VCU STAT 210  Lecture9 (45 pages)
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Lecture9
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 Stat 210  Basic Practice of Statistics
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STAT 210 Lecture 9 Describing Distributions September 15 2017 Practice Problems Pages 65 through 68 Relevant problems III 1 through III 13 Recommended problems Look in solutions at the constructed stem and leaf plots and histograms practice describing the distributions Additional Reading and Examples Read pages 61 through 64 Top Hat 2 Motivating Example 1 Many 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 99 48 59 53 29 Motivating Example 2 Virginia 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 Statistics The 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 Hat Descriptive Statistics When 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 distribution Shapes of Distributions The following slides introduce various shapes that distributions can take Symmetric Distribution A distribution where the right and left sides of the distribution are mirror images of each other is said to be symmetric Symmetric Distribution Symmetric Distribution Symmetric Distribution Normal Distribution A 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 distribution and is the basis for many statistical inference procedures Skewed Left Distribution General bell shape with a long tail to the left Skewed Left Distribution Example grades on a statistics test are often skewed to the left Skewed Right Distribution General bell shape with a long tail to the right Skewed Right Distribution Example the distribution of a group of people s annual income is often skewed to the right Bimodal Distribution A distribution with two significant peaks Bimodal Distribution Trimodal Distribution A distribution with three significant peaks Trimodal Distribution Center and Spread We 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 Features Unusual 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 Outlier An outlier is an observation that stands out from the other observations an extreme value and that often creates a skewed distribution Describing a Distribution 1 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 12 Example 7 0 1 2 3 4 5 6 7 8 9 8 7 3 4 8 8 7 9 7 6 5 4 8 2 0 8 0 5 4 5 5 3 3 6 4 8 7 6 3 9 2 1 4 5 1 2 3 where 3 7 37 Example 12 Example 7 0 1 2 3 4 5 6 7 8 9 8 where 3 7 37 7 3 4 8 8 7 9 7 6 5 4 8 2 0 8 0 5 4 5 5 3 3 6 4 8 7 6 3 9 2 1 4 5 1 2 3 Skewed left center around 80 range from 8 to 95 at least one outlier 8 Example 12 0 1 2 3 4 5 6 7 8 9 95 Example 8 27 59 48 68 56 59 38 50 52 03 80 79 43 79 42 05 94 01 36 where 1 27 12 7 Example 12 0 1 2 3 4 5 6 7 8 9 95 Example 8 27 59 48 68 56 59 38 50 52 03 80 79 43 79 42 05 94 01 36 where 1 27 12 7 Bimodal center around 50 0 data ranges from 9 5 to 93 6 no outliers Example 12 Example 9 0 0 1 1 2 2 3 3 4 4 5 5 3 0 2 2 7 8 8 8 9 2 1 1 1 7 7 9 6 8 9 6 0 2 3 2 1 2 3 1 2 5 8 6 5 8 2 1 3 7 5 1 where 1 2 12 Example 9 0 0 1 1 2 2 3 3 4 4 5 5 Example 12 3 0 where 1 2 12 2 2 7 8 8 8 9 2 1 1 1 7 7 9 6 8 9 6 0 2 3 2 1 2 3 1 2 5 8 6 5 8 2 1 3 7 5 1 Symmetric center around 30 range from 0 to 51 no outliers Example 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 39 40 4 41 42 43 44 45 46 47 Example 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 39 40 4 41 42 43 44 45 46 47 The distribution is relatively symmetrical possibly multi modal with data ranging from 274 to 414 a center around 350 and no obvious outliers Example 10 27 28 33 34 36 38 40 41 42 43 44 45 46 47 29 30 3 31 32 4 9 8 2 6 1 35 3 2 37 7 9 8 0 0 5 4 2 3 6 39 8 8 6 9 3 8 1 0 0 4 2 8 where 43 2 432 0 Example 10 27 28 29 30 3 The distribution is 31 32 4 9 symmetrical possibly …
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