Slide 1Test 1 ResultsPractice ProblemsAdditional Reading and ExamplesSlide 5Motivating ExampleDescriptive StatisticsSlide 8Descriptive StatisticsShapes of DistributionsSymmetric DistributionSymmetric DistributionSymmetric CurveSymmetric (Normal) CurveNormal DistributionSkewed Left DistributionSkewed Left DistributionSkewed Right DistributionSkewed Right DistributionBimodal DistributionBimodal DistributionTrimodal DistributionTrimodal DistributionCenter and SpreadUnusual FeaturesOutlierDescribing a DistributionExample 12Example 12Example 12Example 12Example 12Example 12Slide 34Slide 35Slide 36Slide 37Example 12Example 12Motivating Example 1Motivating Example 1Motivating Example 1Motivating Example 1Motivating Example 1Slide 45Slide 46Slide 47Slide 48STAT 210Lecture 9Describing DistributionsSeptember 15, 2016Test 1 ResultsMean: 78.7 Median: 80Max: 100 Min: 42 n: 144Score Frequency Relative Frequency90’s 40 27.78%80’s 33 22.92%70’s 33 22.92%60’s 16 11.11% <60 22 15.28%Practice ProblemsPages 65 through 68Relevant problems: III.1 through III.13Recommended problems: Look in the back at the solutions for the constructed stem-and-leaf plots and histograms, and practice describing the distributions.Additional Reading and ExamplesRead pages 61 through 64ClickerMotivating ExampleVirginia Blood Services and VCU often partner to give students and faculty an opportunity to donate blood, thus saving lives.A random sample of 64 adults, age 40 or older, were selected and the number of times each person has given blood determined. Given data presented in a stem-and-leaf plot, how would you describe the distribution?Descriptive StatisticsThe branch of statistics concerned with numerical and graphical techniques for analyzing and describing one or more characteristics of a population and for comparing characteristics among populations.Clicker2Descriptive StatisticsWhen describing a distribution we describe four things:(1) the center of the distribution(2) the spread of the distribution(3) the shape of the distribution(4) any unusual features in the distributionShapes of DistributionsThe following slides introduce various shapes that distributions can take.Symmetric DistributionA distribution where the right and left sides of thedistribution are mirror images of each other is said to be symmetric.Symmetric DistributionSymmetric CurveSymmetric (Normal) CurveNormal DistributionA symmetric distribution has the right and left sides of the distribution being mirror images of each other. One type of symmetric distribution is a bell-shaped curve called a normal distribution, as seen on the last slide.This is the most commonly used type of distributionand is the basis for many statistical inference procedures.Skewed Left DistributionGeneral bell-shape, with a long tail to the leftSkewed Left DistributionExample: grades on a statistics test are often skewed to the left.Skewed Right DistributionGeneral bell-shape, with a long tail to the right.Skewed Right DistributionExample: the distribution of a group of people’s annual income is often skewed to the right.Bimodal DistributionA distribution with two significant peaks.Bimodal DistributionTrimodal DistributionA distribution with three significant peaks.Trimodal DistributionCenter and SpreadWe will learn numerical methods for calculating the center and spread of a distribution in the next chapter. For the purpose of the current section we will estimate the approximate center and use the range of the data as a measurement of spread.Unusual FeaturesUnusual features include things that create distributions that are not symmetric (normal). This can include high concentrations of data, gaps in the distribution, and extreme values at the tails of the distribution (called outliers).OutlierAn outlier is an observation that stands out from the other observations (an extreme value) and that often creates skewed distributions.Describing a Distribution1. Shape (using the terms just introduced)2. Center (for now a guess)3. Spread (for now use the range)4. Unusual features (outliers, gaps, high concentrations of data)Example 12Example 7:0 8 where 3|7 = 37123 74 3,4,85 6 87 7,9,7,6,5,4,88 2,0,8,0,5,4,5,5,3,3,6,4,8,7,6,3,99 2,1,4,5,1,2,3Example 12Example 7:0 8 where 3|7 = 37123 74 3,4,85 6 87 7,9,7,6,5,4,88 2,0,8,0,5,4,5,5,3,3,6,4,8,7,6,3,99 2,1,4,5,1,2,3Skewed left, center around 80, range from 8 to 95, at least one outlier (8)Example 120 95 Example 81 27,592 48,68,563 59,38 4 50,525 036 807 79,438 79,42,05,949 01,36 where 1 | 27 = 12.7Example 120 95 Example 81 27,592 48,68,563 59,38 4 50,525 036 807 79,438 79,42,05,949 01,36 where 1 | 27 = 12.7Bimodal, center around 50.0, data ranges from 9.5 to 93.6, no outliersExample 12Example 9:0 . 3, 0 where 1|2 = 120 * 1 . 2, 21 * 7, 8, 8, 8, 92 . 2, 1, 1, 12 * 7, 7, 9, 6, 8, 9, 63 . 0, 2, 3, 2, 1, 2, 3, 1, 23 * 5, 8, 6, 5, 84 . 2, 1, 34* 7, 55 . 15*Example 12Example 9:0 . 3, 0 where 1|2 = 120 * 1 . 2, 21 * 7, 8, 8, 8, 92 . 2, 1, 1, 12 * 7, 7, 9, 6, 8, 9, 63 . 0, 2, 3, 2, 1, 2, 3, 1, 23 * 5, 8, 6, 5, 84 . 2, 1, 34* 7, 55 . 15* Symmetric, center around 30, range from 0 to 51, no outliersExample 10 4 27 7, 8 28 9 29 1 30 6, 2, 4, 1 31 4, 8 32 7, 0, 4, 4, 6, 7 33 34 5, 1, 3, 4 35 3, 1, 9, 8, 0, 9 36 8, 3, 5, 9 37 2, 5, 1, 6, 0 38 2, 9, 6 3940 4 41424344454647Example 10 4 27 7, 8 28 9 29 1 30 6, 2, 4, 1 31 4, 8 32 7, 0, 4, 4, 6, 7 33 34 5, 1, 3, 4 35 3, 1, 9, 8, 0, 9 36 8, 3, 5, 9 37 2, 5, 1, 6, 0 38 2, 9, 6 3940 4 41424344454647The distribution is relatively symmetrical, with data ranging from 274 to 414, a center around 350, and no obvious outliers.27 28 29 30 3 31 32 4, 9 33 8 34 2, 6, 1 35 3 36 2 37 7, 9, 8, 0, 0 38 5, 4, 2, 3, 6 39 8, 840 6, 9, 3 41 8, 1, 0, 042 443 244 8 where 43 | 2 = 43245 46 47 0Example 1027 28 29 30 3 31 32 4, 9 33 8 34 2, 6, 1 35 3 36 2 37 7, 9, 8, 0, 0 38 5, 4, 2, 3, 6 39 8, 840 6, 9, 3 41
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