10.2 Analyzing Data

1
10.2 Analyzing Data

• Mode, Median, Mean
• and a Box with Whiskers.

2
Statistics

• A Statistic is a quantity that is computed from a
  data set.

3
Statistics

• A Statistic is a quantity that is computed from a
  data set.
• There are thousands of examples of statistics
  that are used in many different applications.

4
Statistics

• A Statistic is a quantity that is computed from a
  data set.
• There are thousands of examples of statistics
  that are used in many different applications.
• Some common statistics mathematical
  average(mean), standard deviation, variance,
  mode, and median.

5
Central Tendency

• A Measure of Central Tendencies is a statistic
  that is used to somehow find an average of a
  data set.

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Central Tendency

• A Measure of Central Tendencies is a statistic
  that is used to somehow find an average of a
  data set.
• The three measures that we will use to find
  central tendencies are

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Central Tendency

• A Measure of Central Tendencies is a statistic
  that is used to somehow find an average of a
  data set.
• The three measures that we will use to find
  central tendencies are
• The Mode

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Central Tendency

• A Measure of Central Tendencies is a statistic
  that is used to somehow find an average of a
  data set.
• The three measures that we will use to find
  central tendencies are
• The Mode
• The Median

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Central Tendency

• A Measure of Central Tendencies is a statistic
  that is used to somehow find an average of a
  data set.
• The three measures that we will use to find
  central tendencies are
• The Mode
• The Median
• The Mean

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The Mode

• The number which appears the most often in the
  data list is called The Mode.

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The Mode

• The number which appears the most often in the
  data list is called The Mode.
• A data set can have more than one mode, if two
  (or more) numbers have the same frequency and
  appear more frequently than the other numbers,
  then they are both the mode. If all the numbers
  have the same frequency, then there is no mode.

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The Mode

• The number which appears the most often in the
  data list is called The Mode.
• A data set can have more than one mode, if two
  (or more) numbers have the same frequency and
  appear more frequently than the other numbers,
  then they are both the mode. If all the numbers
  have the same frequency, then there is no mode.
• Example
• 8, 8, 9, 10, 11, 11, 11, 15, 17, 23

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The Mode

• The number which appears the most often in the
  data list is called The Mode.
• A data set can have more than one mode, if two
  (or more) numbers have the same frequency and
  appear more frequently than the other numbers,
  then they are both the mode. If all the numbers
  have the same frequency, then there is no mode.
• Example
• 8, 8, 9, 10, 11, 11, 11, 15, 17, 23
• The mode of this list of numbers is 11.

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The Median

• To find the Median of a set of numbers, put them
  in numerical order. If the number of data points
  in your set is odd, there will be one number in
  the middle, that number is the median

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The Median

• To find the Median of a set of numbers, put them
  in numerical order. If the number of data points
  in your set is odd, there will be one number in
  the middle, that number is the median
• 5, 7, 9, 11,13

Median
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The Median

• To find the Median of a set of numbers, put them
  in numerical order. If the number of data points
  in your set is odd, there will be one number in
  the middle, that number is the median
• 5, 7, 9, 11,13
• If the number of data points in the set is even,
  the median is the mathematical average of the two
  middle numbers

Median
17
The Median

• To find the Median of a set of numbers, put them
  in numerical order. If the number of data points
  in your set is odd, there will be one number in
  the middle, that number is the median
• 5, 7, 9, 11,13
• If the number of data points in the set is even,
  the median is the mathematical average of the two
  middle numbers
• 9, 10, 12, 14

Median
Median is 11
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The Mean

• The Mean of a set of numbers is the mathematical
  average of the set.

