Slide 1Practice ProblemsAdditional Reading and ExamplesSlide 4Motivating Example 1Motivating Example 2Descriptive StatisticsSlide 8Descriptive StatisticsShapes of DistributionsSymmetric DistributionSymmetric DistributionSymmetric DistributionSymmetric DistributionNormal 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 1 SolutionSlide 44Slide 45STAT 210Lecture 9Describing DistributionsSeptember 15, 2017Practice ProblemsPages 65 through 68Relevant problems: III.1 through III.13Recommended problems: Look in solutions at theconstructed stem-and-leaf plots and histograms, practice describing the distributions.Additional Reading and ExamplesRead pages 61 through 64Top Hat 2Motivating Example 1Many college students enjoy playing video games, and a study was created to estimate what percentage of all students at this university who include playing video games on their list of top three things to do. For those who play video games, it is also of interest to estimate the typical cost of the video games, and to describe the distribution of video game costs.The cost of the last video game that was purchased by a sample of 20 students is given below. 59.95 35.99 41.29 38.99 45.39 24.95 29.59 32.95 40.50 47.85 30.99 52.95 25.50 44.45 47.99 34.85 32.00 26.9948.59 53.29Motivating Example 2Virginia 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. The data will be presented later in a stem-and-leaf and used to 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.Top HatDescriptive 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 DistributionSymmetric DistributionNormal 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 a skewed distribution.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, possibly multi-modal, with data ranging from 274 to 414, a center around 350, and no obvious outliers.27 28 29 30 3 31
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