Mean the average sum of the values divided by the number of values Assignment 3 Chapter 1 Describing distributions with numbers pp 27 49 Homework 1 6 1 65 1 66 1 67 Center Spread variability 1 Median M the center value rank order first if two middle values take the average of those two values Variance s2 the sum of squared deviations divided by N 1 Note xi is each individual value in the distribution Standard deviation s the square root of the variance Quartiles 1st Quartile biggest median of the upper half of the distribution 3rd Quartile smallest median of the lower half of the distribution do NOT include the overall median in these computations Five number summary lowest lower quartile median upper quartile highest IQR Inter quartile range The 1st quartile largest minus the 3rd quartile smallest Note it is often easy to get 1st and 3rd mixed up If you want to avoid any error it is just as acceptable to call them upper quartile and lower quartile You know the upper quartile is bigger numbers lower is smaller numbers Box and whisker plots modified boxplot boundaries illustrated below from largest to smallest value 1 Large outliers if they occur shown as dots 2 Line and whisker start at the 3rd quartile 1 5 x IQR above the line if no outliers whisker at largest value 3 The 1st quartile top of the box 4 The median line in the box between the 1st and 3rd quartiles 5 The 3rd quartile bottom of the box 6 Line and whisker start at the 1st quartile 1 5 x IQR below the line if no outliers whisker at smallest value 7 Small outliers if they occur shown as dots Resistant measure Value that does not change central measures or variability a great deal with a few changes of individual values even if those values are large Example medians tend to be more resistant than means Linear transformations xnew a bx 1 Using the distribution 2 3 5 6 7 9 10 compute the mean 2 3 5 6 7 9 10 42 42 7 6 2 Using the distribution 2 3 5 6 7 9 10 compute the median M 2 3 5 6 7 9 10 median 6 3 Using the distribution 2 3 5 6 7 9 10 compute the variance s2 4 3 1 0 1 3 4 7 1 522 8 6667 16 9 1 0 1 9 16 6 8 6667 s 2 944 4 Using the distribution 2 3 5 6 7 9 10 compute the standard deviation s Mean 15K 20K 25K 20K 35K Mean 15k Median 20k 15K 20K 25K 35K Median 20K Mean 15K 20K 25K 20K 35K 400 billion Mean 67 billion Median 20k 15K 20K 25K 35K 400 billion Median 22 5K 1 65 p 48 The following chart notes the sale of Double stout beer in 12 English cities between 1913 pre World War 1 and 1925 post World War 1 The 12 numbers in the chart are percentage in 1913 beer sales of the sale of beer in 1925 For instance Bristol 94 means that sale of beer in 1925 was 94 of the sales in 1913 hence the sale of beer in Bristol decreased The Liverpool 140 means that 1925 sales were 140 of the sales in 1913 hence the sale of beer in Liverpool increased 66 140 428 Manchester 190 Newcastle on Tyne 118 Scottish 24 Bristol Cardiff English Agents 94 112 78 English O English P English R 68 46 111 Glasgow Liverpool London a Mean of these data show work 24 46 66 68 78 94 111 112 118 140 190 428 1475 Mean 1475 12 122 9 b Median of these data show work 24 46 66 68 78 94 111 112 118 140 190 428 94 111 2 102 5 The median is 102 5 c Which measure does a better job at identifying a center value The median because it is less influenced by the outlier 428 1 66 p 48 Refer to the data in the previous problem a compute the standard deviation for these data The variance s2 12 1 11 178 447 122127 6 11 11 178 447 24 122 9 46 122 9 68 122 9 78 122 9 94 122 9 111 122 9 112 122 9 118 122 9 140 122 9 190 122 9 428 122 9 The standard deviation s 5 511 178 447 105 728 24 46 66 68 78 94 111 112 118 140 190 428 b compute the quartiles 111 112 118 140 190 428 118 140 2 129 first quartile Median 102 5 24 46 66 68 78 94 66 68 2 67 third quartile c provide a 5 number summary of these data 24 67 102 5 129 428 1 67 p 48 Is or are there outliers in the data from problem 5 d which provides a better description of the spread of data Probably the 5 number summary based on the median because the mean and standard deviation are skewed by the outlier 428 a compute the inter quartile range IQR for these data using answers from the previous question IQR Q1 Q3 IQR 129 67 62 b compute 1 5 x IQR for these data 1 5 x 62 93 c Construct a modified boxplot to see if the data contain outliers Note whiskers are 1 5 IQR above the first quartile and below the 3rd quartile The values for the boxplot Outlier s 428 Upper boundary of whisker 129 93 22 First quartile 129 Median 102 5 Third quartile 67 Lower bound of whisker 67 93 24 Note If the lower whisker bound is lower than the smallest value the boundary will be the smallest value Outlier 428 425 400 375 350 325 300 275 250 225 200 175 150 125 100 75 50 25 0 25 5 Using the formula for conversion of metrics xnew a bx convert the five highest high jumps of 2017 from centimeters to inches Centimeter values 241 240 238 238 237 Formula height inches 0 3937 centimeters 241 3837 94 88 or 7 10 88 240 3937 94 49 or 7 10 49 238 3937 93 70 or 7 9 70 238 3937 93 70 or 7 9 70 237 3937 93 31 or 7 9 31 6 Using the conversion formula change the following Celsius temperatures 23 37 42 15 25 to Fahrenheit The formula is temperature fahrenheit 32 1 8 celsius 32 1 8 23 32 41 4 73 0 Fahrenheit 32 1 8 37 32 66 6 98 6 Fahrenheit 32 1 8 42 32 75 6 107 6 Fahrenheit 32 1 8 15 32 27 59 0 Fahrenheit 32 1 8 25 32 45 13 0 Fahrenheit