Displaying and Describing Quantitative Data Displaying and Describing Quantitative Data Summarizing numerical data Histograms Stem and Leaf plots Shape and Skewness Center Mean vs Median Boxplots 5 number summary Measuring the spread Frequency Distribution Continuous Data Continuous Data may take on any value in Summarized in a grouped data frequency some interval table Example A manufacturer of insulation randomly selects 20 winter days and records the daily high temperature 24 35 17 21 24 37 26 46 58 30 32 13 12 38 41 43 44 27 53 27 NOTE Temperature is a continuous variable because it could be measured to any degree of precision desired Building a Frequency Table Continuous Data 1 Determine the number of categories classes bins 2 Establish class width Minimum width is the range of the data Largest data point Smallest data point Range 3 Set the class boundaries 4 Determine the frequency in each class Count the number of data points in each category How Many Categories Many Narrow class intervals May yield a very jagged distribution with gaps from empty classes Can give a poor indication of how frequency varies across classes Few Wide class intervals May compress variation too much and yield a blocky distribution Can obscure important patterns of variation y c n e u q e r F y c n e u q e r F 3 5 2 5 1 5 0 5 3 2 1 0 12 10 8 6 4 2 0 4 8 2 1 6 1 0 2 4 2 8 2 2 3 6 3 0 4 4 4 8 4 2 5 6 5 0 6 e r o M Temperature 0 30 60 More Temperature X axis labels are upper class endpoints General Guidelines Number of Data Points Number of Classes under 50 5 7 50 100 6 10 100 250 7 12 over 250 10 20 Class widths can typically be reduced as the number of observations increases Distributions with numerous observations are more likely to be smooth and have gaps filled since data are plentiful Considerations Continuous Data Must be mutually exclusive Must be all inclusive Bins should be of equal width Avoid empty categories How should the endpoints be determined Often by trial and error The goal is to create a distribution that is neither too jagged nor too blocky You want to appropriately show the pattern of variation Example Continuous Data Sort raw data from low to high 12 13 17 21 24 24 26 27 27 30 32 35 37 38 41 43 44 46 53 58 Find range 58 12 46 Select number of classes 5 usually between 5 and 20 Compute class width 10 46 5 then round off Determine class boundaries 10 20 30 40 50 Sometimes class midpoints are reported 15 25 35 45 55 Count the number of values in each class Frequency Distribution Example 12 13 17 21 24 24 26 27 27 30 32 35 37 38 41 43 44 46 53 58 Data from low to high Frequency Distribution Class Frequency Relative Frequency 10 but under 20 3 15 20 but under 30 6 30 5 25 30 but under 40 40 but under 50 4 20 2 10 50 but under 60 20 1 00 Total Histogram Example Data in ordered array 12 13 17 21 24 24 26 27 27 30 32 35 37 38 41 43 44 46 53 58 His togram 6 5 4 y c n e u q e r F 7 6 5 4 3 2 1 0 3 0 5 0 10 20 30 40 50 60 25 36 Class Midpoints Class Endpoints Bins 45 15 55 More No gaps between bars since continuous data 2 0 Frequency Histograms Visual representation of the frequency table The classes intervals bins are shown on the horizontal axis Frequency is measured on the vertical axis Bars of the appropriate heights can be used to represent the number of observations within each class Shows the center of the data and the spread Stem and Leaf Plots A quick and dirty histogram 11 24 24 25 27 30 30 31 32 33 44 46 47 50 52 11 2 4457 3 4 5 00123 467 02 The parts 1 2 3 4 5 1 4457 00123 467 02 Stems Leaves Key 1 1 stands for 11 Splitting Stems You want about 7 to 10 stems depending on how much data you have Create more stems be splitting them 7 7 8 8 02 8899 0012233 555555566 7 0 to 7 4 7 5 to 7 9 8 0 to 8 4 8 5 to 8 9 07 27 7 7 7 8 8 8899 001 2233 Or More 7 0 and 7 1 7 2 and 7 3 7 4 and 7 5 7 6 and 7 7 7 8 and 7 9 8 0 and 8 1 8 2 and 8 3 Key 1 3 means 13 80 1 2 3 34567889 00111126668999 1358 20 18 16 29 26 14 21 26 24 21 22 8 28 21 31 20 29 33 26 18 21 19 38 21 13 29 17 15 35 26 80 1 1 2 2 3 3 34 567889 0011112 6668999 13 58 Normal Body Temperature 96 3 96 7 96 9 97 0 97 1 97 1 97 1 97 2 97 3 97 4 97 4 97 4 97 4 97 5 97 5 97 6 97 6 97 6 97 7 97 8 97 8 97 8 97 8 97 9 97 9 98 0 98 0 98 0 98 0 98 0 98 0 98 1 98 1 98 2 98 2 98 2 98 2 98 3 98 3 98 4 98 4 98 4 98 4 98 5 98 5 98 6 98 6 98 6 98 6 98 6 98 6 98 7 98 7 98 8 98 8 98 8 98 9 99 0 99 0 99 0 99 1 99 2 99 3 99 4 99 5 3 79 96 96 97 97 98 98 99 99 0111234444 556667888899 000000112222334444 55666666778889 0001234 5 Symmetric or Skewed Is the distribution symmetric A distribution is symmetric if the right and left sides of the histogram are approximately mirror images of each other or right skewed left skewed A distribution is skewed to the right if the right side of the histogram side with larger values extends much farther out than the left side It is skewed to the left if the left side of the histogram extends much farther out than the right side Other Distributions Bimodal Distribution Uniform Distribution Outliers Data points that don t seem to fit in the distribution Far to the left or right in the graph Quantitative Summaries Mean Median 5 number summary Boxplots Measuring Spread Describing Quantitative Data Where is it What is its center What is the spread or variability How much noise is in the data What is the shape of the distribution Is it symmetric Measuring these attributes Center of the Distribution The average salary height etc Mean add up the data and divide …
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