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The Mean

• The Mean of a set of numbers is the mathematical
  average of the set.
• For this set 3, 5, 7, 10, 14, 15

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The Mean

• The Mean of a set of numbers is the mathematical
  average of the set.
• For this set 3, 5, 7, 10, 14, 15
• The mean would be (357101415)/6 9

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The Mean

• The Mean of a set of numbers is the mathematical
  average of the set.
• For this set 3, 5, 7, 10, 14, 15
• The mean would be (357101415)/6 9
• Note, the mean is usually not equal to the
  median. The median of the above set would be
  (710)/2 8.5

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Example

• Find the Mode, Median, and Mean of the following
  set

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Example

• Find the Mode, Median, and Mean of the following
  set
• First thing to do is put the list in numerical
  order

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Example

• Find the Mode, Median, and Mean of the following
  set
• First thing to do is put the list in numerical
  order
• Then we can see there are two numbers that appear
  twice, so there are two modes 18.5 19.5

25
Example

• Find the Mode, Median, and Mean of the following
  set
• First thing to do is put the list in numerical
  order
• Then we can see there are two numbers that appear
  twice, so there are two modes 18.5 19.5
• Since there are ten numbers in the list the
  median is the average of the fifth and sixth
  numbers (19.520.1)/2 19.8

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Example

• Find the Mode, Median, and Mean of the following
  set
• First thing to do is put the list in numerical
  order
• Then we can see there are two numbers that appear
  twice, so there are two modes 18.5 19.5
• Since there are ten numbers in the list the
  median is the average of the fifth and sixth
  numbers (19.520.1)/2 19.8
• Finally, the mean is

(18.018.518.519.519.520.121.322.324.527.2
) 10
20.94
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Quartiles

• The Quartiles of a data set give us break downs
  of the data into four groups with approximately
  the same number of points.

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Quartiles

• The Quartiles of a data set give us break downs
  of the data into four groups with approximately
  the same number of points.
• The Lower Quartile(LQ) of a data set is the
  median of lower half of the ordered data.

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Quartiles

• The Quartiles of a data set give us break downs
  of the data into four groups with approximately
  the same number of points.
• The Lower Quartile(LQ) of a data set is the
  median of lower half of the ordered data. Called
  the 1st quartile.
• The Upper Quartile(UQ) of a data set is the
  median of the upper half of the data. Called the
  3rd quartile
• The median of the entire set of data is called
  the 2nd quartile

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Quartiles

• The Quartiles of a data set give us break downs
  of the data into four groups with approximately
  the same number of points.
• The Lower Quartile(LQ) of a data set is the
  median of lower part of the ordered data.
• The Upper Quartile(UQ) of a data set is the
  median of the upper part of the data.
• 2, 3, 4, 5, 6, 7, 8

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Quartiles

• The Quartiles of a data set give us break downs
  of the data into four groups with approximately
  the same number of points.
• The Lower Quartile(LQ) of a data set is the
  median of lower part of the ordered data.
• The Upper Quartile(UQ) of a data set is the
  median of the upper part of the data.
• 2, 3, 4, 5, 6, 7, 8

median
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Quartiles

• The Quartiles of a data set give us break downs
  of the data into four groups with approximately
  the same number of points.
• The Lower Quartile(LQ) of a data set is the
  median of lower part of the ordered data.
• The Upper Quartile(UQ) of a data set is the
  median of the upper part of the data.
• 2, 3, 4, 5, 6, 7, 8

Lower Part
Upper Part
median
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Quartiles

• The Quartiles of a data set give us break downs
  of the data into four groups with approximately
  the same number of points.
• The Lower Quartile(LQ) of a data set is the
  median of lower part of the ordered data.
• The Upper Quartile(UQ) of a data set is the
  median of the upper part of the data.
• 2, 3, 4, 5, 6, 7, 8

Lower Quartile
Lower Part
Upper Part
median
34
Quartiles

• The Quartiles of a data set give us break downs
  of the data into four groups with approximately
  the same number of points.
• The Lower Quartile(LQ) of a data set is the
  median of lower part of the ordered data.
• The Upper Quartile(UQ) of a data set is the
  median of the upper part of the data.
• 2, 3, 4, 5, 6, 7, 8

Lower Quartile
Upper Quartile
Lower Part
Upper Part
median
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Example

• Find the median, and upper and lower quartiles of
  this set
• 22, 19, 27, 32, 38, 25, 32, 26

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Example

• Find the median, and upper and lower quartiles of
  this set
• 22, 19, 27, 32, 38, 25, 32, 26
• Again, first step, order the data
• 19, 22, 25, 26, 27, 32, 32, 38

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Example

• Find the median, and upper and lower quartiles of
  this set
• 22, 19, 27, 32, 38, 25, 32, 26
• Again, first step, order the data
• 19, 22, 25, 26, 27, 32, 32, 38
• So, there are eight numbers, the median is the
  average of the fourth and fifth numbers. The
  lower quartile is the median of the first four
  numbers, and the upper quartile is the median of
  the last four numbers.

