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Statistics or What’s normal about the normal curve, and what’s standard about the standard deviation, and what co-relates 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-relates in a correlation? ] Statistics: Intro What’s normal about the normal curve(s)? • The normal curve is not a single curve, but a class of curve of property distribution /probability, that share properties in common • There are a number of ways of mathematically defining and estimating the normal distribution • The actual definition (which you don’t need to know) is: Statistics: Intro What’s normal about the normal curve(s)? • The main questions 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? Statistics: Intro What is the normal curve? • The normal curve has the following properties: – It is bell-shaped – It is symmetric – The total area under the curve is 1. – The normal curve extends indefinitely in both directions, getting infinitely close to zero in either direction. Statistics: Intro 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. 171-202. 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? 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)!) * pr * (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 outcomes are mutually exclusive – 4. The events are randomly selected Statistics: Intro Let’s try it out (Example 6.3 from our first probability 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) = 0.138 - the odds of rolling 7 second are 5/6 * 1/6 (independent events) = 0.138 - since these two outcomes are mutually exclusive, we can add them to get 0.138 + 0.138 = 0.277 Statistics: Intro The generalization (Example 6.3 from last class) • What are the odds of there being exactly one seven out of two rolls? An event of probability p happens r times out of n tries: P(r) = n!/(r! (n - r)!) * pr * (1 - p) n-r p = 1/6; N = 2; r = 1 2!/(1!1!)*1/61*5/61 = 0.277 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 http://ccl.northwestern.edu/cm/index.html 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 From: http://www.psychstat.smsu.edu/introbook/sbk00.htm • 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 Statistics: Intro The standard deviation (SD) • The definition of SD takes the second approach: it keeps the size of each division constant, but it 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. [Why do we square the difference?] Statistics: Intro Z-scores • If we express differences by dividing them by SDs, we have zscores: 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 Wayne Gretzky was better hockey player than Tiger Woods is a golfer (a prize for the person who proves on or the other!) • If a person with a WAIS IQ of 140 is rarer(= less probable) 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 We need first to know: What is covariance? • Covariance is closely related to variance (which is, recall, the average of the squared deviations from a mean) • The covariance of two features X and Y measures their tendency to vary together, i.e., to co-vary. • It is defined as the average of (differences from the mean for X multiplied by the differences from the mean for Y) – That is: the average of the products of the deviations from the mean of X and Y – In variance (one variables), we square the differences from the mean – In covariance (two variables), we multiply one difference from the mean by the other difference from the mean Statistics: Intro We need first to know: What is covariance? • Covariance is the average of the products of the deviations from the mean of X and Y • Properties: – If X and Y tend to increase together, then c(X,Y) > 0 – If one tends to decrease when the other increases, then c(X,Y) < 0 – If X and Y are independent, then c(X,Y) = 0 – | c(X,Y) | <= the product of the standard deviations of X and Y Statistics: Intro What is a correlation? • R = The covariance of x and y / the product of the SDs of X and Y • It is measure of how (the product of) item-by-item differences from the two means relates to (the product of ) their overall average differences • When X and Y are related, covariance close to the product of the SDs of X and Y, so R will be close to 1. • When X and Y are unrelated, the differences from the means by item will depart from the average differences from the mean: c(x,y) < SD(x) * SD(y) Statistics: Intro Visual help • Check out the normal curve and correlation real-time demos (as well as infinite 2-dice problems!) at: http://noppa5.pc.helsinki.fi/koe/ Statistics: Intro