Slide 1Practice ProblemsAdditional Reading and ExamplesSlide 4Motivating ExampleMotivating Example - SolutionDescriptive StatisticsSlide 8Descriptive StatisticsBack-to-Back Stem-and-Leaf PlotsExample 10Example 10Slide 13Slide 14Slide 15HistogramsHistogramsHistogramsHistogramsHistogramsHistogramsHistogramsHistogramsHistogramsHistogramsHistogramsSlide 27Example 11Example 11Example 11Example 11Example 11Example 11Example 11Example 11Example 11Example 11Example 11Example 11Slide 40Motivating ExampleMotivating Example - SolutionMotivating Example - SolutionMotivating Example - SolutionSTAT 210Lecture 8III. Graphically Displaying DistributionsSeptember 13, 2017Practice ProblemsPages 64 through 67Relevant problems: III.2, III.8 – III.12 and III.13 (b)Recommended problem: III.2, III.8 and III.13 (b)The “use the plot to describe the distribution” part ofthe problems can be postponed until after Friday’slecture.Additional Reading and ExamplesRead pages 60 through 64Top HatMotivating ExampleMany 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. Construct an appropriate stem-and-leaf plot to display this data.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 - Solution2 4952 959, 550, 6993 295, 099, 485, 2003 599, 8994 129, 050, 4454 539, 785, 799, 8595 295, 3295 995 where 4 | 129 = $41.29Descriptive 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 distributionBack-to-Back Stem-and-Leaf PlotsUsed to display two sets of data side-by-side, making iteasier to compare the distributions.Use a common column of stems, with one distribution displayed to the right of the stems (as we have been doing) and one distribution displayed to the left of the stems.Example 10Example 10The data ranges from 274 to 470. It is best to use a standard stem-and-leaf plot with two digit stems ranging from 27 to 47.27 28 2930 31 32 33 34 35 3637 38 3940414243444546474 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 414243444546474 27 7, 8 28 9 29 1 30 3 6, 2, 4, 1 31 4, 8 32 4, 9 7, 0, 4, 4, 6, 7 33 8 34 2, 6, 1 5, 1, 3, 4 35 3 3, 1, 9, 8, 0, 9 36 2 8, 3, 5, 9 37 7, 9, 8, 0, 0 2, 5, 1, 6, 0 38 5, 4, 2, 3, 6 2, 9, 6 39 8, 840 6, 9, 3 4 41 8, 1, 0, 042 443 244 8 where 43 | 2 = 43245 46 47 0HistogramsHistograms are a second graphical technique for displaying quantitative data so that the distribution can be described. Unlike the stem-and-leaf plot, a histogram does not retain the original data.We will only consider histograms with equal class widths.Histograms1. Determine the number of class intervals to use. One rule is to calculate the square root of the sample size, and round up. Example: If the sample consists of 210 subjects, then the square root of 210 is 14.5, which is rounded up to 15.Histograms2. Determine the range of the data by subtracting the smallest observation from the largest observation.Example: If the smallest observation is 202 and the largest observation is 496, then the range is 496 - 202 = 294.Histograms3. Divide the range by the number of class intervals and round to a convenient number. This will be the equal class width.Example: Range = 294 Number of intervals = 15 294/15 = 19.6 round to 20 Use a class width of 20 for each interval.Histograms4. The lower limit of the first interval should be a multiple of the class width and should be chosen such that the smallest observation is contained in the first interval.Example: Multiples of 20 are: 0, 20, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220, 240, 260, ….. The smallest observation is 202. Choose the lower limit of the first interval to be 200.HistogramsThe rest of the intervals are obtained by adding the class width (20 in the example) to this first lower limit value.Example: 200 - 220 220 - 240 240 - 260 ... 480 - 500HistogramsDue to rounding in previous steps, it is possible for theactual number of intervals to be one fewer or one morethan that specified in step 1.Histograms5. Count the number of observations falling in each interval. These counts are referred to as frequencies (or class frequencies). The frequencies should sum to the number of observations in the data set.As a rule, an observation that falls on the boundary of twointervals should be placed in the second interval, not the first.Example: 220 goes in the interval 220 - 240, not 200 - 220Histograms6. Determine the relative frequency for each class interval by dividing the class frequency by the total number of observations and multiplying by 100. The relative frequencies are the percentages of the observations in each interval and should sum to 100%. Example: If the frequency in the 200 - 220 class is 4 and the number of observations is 210, then the relative frequency is (4/210)*100% = 1.9%HistogramsThe class intervals, class frequencies, and relativefrequencies are often displayed in a frequency table.Histograms7. Construct the histogram. On the horizontal axis, mark and label the class intervals. On the vertical axis, mark and label the class frequencies (to create a frequency histogram) or the relative frequencies (to create a relative frequency histogram). Over each class interval draw a rectangle whose height equals the correct frequency or relative
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