STAT 1053 Week 3 Notes
STAT 1053 Week 3 Notes STAT 1053
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This 5 page Class Notes was uploaded by Morgan Routman on Wednesday January 28, 2015. The Class Notes belongs to STAT 1053 at George Washington University taught by Professor Balaji in Winter2015. Since its upload, it has received 122 views. For similar materials see Intro to Statistics for Social Sciences in Statistics at George Washington University.
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Date Created: 01/28/15
STAT 1053 Week 3 Chapter 2 cont 12815 1009 PM 0 Sometimes the distribution of data is symmetrical and bellshaped o For these distributions the mean and standard deviation can describe the data well 0 Most observations will lie near the center mean of the curve The Empirical Rule can be used to describe these occurrences Interval Empirical Rule Chebyshev s Rule xbar d s No information xbars xbars xbar i 25 At least 75 of data xbar i 35 At least 89 of data All real numbers x such that altxltb All real numbers x such that anSb All real numbers x such that x20 0 All real numbers a b ab xbar s xbar 5 Example In a class of 500 students the mean score of an exam is 85 and the standard deviation is 5 with a symmetrical distribution Using the Empirical Rule 68 scored between a 74 and 89 95 scored between 74 and 94 and 997 scored between 69 and 99 Find the following a percentage of scores between 84 and 89 b percentage of scores between 79 and 94 c percentage of scores above 94 SOLUTIONS a Since 68 of the data lies between 74 and 89 and were looking for only half of that we can divide 68 by 2 to get the answer of 34 b As well we are looking for half of the 7494 range for b so we can divide 95 by 2 and add that to the 34 we found in the prior section to get the answer or 815 c For the last portion we divide the last 5 of the data by 2 to get the answer of 25 o Applicable to any data Empirical Rule only works on a Bell Curve Example Using the same data set above and the Chebysheve Rule what proportion of students scored a between 7494 and b lower than 69 SOLUTIONS a At least 75 b At most 11 Measures of Relative Standing 0 Measures relative position of an observation in comparison to the mean 0 Represents the distance between the given observation x and the mean Expressed in terms of Standard Deviation 0 Defined as z observation meanStandard Deviation Example 3 5 9 11 12 Find the zscore of 9 given that the mean is 8 and the Standard Deviation is 387 9 8387 0258 An extreme observation that doesn t match a given pattern a To use a zscore to find outliers is quite simple O 0 Mark in the number line that tells what percent of the data is to the left or the right Makes use of the Pth percentile o The median is an example 0 Partitions the data into four equal parts 25 25 25 25 QlQL QZMedian Q3QU L Lower U Upper From the quartiles we can make a boxplot STAT 1053 Week 3 Cont 12815 1009 PM 0 Find the median of the full data set 0 This is Q2 0 Find the median of all of the data points that are lesser than Q2 0 This is Q1 0 Find the median of all of the data points that are greater than Q2 0 This is Q3 Example 2 2 3 3 4 5 6 7 8 11 Median is 4 9 Q2 Median of all points lesser than 4 is 25 9 Q1 Median of all points greater than 4 is 75 9 Q3 0 Measures the spread of the data and gives the range of the middle 50 of data points 0 Q3 Q1 to find the range Ie for prior data set the IQR would be 5 Not sensitive to extreme observations or outliers Another graphical representation of distance using a five number summary 0 The Point at a distance of 15 IQR from each of the two ends of the Box Plot the hingemarks are To find the Inner Fences n Q1 15xIQR n Q3 15xIQR o mark where outliers begin to become extreme To find the Outer Fences D Q1 3XIQR D Q3 3xIQR Observations outside the outer fences are considered extreme outliers and are Observations marked inside the outer fences but outside the inner fences between them are considered mild outliers and are When looking at a Box Plot if one whisker or end of the line that is connected to the box is longer than the other it denotes the graph being asymmetrical In the the and if EXAMPLE Graph a box plot from the following data 137 141 142 148 150 151 153 155 Median 148 1502 149 9 Q2 Median for lower values 141 1422 1415 9 Q1 Median for higher values 151 1532 152 9 Q3 Minimum from data 137 Maximum from data 155 IQR155 137 018 Inner Fences 1415 15 x 018 1145 152 15 x 018 179 All data falls within inner fences No Outliers I I 137 1415 149 152 155