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Analyzing Data Sets: Range, Median, and Interquartile Range

This text provides an analysis of various data sets, including the range, median, and interquartile range. It compares the spreads of different sets and identifies patterns.

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Analyzing Data Sets: Range, Median, and Interquartile Range

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  1. 1. Find the range, median and interquartile range of the following data sets: 13, 15, 9, 35, 25 (i) Order the set: 9, 13, 15, 25, 35 Range: 35 – 9 = 26 (highest – lowest) Median: 9, 13, 15, 25, 35 15 IQR: Middle of the bottom half is between 9 and 13 Middle of the top half is between 25 and 35 IQR = Q3 – Q1 = 30 – 11 = 19

  2. 1. Find the range, median and interquartile range of the following data sets: 6, 1, 3, 8, 5, 11, 1, 5 (ii) Order the set: 1, 1, 3, 5, 5, 6, 8, 11 Range: 11 – 1 = 10 (highest – lowest) Median: IQR: Middle of the bottom half is between 1 and 3 Middle of the top half is between 6 and 8 IQR = Q3 – Q1 = 7 – 2 = 5

  3. 1. Find the range, median and interquartile range of the following data sets: 5, 9, 17, 25, 36, 45 (iii) Order the set: 5, 9, 17, 25, 36, 45 Range: 45 – 5 = 40 (highest – lowest) Median: IQR: Q1 = 9 Middle of the bottom half Q3 = 36 Middle of the top half IQR = Q3 – Q1 = 36 – 9 IQR = 27

  4. 1. Find the range, median and interquartile range of the following data sets: 8, 11, 32, 29, 9, 34 (iv) Order the set: 8, 9, 11, 29, 32, 34 Range: 34 – 8 = 26 Median: (Between 11 and 29) IQR: Q1 = 9 Middle of the bottom half Q3 = 32 Middle of the top half IQR = Q3 – Q1 = 32 – 9 IQR = 23

  5. 1. Find the range, median and interquartile range of the following data sets: 3, 14, 28, 22, 5, 9 (v) Order the data: 3, 5, 9, 14, 22, 28 Range: 28 – 3 = 25 Median: Between 9 and 14

  6. 1. Find the range, median and interquartile range of the following data sets: 3, 14, 28, 22, 5, 9 (v) Order the data: 3, 5, 9, 14, 22, 28 IQR: Q1 = 5 Middle of the bottom half Q3 = 22 Middle of the top half IQR = Q3 – Q1 = 22 – 5 IQR = 17

  7. 2. The ages of two families are shown below: Smith family: 1, 4, 7, 12, 40, 46 Dunne family: 3, 7, 15, 18, 44, 48 Compare the spread of the ages in both families under the headings: Range (i) Range: Smith Family = 46 – 1 = 45 Dunne Family = 48 – 3 = 45 Both families have the same range of ages.

  8. 2. The ages of two families are shown below: Smith family: 1, 4, 7, 12, 40, 46 Dunne family: 3, 7, 15, 18, 44, 48 Compare the spread of the ages in both families under the headings: Median (ii) Median: Smith Family = Dunne Family = The Dunne Family has a higher median age than the Smith Family. This indicates the children are of an older age which we can see is the case.

  9. 2. The ages of two families are shown below: Smith family: 1, 4, 7, 12, 40, 46 Dunne family: 3, 7, 15, 18, 44, 48 Compare the spread of the ages in both families under the headings: Interquartile range (iii) Interquartile range: Smith: IQR: Q1 = 4 Q3 = 40 IQR = Q3 – Q1 = 40 – 4 IQR = 36

  10. 2. The ages of two families are shown below: Smith family: 1, 4, 7, 12, 40, 46 Dunne family: 3, 7, 15, 18, 44, 48 Compare the spread of the ages in both families under the headings: Interquartile range (iii) Dunne: IQR: Q1 = 7 Q3 = 44 IQR = Q3 – Q1 = 44 – 7 IQR = 37 The IQR is very similar in both families which indicates that without the extreme lowest and highest ages the range of families ages are much more similar. The spread of the middle 50% of the ages is similar for both families.

  11. 3. The numbers of goals scored in 10 games by the top two teams in an inter-schools soccer league are shown below: School A: 3, 3, 2, 1, 3, 2, 1, 1, 2, 3 School B: 1, 5, 2, 2, 4, 1, 3, 3, 2, 4 Compare the spread of the scores of each team under the following headings: (a) (i) Range Range : School A = 3 – 1 = 2 School B = 5 – 1 = 4 School B has a significantly higher range of results than School A which indicates more varied sporting performances.

  12. 3. The numbers of goals scored in 10 games by the top two teams in an inter-schools soccer league are shown below: School A: 3, 3, 2, 1, 3, 2, 1, 1, 2, 3 School B: 1, 5, 2, 2, 4, 1, 3, 3, 2, 4 Compare the spread of the scores of each team under the following headings: (a) (ii) Median Median : School A: 1, 1, 1, 2, 2, 2, 3, 3, 3, 3 n = 10 (Between 5th and 6th position)

  13. 3. The numbers of goals scored in 10 games by the top two teams in an inter-schools soccer league are shown below: School A: 3, 3, 2, 1, 3, 2, 1, 1, 2, 3 School B: 1, 5, 2, 2, 4, 1, 3, 3, 2, 4 Compare the spread of the scores of each team under the following headings: (a) (ii) Median Median : School B: 1, 1, 2, 2, 2, 3, 3, 4, 4, 5 n = 10 (Between 5th and 6th position) School B has a higher median score which indicates that despite having more varied results they have more higher results than school A.

