1. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 5 – Slide 1 of 21 Chapter 3 Section 5 The Five-Number Summary And Boxplots

2. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 5 – Slide 2 of 21 Chapter 3 – Section 5 ●Learning objectives  Compute the five-number summary  Draw and interpret boxplots 1 2

3. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 5 – Slide 3 of 21 Chapter 3 – Section 5 ●Learning objectives  Compute the five-number summary  Draw and interpret boxplots 1 2

4. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 5 – Slide 4 of 21 Chapter 3 – Section 5 ●Traditional statistics is to collect data to analyze / test a particular conjecture  Is there a correlation between Measurement 1 and Measurement 2?  Is Drug A more effective than Drug B? ●Traditional statistics is to collect data to analyze / test a particular conjecture  Is there a correlation between Measurement 1 and Measurement 2?  Is Drug A more effective than Drug B? ●A new approach, Exploratory Data Analysis (EDA) examines data to look for patterns  Are there patterns when comparing Group I to Group II to Group III to Group IV?  Are there patterns in people’s spending?

5. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 5 – Slide 5 of 21 Chapter 3 – Section 5 ●The five-number summary is the collection of  The smallest value  The first quartile (Q 1 or P 25 )  The median (M or Q 2 or P 50 )  The third quartile (Q 3 or P 75 )  The largest value ●These five numbers give a concise description of the distribution of a variable

6. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 5 – Slide 6 of 21 Chapter 3 – Section 5 ●The median  Information about the center of the data  Resistant ●The median  Information about the center of the data  Resistant ●The first quartile and the third quartile  Information about the spread of the data  Resistant ●The median  Information about the center of the data  Resistant ●The first quartile and the third quartile  Information about the spread of the data  Resistant ●The smallest value and the largest value  Information about the tails of the data  Not resistant

7. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 5 – Slide 7 of 21 Chapter 3 – Section 5 ●Compute the five-number summary for 1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 54 ●Compute the five-number summary for 1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 54 ●Calculations  The minimum = 1  Q 1 = P 25, the index i = 3.75, Q 1 = (4 + 7) / 2 = 5.5  M = Q 2 = P 50 = (16 + 19) / 2 = 17.5  Q 3 = P 75, the index i = 11.25, Q 3 = (27 + 31) / 2 = 29  The maximum = 54 ●Compute the five-number summary for 1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 54 ●Calculations  The minimum = 1  Q 1 = P 25, the index i = 3.75, Q 1 = (4 + 7) / 2 = 5.5  M = Q 2 = P 50 = (16 + 19) / 2 = 17.5  Q 3 = P 75, the index i = 11.25, Q 3 = (27 + 31) / 2 = 29  The maximum = 54 ●The five-number summary is 1, 5.5, 17.5, 29, 54

8. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 5 – Slide 8 of 21 Chapter 3 – Section 5 ●Learning objectives  Compute the five-number summary  Draw and interpret boxplots 1 2

9. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 5 – Slide 9 of 21 Chapter 3 – Section 5 ●The five-number summary can be illustrated using a graph called the boxplot ●An example of a (basic) boxplot is ●The middle box shows Q 1, Q 2, and Q 3 ●The horizontal lines (sometimes called “whiskers”) show the minimum and maximum

10. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 5 – Slide 10 of 21 Chapter 3 – Section 5 ●To draw a (basic) boxplot:  Calculate the five-number summary ●To draw a (basic) boxplot:  Calculate the five-number summary  Draw a horizontal line that will cover all the data from the minimum to the maximum ●To draw a (basic) boxplot:  Calculate the five-number summary  Draw a horizontal line that will cover all the data from the minimum to the maximum  Draw a box with the left edge at Q 1 and the right edge at Q 3 ●To draw a (basic) boxplot:  Calculate the five-number summary  Draw a horizontal line that will cover all the data from the minimum to the maximum  Draw a box with the left edge at Q 1 and the right edge at Q 3  Draw a line inside the box at M = Q 2 ●To draw a (basic) boxplot:  Calculate the five-number summary  Draw a horizontal line that will cover all the data from the minimum to the maximum  Draw a box with the left edge at Q 1 and the right edge at Q 3  Draw a line inside the box at M = Q 2  Draw a horizontal line from the Q 1 edge of the box to the minimum and one from the Q 3 edge of the box to the maximum

11. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 5 – Slide 11 of 21 Chapter 3 – Section 5 ●To draw a (basic) boxplot Voila! Draw the middle box Draw the minimum and maximum Draw in the median

12. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 5 – Slide 12 of 21 Chapter 3 – Section 5 ●An example of a more sophisticated boxplot is ●The middle box shows Q 1, Q 2, and Q 3 ●The horizontal lines (sometimes called “whiskers”) show the minimum and maximum ●The asterisk on the right shows an outlier (determined by using the upper fence)

13. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 5 – Slide 13 of 21 Chapter 3 – Section 5 ●To draw this boxplot (in a slightly different way than the text)  Draw the center box and mark the median, as before ●To draw this boxplot (in a slightly different way than the text)  Draw the center box and mark the median, as before  Compute the upper fence and the lower fence ●To draw this boxplot (in a slightly different way than the text)  Draw the center box and mark the median, as before  Compute the upper fence and the lower fence  Temporarily remove the outliers as identified by the upper fence and the lower fence (but we will add them back later with asterisks) ●To draw this boxplot (in a slightly different way than the text)  Draw the center box and mark the median, as before  Compute the upper fence and the lower fence  Temporarily remove the outliers as identified by the upper fence and the lower fence (but we will add them back later with asterisks)  Draw the horizontal lines to the new minimum and new maximum ●To draw this boxplot (in a slightly different way than the text)  Draw the center box and mark the median, as before  Compute the upper fence and the lower fence  Temporarily remove the outliers as identified by the upper fence and the lower fence (but we will add them back later with asterisks)  Draw the horizontal lines to the new minimum and new maximum  Mark each of the outliers with an asterisk

14. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 5 – Slide 14 of 21 Chapter 3 – Section 5 ●To draw this boxplot Draw the middle box and the median Draw in the fences, remove the outliers (temporarily) Draw the minimum and maximum Draw the outliers as asterisks

15. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 5 – Slide 15 of 21 Chapter 3 – Section 5 ●The distribution shape and boxplot are related  Symmetry (or lack of symmetry)  Quartiles  Maximum and minimum ●The distribution shape and boxplot are related  Symmetry (or lack of symmetry)  Quartiles  Maximum and minimum ●Relate the distribution shape to the boxplot for  Symmetric distributions  Skewed left distributions  Skewed right distributions

16. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 5 – Slide 16 of 21 Chapter 3 – Section 5 ●Symmetric distributions DistributionBoxplot Q 1 is equally far from the median as Q 3 is The median line is in the center of the box Q1Q1 MQ3Q3 Q1Q1 MQ3Q3 DistributionBoxplot Q 1 is equally far from the median as Q 3 is The median line is in the center of the box The min is equally far from the median as the max is The left whisker is equal to the right whisker Q1Q1 MQ3Q3 MinMaxQ1Q1 MQ3Q3 MinMax

17. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 5 – Slide 17 of 21 Chapter 3 – Section 5 ●Skewed left distributions DistributionBoxplot Q 1 is further from the median than Q 3 is The median line is to the right of center in the box Q1Q1 MQ3Q3 Q1Q1 MQ3Q3 DistributionBoxplot Q 1 is further from the median than Q 3 is The median line is to the right of center in the box The min is further from the median than the max is The left whisker is longer than the right whisker MinMaxQ1Q1 MQ3Q3 MinMaxQ1Q1 MQ3Q3

18. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 5 – Slide 18 of 21 Chapter 3 – Section 5 ●Skewed right distributions DistributionBoxplot Q 1 is closer to the median than Q 3 is The median line is to the left of center in the box Q1Q1 MQ3Q3 Q1Q1 MQ3Q3 DistributionBoxplot Q 1 is closer to the median than Q 3 is The median line is to the left of center in the box The min is closer to the median than the max is The left whisker is shorter than the right whisker MinMaxQ1Q1 MQ3Q3 MinMaxQ1Q1 MQ3Q3

19. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 5 – Slide 19 of 21 Chapter 3 – Section 5 ●We can compare two distributions by examining their boxplots ●We draw the boxplots on the same horizontal scale ●We can compare two distributions by examining their boxplots ●We draw the boxplots on the same horizontal scale  We can visually compare the centers  We can visually compare the spreads  We can visually compare the extremes

20. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 5 – Slide 20 of 21 Chapter 3 – Section 5 ●Comparing the “flight” with the “control” samples Center Spread

21. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 5 – Slide 21 of 21 Summary: Chapter 3 – Section 5 ●5-number summary  Minimum, first quartile, median, third quartile maximum  Resistant measures of center (median) and spread (interquartile range) ●Boxplots  Visual representation of the 5-number summary  Related to the shape of the distribution  Can be used to compare multiple distributions