# VCU STAT 210 - Lecture8(2) (2) (45 pages)

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## Lecture8(2) (2)

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- Pages:
- 45
- School:
- Virginia Commonwealth University
- Course:
- Stat 210 - Basic Practice of Statistics

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STAT 210 Lecture 8 Graphically Displaying Distributions September 14 2016 Practice Problems Pages 65 through 68 Relevant problems III 2 III 8 III 12 and III 13 b Recommended problems III 2 III 8 and III 13 b The use the plot to describe the distribution part of the problems can be postponed until after tomorrow s lecture Additional Reading and Examples Read pages 61 through 64 Clicker Motivating Example Several students when asked to list their three favorite things to do included video games on their list Suppose you are interested in the typical cost of video games What is another method other than a stem and leaf plot you could use to display and describe the distribution of video game costs for a sample of 20 video games 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 Clicker 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 Back to Back Stem and Leaf Plots Used to display two sets of data side by side making it easier 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 10 Example 10 The 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 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 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 4 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 8 40 6 9 3 4 41 8 1 0 0 42 4 43 2 44 8 where 43 2 432 45 46 47 0 Histograms Histograms are a second graphical technique for displaying quantitative data so that the distribution can be described Unlike the stemand leaf plot a histogram does not retain the original data We will only consider histograms with equal class widths Histograms 1 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 Histograms 2 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 Histograms 3 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 Histograms 4 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 Histograms The rest of the intervals are obtained by adding the class width 20 in this example to the first lower limit value Example 200 220 220 240 240 260 480 500 Histograms Due to rounding in previous steps it is possible for the actual number of intervals to be one fewer or one more than that specified in step 1 Histograms 5 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 two intervals should be placed in the second interval not the first Example 220 goes in the interval 220 240 not 200 220 Histograms 6 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 Histograms 6 The class intervals class frequencies and relative frequencies are often displayed in a frequency table Histograms 7 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 frequency Clicker Example 11 1 The sample size is 60 and the square root of 60 is 7 75 Hence we want to use 8 class intervals Example 11 1 The sample size is 60 and the square root of 60 is 7 75 Hence we want to use 8 class intervals 2 Range Clicker Example 11 1 The sample size is 60 and the square root of 60 is 7 75 Hence we want to use 8 class intervals 2 Range 140 52 88 Example 11 1 The sample size is 60 and the square root of 60 is 7 75 Hence we want to use 8 class intervals 2 Range 140 52 88 3 Range 8 88 8 11 Use a class width of 11 Example 11 1 The sample size is 60 and the square root of 60 is 7 75 Hence we want to use 8 class intervals 2 Range 140 52 88 3 Range 8 88 8 11 Use a class width of 11 4 Multiples of 11 are 0 11 22 33 44 55 66 Smallest observation is 52 So the lower limit of the first interval is 44 Example 11 Step 4 Start with 44 then keep adding class width 11 Class Intervals 44 55 55 66 66 77 77 88 88 99 99 110 110 121 121 132 132 143 Example 11 …

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