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10.2 Analyzing Data

10.2 Analyzing Data. Mode, Median, Mean… …and a Box with Whiskers. Statistics. A Statistic is a quantity that is computed from a data set. Statistics. A Statistic is a quantity that is computed from a data set.

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10.2 Analyzing Data

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  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.

  6. 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:

  7. 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

  8. 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

  9. 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

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

  11. 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.

  12. 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

  13. 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.

  14. 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:

  15. 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

  16. 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

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

  19. 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

  20. 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: (3+5+7+10+14+15)/6 = 9

  21. 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: (3+5+7+10+14+15)/6 = 9 • Note, the mean is usually not equal to the median. The median of the above set would be (7+10)/2 = 8.5

  22. Example • Find the Mode, Median, and Mean of the following set:

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

  24. 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: (19.5+20.1)/2 = 19.8

  26. 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: (19.5+20.1)/2 = 19.8 • Finally, the mean is: (18.0+18.5+18.5+19.5+19.5+20.1+21.3+22.3+24.5+27.2) 10 = 20.94

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

  28. 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.

  29. 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

  30. 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

  31. 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

  32. 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

  33. 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

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

  36. 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

  37. 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.

  38. 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 = (26+27)/2 = 26.5

  39. 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 = (26+27)/2 = 26.5 • Lower Quartile = (22+25)/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 = (26+27)/2 = 26.5 • Lower Quartile = (22+25)/2 = 23.5 • Upper Quartile = (32+32)/2 = 32

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

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

  43. 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

  44. 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 = (65+63)/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 = (65+63)/2 = 64 • LQ = (55+57)/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 = (65+63)/2 = 64 • LQ = (55+57)/2 = 56 • IQR = 64 – 56 = 8

  48. 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.

  49. 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 = (65+63)/2 = 64 • LQ = (55+57)/2 = 56 • IQR = 64 – 56 = 8

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