The Five-Number SummaryAnd Boxplots

Chapter 3 – Section 5

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

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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 (Q1 or P25)

The median (M or Q2 or P50)

The third quartile (Q3 or P75)

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 for1, 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 (16 + 19) / 2 = 17.5 Q1 =, Find the middle value of lower half, Q1 = 7

Q3 = Find the middle value of upper half, Q3 = 27

The maximum = 54

● The five-number summary is1, 7, 17.5, 27, 54

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

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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 Q1, Q2, and Q3

● 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 Q1 and the right edge at Q3

Draw a line inside the box at M = Q2

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● 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 Q1 and the right edge at Q3

Draw a line inside the box at M = Q2

Draw a horizontal line from the Q1 edge of the box to the minimum and one from the Q3 edge of the box to the maximum

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● 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 Q1 and the right edge at Q3

Draw a line inside the box at M = Q2

Draw a horizontal line from the Q1 edge of the box to the minimum and one from the Q3 edge of the box to the maximum

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● 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 Q1, Q2, and Q3

● 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 distributionsDistribution Boxplot

Q1 is equally far from the median as Q3 is

The median line is in thecenter of the box

Q1 M Q3Q1 M Q3

Distribution Boxplot

Q1 is equally far from the median as Q3 is

The median line is in thecenter of the box

The min is equally far fromthe median as the max is

The left whisker is equalto the right whisker

Q1 M Q3Min MaxQ1 M Q3Min Max

● Skewed left distributionsDistribution Boxplot

Q1 is further from the median than Q3 is

The median line is to theright of center in the box

Q1 MQ3Q1 MQ3

Distribution Boxplot

Q1 is further from the median than Q3 is

The median line is to theright of center in the box

The min is further from the median than the max is

The left whisker is longerthan the right whisker

Min MaxQ1 MQ3Min MaxQ1 MQ3

● Skewed right distributionsDistribution Boxplot

Q1 is closer to the medianthan Q3 is

The median line is to theleft of center in the box

Q1M Q3 Q1M Q3

Distribution Boxplot

Q1 is closer to the medianthan Q3 is

The median line is to theleft of center in the box

The min is closer to the median than the max is

The left whisker is shorterthan the right whisker

Min MaxQ1M Q3Min MaxQ1M Q3

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

CenterSpread

Space rats and blood cell mass. Rats were sent into space known as the flightgroup. A control group was kept on earth. The mass of their red blood cells wasmeasured. 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

FINISHED

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