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Example

• Find the median, and upper and lower quartiles of
  this set
• 22, 19, 27, 32, 38, 25, 32, 26
• Again, first step, order the data
• 19, 22, 25, 26, 27, 32, 32, 38
• So, there are eight numbers, the median is the
  average of the fourth and fifth numbers. The
  lower quartile is the median of the first four
  numbers, and the upper quartile is the median of
  the last four numbers.
• Median (2627)/2 26.5

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Example

• Find the median, and upper and lower quartiles of
  this set
• 22, 19, 27, 32, 38, 25, 32, 26
• Again, first step, order the data
• 19, 22, 25, 26, 27, 32, 32, 38
• So, there are eight numbers, the median is the
  average of the fourth and fifth numbers. The
  lower quartile is the median of the first four
  numbers, and the upper quartile is the median of
  the last four numbers.
• Median (2627)/2 26.5
• Lower Quartile (2225)/2 23.5

40
Example

• Find the median, and upper and lower quartiles of
  this set
• 22, 19, 27, 32, 38, 25, 32, 26
• Again, first step, order the data
• 19, 22, 25, 26, 27, 32, 32, 38
• So, there are eight numbers, the median is the
  average of the fourth and fifth numbers. The
  lower quartile is the median of the first four
  numbers, and the upper quartile is the median of
  the last four numbers.
• Median (2627)/2 26.5
• Lower Quartile (2225)/2 23.5
• Upper Quartile (3232)/2 32

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Interquartile Range

• The distance between the upper and lower
  quartiles is called the Interquartile Range(IQR).
  Called the 4th quartile.

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Interquartile Range

• The distance between the upper and lower
  quartiles is called the Interquartile Range(IQR).
• So, IQR Upper Quartile - Lower Quartile

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Interquartile Range

• The distance between the upper and lower
  quartiles is called the Interquartile Range(IQR).
• So, IQR Upper Quartile - Lower Quartile
• Take the ordered Set
• 28, 55, 57, 58, 61, 61, 63, 65, 83

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Interquartile Range

• The distance between the upper and lower
  quartiles is called the Interquartile Range(IQR).
• So, IQR Upper Quartile - Lower Quartile
• Take the ordered Set
• 28, 55, 57, 58, 61, 61, 63, 65, 83
• Median 61

45
Interquartile Range

• The distance between the upper and lower
  quartiles is called the Interquartile Range(IQR).
• So, IQR Upper Quartile - Lower Quartile
• Take the ordered Set
• 28, 55, 57, 58, 61, 61, 63, 65, 83
• Median 61
• UQ (6563)/2 64

46
Interquartile Range

• The distance between the upper and lower
  quartiles is called the Interquartile Range(IQR).
• So, IQR Upper Quartile - Lower Quartile
• Take the ordered Set
• 28, 55, 57, 58, 61, 61, 63, 65, 83
• Median 61
• UQ (6563)/2 64
• LQ (5557)/2 56

47
Interquartile Range

• The distance between the upper and lower
  quartiles is called the Interquartile Range(IQR).
• So, IQR Upper Quartile - Lower Quartile
• Take the ordered Set
• 28, 55, 57, 58, 61, 61, 63, 65, 83
• Median 61
• UQ (6563)/2 64
• LQ (5557)/2 56
• IQR 64 56 8

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Outliers

• An Outlier is a number that is so far above the
  data set or below most of the data set as to be
  considered abnormal and therefore of questionable
  accuracy.

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Outliers

• An Outlier is a number that is so far above the
  data set or below most of the data set as to be
  considered abnormal and therefore of questionable
  accuracy.
• For many purposes an outlier is defined to be any
  data point that is more than 1.5 IQRs below the
  lower quartile or above the upper quartile.

