores If we express differences by dividing them by SDs, we have z-scores: standard units of difference from the mean THESE Z-SCORES WILL COME IN EXTREMELY USEFUL! –For example, we might want to know: If a 12-foot elephant is taller (compared to the height of average elephants) than a 230 pound man is heavy (compared to weight of average men) If a person with a WAIS IQ of 140 is rarer than a person with a GPA of 3.9 —Etc.

Statistics: Intro What co-relates in a correlation? In a correlation, we want to find the equation for the (one and only) line (the line of regression) which describes the relation between variables with the least error. –This is done mathematically, but the idea is simply that we draw a line such that the squared distances on two (or more) dimensions of points from the line would not be less for any other line

Statistics: Intro What co-relates in a correlation? R = The covariance of x and y / the product of the SDs of X and Y Covariance is related to variance = the mean value of all the pairs of differences from the mean for X multiplied by the differences from the mean for Y (the mean product of differences from the means) When X and Y are related, large numbers will be systematically multiplied by large numbers with the same sign (for differences on both sides of the mean) = covariance will be large & close to the product of the SDs of X and Y, so R will be close to 1.