3. Z-scores Standardizing Process by which a measurement is compared to other measurements from the same data set Measures relative distance from the mean of a distribution Although data sets need not be symmetric to use z-scores, it is often preferable
4. Z-scores x-bar = mean of distribution s = standard deviation of distribution The importance of the above formula cannot be emphasized enough. Please memorize!
5. Z-scores The z-score of the mean of a distribution is (by definition) ‘0’ Positive z-scores indicate data greater than the mean Negative z-scores indicate data below the mean
6. Z-scores If z = 1, then the measurement is exactly one standard deviation above the mean, or x = x-bar + s Remember: most of the data in a distribution lie within one standard deviation of the mean That is , z = -1, z = +1
7. Percentiles Relative position is also frequently measured using percentiles (%-iles) Ex. if an SAT score is at the 63rd percentile (has a percentile rank of 63%) … 63% of all test takers had the same score or less on the SAT that year 37% of all test takes had a greater score on the SAT that year
8. Chebychev’s Inequality In any distribution, the percent of observations that lie within k standard deviations of the mean is at least:
9. Density Curves Density curves graphical representation of a distribution Bound below by the x-axis The curve is always above the x-axis Bound above by the “curve” i.e. bell curve Area under the curve is always 1.00
11. Density Curves Area under curve above the median is 0.5 Area under the curve below the median is 0.5 Area to the left of an observation is the same as percentile rank The mean is the “balancing point” of the distribution
12. Density Curves Mean and Median Symmetric distribution Mean and Median are the same Left Skewed distribution Mean < Median The mean is to the left of the median Right Skewed distribution Median < Mean The mean is to the right of the median
13. Density Curves REMEMBER Area is the same as “percent of measurements” Area to the left of a measurement (x) is “percent of data whose measurement is less than x” Area to the right of a measurement (x) is “percent of data whose measurement is greater than x” In many abstract problems, area is computed geometrically
16. The Normal Distribution The Normal Distribution is a specific density curve with the following properties Symmetric Single peak Has a “bell shaped curve” At z = ±1, the curve has points of inflection (ask the calculus students if you don’t know)
17. The Normal Distribution 68-95-99.7 Rule Approximately 68% of the observations are within 1 standard deviation of the mean Approx. 95% of the observations are within 2 std. devs. Approx. 99.7% of the observations are within 3 std. devs. NOTE: these are approximate, we will learn the actual areas soon!
18. The Standard Normal Distribution A standard Normal distribution has a mean of ‘0’ and a standard deviation of ‘1’ μ = 0; σ = 1 This is often abbreviated as “N(0, 1)”
19. Normal Distribution “What we do” When a distribution is recognized as “Normal” or “approximately Normal,” Transform the distribution into the standard Normal distribution Use either Table A (back of the book) or your calculator to calculate area/percentage
20. Normal distribution Remember that formula? Kind of? Use this formula to find the z-score of a measurement. Always round z-scores to hundredth place (2 decimals) This is just convention
21. Using your calculator TI-83/84 Calculate the z-score Key in [2nd] [vars] Note: 2ndvars is the “distributions menu” Locate and select “normCdf” Area = normCdf(a, b) where ‘a’ = left end-point and ‘b’ = right end-point Use “-1E99” for negative infinity Use “1E99” for positive infinity
22. Using your calculator TI-89 Calculate the z-score Key in [CATALOG] [F3] Note: This brings you to the “Flash Apps” functions Locate and select “normCdf( … TIStat” Area to the left = normCdf(a, b) where ‘a’ = left end-point and ‘b’ = right end-point Use “-1E99” for negative infinity Use “1E99” for positive infinity
23. Solving Problems Involving Normal Distributions State problem in terms of ‘x,’ the observed variable. Draw and shade the distribution of ‘x.’ Standardize (find the z-score). Use proper notation!Draw and shade the Normal distribution. Find the area using Table A or your calculator.Remember that the area under curve is ‘1.’ Write up a conclusion in context of the problem.
24. Solving Problems Involving Normal Distributions “The level of cholesterol in the blood is important because high cholesterol levels may increase the risk of heart disease. The distribution of blood cholesterol levels in a large population of people of the same age and sex is is roughly Normal. For 14-year old boys, the mean is 170 mg/dL and the standard deviation is 30 mg/dL. What percent of 14-year old boys have more than 240 mg/dL of cholesterol?” “WHEW. This is one problem?”
25. Solving Problems Involving Normal Distributions 1. State problem in terms of ‘x,’ the observed variable. Draw and shade the distribution of ‘x.’ “We would like to know the proportion of 14-year old boys who have a cholesterol greater than 240 mg/dL.”
26. Solving Problems Involving Normal Distributions State problem in terms of ‘x,’ the observed variable. Draw and shade the distribution of ‘x.’ (While I appreciate a great drawing here, don’t drive yourself crazy!) Always notate the location of 1 std dev above and/or below
27. Solving Problems Involving Normal Distributions 2. Standardize (find the z-score). roper notation!DraUse pw and shade the Normal distribution.
28. Solving Problems Involving Normal Distributions Standardize (find the z-score). Use proper notation!Draw and shade the Normal distribution.
29. Solving Problems Involving Normal Distributions Find the area using Table A or your calculator.Remember that the area under curve is ‘1.’ <calculator> normCdf(2.33,1E99) </calculator> (Calculations continued from step 2) Note: keep all computation of area rounded to ten-thousandth place
30. Solving Problems Involving Normal Distributions 4. Write up a conclusion in context of the problem. “ Approximately 0.99% of 14 year old boys have a cholesterol level greater that 240 mg/dL”
31. Solving Problems Involving Normal Distributions This skill is one of the foundation of a Statistics course; you must learn to do this well. Although it may seem superfluous to draw the normal distribution for every problem, the AP readers always look for it! For some reason they think that it shows you understand what’s going on with the problem. (I don’t get it either) If you do not write a conclusion in context of the problem, you will get BURIED. If you use procedures for Normal Distributions for non-Normal distributions, you will get BURIED!!
33. Assessing Normality Many times, we are given data with no indication that it is Normal. Usually it would be NICE if the data was normal, or close to normal. We could then use the procedures we just learned! Remember, you will get BURIED if you use Normal procedures on non-Normal data!
34. Assessing Normality Method 1: construct a histogram or stem-plot You can state that your data is “approximately Normal” if your histogram is single-peaked and symmetric about the mean
35. Assessing Normality Method 2: Construct a “Normal Probability Plot” steps: 1) find the z-score for each measurement 2) create a scatterplot of measurement vs. z-score (note: this can be automated using your calculator)
36. Assessing Normality Method 2: Construct a “Normal Probability Plot” You can state that your data is “approximately Normal” if your Normal Probability Plot is approximately linear.
38. Normal Prob Plot on the TI84 Enter the data into list1 [2nd] -> [Y=] (STATPLOT) Select plot #1 Turn on Select “normal probability plot Set: “Data list: L1”
39. Normal Prob Plot on the TI84 Enter the data into list1 [2nd] -> [Y=] (STATPLOT) Select plot #1 Turn on Select “normal probability plot Set: “Data list: L1”
40. Normal Prob Plot on the TI84 Enter the data into list1 [2nd] -> [Y=] (STATPLOT) Select plot #1 Turn on Select “normal probability plot Set: “Data list: L1” [ZOOM] -> [9] (ZOOMSTAT)
41. Normal Prob Plot on the TI84 Enter the data into list1 [2nd] -> [Y=] (STATPLOT) Select plot #1 Turn on Select “normal probability plot Set: “Data list: L1” [ZOOM] -> [9] (ZOOMSTAT)
