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## Variance Formula

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**Probability**A. The importance of probability Hypothesis testing and statistical significance Probabilisticcausation - because error always exists in our sampling (sampling error) we can only deal with probabilities of being correct or incorrect in our conclusions**The Normal Curve**• The normal curve represents a probability distribution • The mean and standard deviation, in conjunction with the normal curve allow for more sophisticated description of the data and (as we see later) statistical analysis • For example, a school is not that interested in the raw GRE score, it is interested in how you score relative to others.**Even if the school knows the average (mean) GRE score, your**raw score still doesn’t tell them much, since in a perfectly normal distribution, 50% of people will score higher than the mean. • This is where the standard deviation is so helpful. It helps interpret raw scores and understand the likelihood of a score. • So if I told you if I scored 710 on the quantitative section and the mean score is 591. Is that good?**It’s above average, but who cares.**• What if I tell you the standard deviation is 148? • What does that mean? • What if I said the standard deviation is 5? • Calculating z-scores**z-scores & conversions**• What is a z-score? • A measure of an observation’s distance from the mean. • The distance is measured in standard deviation units. • If a z-score is zero, it’s on the mean. • If a z-score is positive, it’s above the mean. • If a z-score is negative, it’s below the mean. • If a z-score is 1, it’s 1 SD above the mean. • If a z-score is –2, it’s 2 SDs below the mean.**Converting raw scores to z scores**What is a z score? What does it represent Z = (x-µ) / σ Z = (710-563)/140 = 147/140 = 1.05 Converting z scores into raw scores X = z σ + µ [(1.05*140)+563=710]**The Normal Curve**• A mathematical model or and an idealized conception of the form a distribution might have taken under certain circumstances. • A sample of means from any distribution has a normal distribution (Central Limit Theorem) • Many observations (height of adults, weight of children in Nevada, intelligence) have Normal distributions**Finding Probabilities under the Normal Curve**So what % of GRE takers scored above and below 710? (Z = 1.05) - Why is this important? Infer the likelihood of a result Confidence Intervals/Margin of Error Inferential Statistics (to be cont.- ch.6-7)**pi = ≈3.14159265**e = ≈2.71 Stuff you don’t need to know:**Powersxa xb = x (a + b)xa ya = (xy)a(xa)b = x (ab) x(a/b) =**bth root of (xa): Example X(1/2) = √Xx(-a) = 1 / xax(a - b) = xa / xb**Midterm**• Next Thursday • Take home • Babbie Chapters 1, 2, 3, 4, 5, 6, 13 • Levin and Fox • Calculate variance and standard deviation • Calculate z-score