The Five-Number Summary And Boxplots

Chapter 3 – Section 5 ●Learning objectives  Compute the five-number summary  Draw and interpret boxplots 1 2

●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?

●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

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

Chapter 3 – Section 5 ●Compute the five-number summary for 1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 54 ●Calculations  The minimum = 1  M = Find the middle value if even number average the two values ( ) / 2 = 17.5  Q 1 =, Find the middle value of lower half, Q 1 = 7  Q 3 = Find the middle value of upper half, Q 3 = 27  The maximum = 54 ●The five-number summary is 1, 7, 17.5, 27, 54

●Learning objectives  Compute the five-number summary  Draw and interpret boxplots 1 2

●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

●To draw a (basic) boxplot: Calculate the five-number summary 1,7, 17.5, 27, 54 Draw a horizontal line that will cover all the data from the minimum to the maximum label a scale on line. 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 1,7, 17.5, 27, 54 Draw a horizontal line that will cover all the data from the minimum to the maximum label a scale on line. 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

●To draw a (basic) boxplot: Calculate the five-number summary 1,7, 17.5, 27, 54 Draw a horizontal line that will cover all the data from the minimum to the maximum label a scale on line. 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

●To draw a (basic) boxplot Draw the middle box Draw the minimum and maximum Draw in the median

●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)

●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

●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

●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

●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

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

●Comparing the “flight” with the “control” samples Center Spread Space rats and blood cell mass. Rats were sent into space known as the flight group. A control group was kept on earth. The mass of their red blood cells was measured. The data is drawn below in 2 boxplots.

●Comparing the “flight” with the “control” samples Notice all of the control group is above the lower 25% of the flight group

●Comparing the “flight” with the “control” samples Almost 75% of the flight group is below the top 50% of the control group.

Summary ●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

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