50
Outliers

• An Outlier is a number that is so far above the
  data set or below most of the data set as to be
  considered abnormal and therefore of questionable
  accuracy.
• For many purposes an outlier is defined to be any
  data point that is more than 1.5 IQRs below the
  lower quartile or above the upper quartile.
• So, for our last example
• 28, 55, 57, 58, 61, 61, 63, 65, 83
• UQ (6563)/2 64
• LQ (5557)/2 56
• IQR 64 56 8

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Outliers

• An Outlier is a number that is so far above the
  data set or below most of the data set as to be
  considered abnormal and therefore of questionable
  accuracy.
• For many purposes an outlier is defined to be any
  data point that is more than 1.5 IQRs below the
  lower quartile or above the upper quartile.
• So, for our last example
• 28, 55, 57, 58, 61, 61, 63, 65, 83
• UQ (6563)/2 64
• LQ (5557)/2 56
• IQR 64 56 8
• So any number below LQ 1.5(IQR) 56 1.5(8)
  44 or any number above UQ 1.5(IQR) 64
  1.5(8) 78 is an outlier.

52
Outliers

• An Outlier is a number that is so far above the
  data set or below most of the data set as to be
  considered abnormal and therefore of questionable
  accuracy.
• For many purposes an outlier is defined to be any
  data point that is more than 1.5 IQRs below the
  lower quartile or above the upper quartile.
• So, for our last example
• 28, 55, 57, 58, 61, 61, 63, 65, 83
• UQ (6563)/2 64
• LQ (5557)/2 56
• IQR 64 56 8
• So any number below LQ 1.5(IQR) 56 1.5(8)
  44 or any number above UQ 1.5(IQR) 64
  1.5(8) 78 is an outlier.
• Therefore the outliers of this data set are 28
  83

53
Box and Whisker Plot

• A Box and Whisker Plot is a graphic that provides
  an easy way of comparing the mean, quartiles, and
  outliers of different data sets.

54
Box and Whisker Plot

• A Box and Whisker Plot is a graphic that provides
  an easy way of comparing the mean, quartiles, and
  outliers of different data sets.
• We put the data set on a number line

10
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30
40
60
50
70
55
Box and Whisker Plot

• A Box and Whisker Plot is a graphic that provides
  an easy way of comparing the mean, quartiles, and
  outliers of different data sets.
• We put the data set on a number line
• We mark the outliers with a star

10
20
30
40
60
50
70
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Box and Whisker Plot

• A Box and Whisker Plot is a graphic that provides
  an easy way of comparing the mean, quartiles, and
  outliers of different data sets.
• We put the data set on a number line
• We mark the outliers with a starmark the highest
  and lowest data points which are not outliers
  with connected black circles

10
20
30
40
60
50
70
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Box and Whisker Plot

• A Box and Whisker Plot is a graphic that provides
  an easy way of comparing the mean, quartiles, and
  outliers of different data sets.
• We put the data set on a number line
• We mark the outliers with a starmark the highest
  and lowest data points which are not outliers
  with connected black circlesput a box between
  the quartiles

10
20
30
40
60
50
70
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Box and Whisker Plot

• A Box and Whisker Plot is a graphic that provides
  an easy way of comparing the mean, quartiles, and
  outliers of different data sets.
• We put the data set on a number line
• We mark the outliers with a starmark the highest
  and lowest data points which are not outliers
  with connected black circlesput a box between
  the quartiles and a line through the median.

10
20
30
40
60
50
70
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Comparing Box and Whisker Plot

• We can use Box and Whisker Plots to compare
  statistics between two data sets
• We can see that in general the scores in the
  first approval poll are lower and not as wide
  spread as those of the second approval poll.

Approval Poll One
Approval Poll Two
60
Example

• Find the Mode, Median, Mean, Upper Quartile and
  Lower Quartile for this data set, and construct a
  Box and Whisker Plot.

61
Example

• Find the Mode, Median, Mean, Upper Quartile and
  Lower Quartile for this data set, and construct a
  Box and Whisker Plot.
• Mode There are no duplicates, so there is no
  mode.

