This document discusses measures of relative standing and density curves in statistics. It introduces z-scores as a standardizing process that measures the relative distance of a data point from the mean of a distribution. It describes key properties of the normal distribution including that 68% of data lies within one standard deviation of the mean. It also provides methods for assessing whether a dataset can be approximated as normally distributed including constructing histograms, stem-plots, and normal probability plots.

1. Chapter 2Describing Location in a Distribution

2. 2.1 Measures of Relative Standing and Density Curves

3. Z-scoresStandardizingProcess by which a measurement is compared to other measurements from the same data setMeasures relative distance from the mean of a distributionAlthough data sets need not be symmetric to use z-scores, it is often preferable

4. Z-scoresx-bar = mean of distributions = standard deviation of distributionThe importance of the above formula cannot be emphasized enough. Please memorize!

5. Z-scoresThe z-score of the mean of a distribution is (by definition) ‘0’Positive z-scores indicate data greater than the meanNegative z-scores indicate data below the mean

6. Z-scoresIf z = 1, then the measurement is exactly one standard deviation above the mean, or x = x-bar + sRemember: most of the data in a distribution lie within one standard deviation of the meanThat is , z = -1, z = +1

7. PercentilesRelative 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 year37% 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 CurvesDensity curvesgraphical representation of a distributionBound below by the x-axisThe curve is always above the x-axisBound above by the “curve”i.e. bell curveArea under the curve is always 1.00

10. Density CurvesMeasurements (as a number line)

11. Density CurvesArea under curve above the median is 0.5Area under the curve below the median is 0.5Area to the left of an observation is the same as percentile rankThe mean is the “balancing point” of the distribution

12. Density CurvesMean and MedianSymmetric distributionMean and Median are the sameLeft Skewed distributionMean < MedianThe mean is to the left of the medianRight Skewed distributionMedian < MeanThe mean is to the right of the median

13. Density CurvesREMEMBERArea 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

14. Assignment 2AP118 #1-6, 9-13

15. 2.2 Normal Distribution

16. The Normal Distribution The Normal Distribution is a specific density curve with the following propertiesSymmetricSingle peakHas 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 Distribution68-95-99.7 RuleApproximately 68% of the observations are within 1 standard deviation of the meanApprox. 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 DistributionA standard Normal distribution has a mean of ‘0’ and a standard deviation of ‘1’μ = 0; σ = 1This 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 distributionUse either Table A (back of the book) or your calculator to calculate area/percentage

20. Normal distributionRemember 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-83/84Calculate the z-scoreKey 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-pointUse “-1E99” for negative infinityUse “1E99” for positive infinity

22. Using your calculator TI-89Calculate the z-scoreKey in [CATALOG] [F3] Note: This brings you to the “Flash Apps” functionsLocate and select “normCdf( … TIStat”Area to the left = normCdf(a, b) where ‘a’ = left end-point and ‘b’ = right end-pointUse “-1E99” for negative infinityUse “1E99” for positive infinity

23. Solving Problems Involving Normal DistributionsState 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 Distributions1. 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 DistributionsState 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 Distributions2. Standardize (find the z-score). roper notation!DraUse pw and shade the Normal distribution.

28. Solving Problems Involving Normal DistributionsStandardize (find the z-score). Use proper notation!Draw and shade the Normal distribution.

29. Solving Problems Involving Normal DistributionsFind 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 Distributions4. 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 DistributionsThis 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!!

32. Assignment 2.2AP137 24-26, 31-36

33. Assessing NormalityMany 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 NormalityMethod 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 NormalityMethod 2: Construct a “Normal Probability Plot” steps: 1) find the z-score for each measurement2) create a scatterplot of measurement vs. z-score(note: this can be automated using your calculator)

36. Assessing NormalityMethod 2: Construct a “Normal Probability Plot”You can state that your data is “approximately Normal” if your Normal Probability Plot is approximately linear.

37. Normal Prob Plot on the TI84Enter the data into list1

38. Normal Prob Plot on the TI84Enter the data into list1[2nd] -> [Y=] (STATPLOT)Select plot #1Turn onSelect “normal probability plotSet: “Data list: L1”

39. Normal Prob Plot on the TI84Enter the data into list1[2nd] -> [Y=] (STATPLOT)Select plot #1Turn onSelect “normal probability plotSet: “Data list: L1”

40. Normal Prob Plot on the TI84Enter the data into list1[2nd] -> [Y=] (STATPLOT)Select plot #1Turn onSelect “normal probability plotSet: “Data list: L1”[ZOOM] -> [9] (ZOOMSTAT)

41. Normal Prob Plot on the TI84Enter the data into list1[2nd] -> [Y=] (STATPLOT)Select plot #1Turn onSelect “normal probability plotSet: “Data list: L1”[ZOOM] -> [9] (ZOOMSTAT)

42. 2.2BPage 156 #39-40