62
Example

• Find the Mode, Median, Mean, Upper Quartile and
  Lower Quartile for this data set, and construct a
  Box and Whisker Plot.
• Mode There are no duplicates, so there is no
  mode.
• Median There are 12 numbers, so we average the
  two middle numbers (67.1 67.3)/2 67.2

63
Example

• Find the Mode, Median, Mean, Upper Quartile and
  Lower Quartile for this data set, and construct a
  Box and Whisker Plot.
• Mode There are no duplicates, so there is no
  mode.
• Median There are 12 numbers, so we average the
  two middle numbers (67.1 67.3)/2 67.2
• Mean Add em up and divide by 12 (807.1)/12
  67.26

64
Example

• Find the Mode, Median, Mean, Upper Quartile and
  Lower Quartile for this data set, and construct a
  Box and Whisker Plot.
• Mode There are no duplicates, so there is no
  mode.
• Median There are 12 numbers, so we average the
  two middle numbers (67.1 67.3)/2 67.2
• Mean Add em up and divide by 12 (807.2)/12
  67.26
• UQ (68.2 68.5)/2 68.35

65
Example

• Find the Mode, Median, Mean, Upper Quartile and
  Lower Quartile for this data set, and construct a
  Box and Whisker Plot.
• Mode There are no duplicates, so there is no
  mode.
• Median There are 12 numbers, so we average the
  two middle numbers (67.1 67.3)/2 67.2
• Mean Add em up and divide by 12 (807.1)/12
  67.26
• UQ (68.2 68.5)/2 68.35
• LQ (65.9 66.2)/2 66.05

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Example Continued

• To construct the Box and Whisker Plot we need to
  find the outliers.

67
Example Continued

• To construct the Box and Whisker Plot we need to
  find the outliers.
• The IQR UQ LQ

68
Example Continued

• To construct the Box and Whisker Plot we need to
  find the outliers.
• The IQR UQ LQ 68.35 66.05 2.3

69
Example Continued

• To construct the Box and Whisker Plot we need to
  find the outliers.
• The IQR UQ LQ 68.35 66.05 2.3
• The outliers are any data points
• less than LQ 2.3(1.5) 66.05 2.3(1.5)
• or greater than UQ 2.3(1.5)

70
Example Continued

• To construct the Box and Whisker Plot we need to
  find the outliers.
• The IQR UQ LQ 68.35 66.05 2.3
• The outliers are any data points
• less than LQ 2.3(1.5) 66.05 2.3(1.5)
  62.6
• or greater than UQ 2.3(1.5) 71.8

71
Example Continued

• To construct the Box and Whisker Plot we need to
  find the outliers.
• The IQR UQ LQ 68.35 66.05 2.3
• The outliers are any data points
• less than LQ (1.5)2.3 66.05 3.45 62.6
• or greater than UQ 2.3(1.5) 71.8
• So, the outlier is 62.0

72
Example Conclusion

• So we have

73
Example Conclusion

• So we have Median 67.2

74
Example Conclusion

• So we have Median 67.2
• LQ 66.05

75
Example Conclusion

• So we have Median 67.2
• LQ 66.05
• UQ 68.35

76
Example Conclusion

• So we have Median 67.2
• LQ 66.05
• UQ 68.35
• Outliers 62.0

77
Example Conclusion

• So we have Median 67.2
• LQ 66.05
• UQ 68.35
• Outliers 62.0

78
Example Conclusion

• So we have Median 67.2
• LQ 66.05
• UQ 68.35
• Outliers 62.0

62.0
79
Example Conclusion

• So we have Median 67.2
• LQ 66.05
• UQ 68.35
• Outliers 62.0

65.6
70.9
62.0
80
Example Conclusion

• So we have Median 67.2
• LQ 66.05
• UQ 68.35
• Outliers 62.0

65.6
70.9
62.0
66.05
68.35
81
Example Conclusion

• So we have Median 67.2
• LQ 66.05
• UQ 68.35
• Outliers 62.0

65.6
70.9
62.0
66.05
67.2
68.35
82
Example Conclusion

• So we have Median 67.2
• LQ 66.05
• UQ 68.35
• Outliers 62.0

65.6
70.9
62.0
66.05
67.2
68.35
LQ
UQ
Outlier
median