  14. 3. The numbers of goals scored in 10 games by the top two teams in an inter-schools soccer league are shown below: School A: 3, 3, 2, 1, 3, 2, 1, 1, 2, 3 School B: 1, 5, 2, 2, 4, 1, 3, 3, 2, 4 Compare the spread of the scores of each team under the following headings: (a) (iii) Interquartile range IQR: School A : Q1 = 1 Q3 = 3 IQR = Q3 – Q1 = 3 – 1 = 2

  15. 3. The numbers of goals scored in 10 games by the top two teams in an inter-schools soccer league are shown below: School A: 3, 3, 2, 1, 3, 2, 1, 1, 2, 3 School B: 1, 5, 2, 2, 4, 1, 3, 3, 2, 4 Compare the spread of the scores of each team under the following headings: (a) (iii) Interquartile range IQR: School B : Q1 = 2 Q3 = 4 IQR = Q3 – Q1 = 4 – 2 = 2 Without the outliers, the schools have the same range.

  16. 3. The numbers of goals scored in 10 games by the top two teams in an inter-schools soccer league are shown below: School A: 3, 3, 2, 1, 3, 2, 1, 1, 2, 3 School B: 1, 5, 2, 2, 4, 1, 3, 3, 2, 4 Compare the spread of the scores of each team under the following headings: (a) (iv) Mean Mean: School A: = 2·1

  17. 3. The numbers of goals scored in 10 games by the top two teams in an inter-schools soccer league are shown below: School A: 3, 3, 2, 1, 3, 2, 1, 1, 2, 3 School B: 1, 5, 2, 2, 4, 1, 3, 3, 2, 4 Compare the spread of the scores of each team under the following headings: (a) (iv) Mean School A: = 2·7 School B has a higher mean score which reflects the effect the outliers have on the mean.

  18. 3. The numbers of goals scored in 10 games by the top two teams in an inter-schools soccer league are shown below: School A: 3, 3, 2, 1, 3, 2, 1, 1, 2, 3 School B: 1, 5, 2, 2, 4, 1, 3, 3, 2, 4 Why is the range not a very good measure of spread? (b) The range is affected by outliers and so is not an accurate measure of spread. Which is the more reliable average, the mean or the median? Explain your reasoning. (c) Median is more reliable because it is not affected by outliers.

  19. 4. The stem-and-leaf diagram below shows the average time (in mins) it took a group of students to solve a maths problem. Key: 1 | 1 = 11 mins Find the range of this data. (i) Range: 56 – 11 = 45 minutes

  20. 4. The stem-and-leaf diagram below shows the average time (in mins) it took a group of students to solve a maths problem. Key: 1 | 1 = 11 mins Find the median time to solve the problem. (ii) Median: Median is in the 10th position, so it is 32 minutes.

  21. 4. The stem-and-leaf diagram below shows the average time (in mins) it took a group of students to solve a maths problem. Key: 1 | 1 = 11 mins Find the interquartile range of this data set. (iii) IQR: 11, 12, 15, 17, (20), 21, 23, 24, 28, (32), 39, 43, 43, 45, (46), 48, 50, 51, 56 Q1 = 20 Q3 = 46 IQR = Q3 – Q1 = 46 – 20 IQR = 26 minutes

  22. 5. The back-to-back stem-and-leaf diagram below shows the results of two different maths classes on the same test Compare the range of both classes. (i) Range: X’s Classes = 92 – 54 = 38 Y’s Classes = 99 – 51 = 48 Ms Y’s classes have a larger range of maths results than X’s.

  23. 5. The back-to-back stem-and-leaf diagram below shows the results of two different maths classes on the same test Compare the interquartile range for both classes. (ii) IQR: Miss X’s Classes: 54, 56, 59, 62 , 65, 67, 68, 71 , 74, 80, 81, 81 , 91, 92, 92 IQR = Q3 – Q1 = 81 – 62 IQR = 19

  24. 5. The back-to-back stem-and-leaf diagram below shows the results of two different maths classes on the same test Compare the interquartile range for both classes. (ii) IQR: Miss Y’s Classes: 51, 55, 64, 68, 74 , 76, 76, 77, 86 | 86, 86, 86, 94, 96 , 97, 98, 99, 99 IQR = Q3 – Q1 = 96 – 74 IQR = 22 Based on the range of the middle 50%, Miss Y’s results is larger than Miss X’s.

  25. 5. The back-to-back stem-and-leaf diagram below shows the results of two different maths classes on the same test Find the mean, correct to one decimal place, of each class. (iii) Mean: X’s Classes: = 72·86 72·9

  26. 5. The back-to-back stem-and-leaf diagram below shows the results of two different maths classes on the same test Find the mean, correct to one decimal place, of each class. (iii) Mean: Y’s Classes: = 81·55 81·6

  27. 5. The back-to-back stem-and-leaf diagram below shows the results of two different maths classes on the same test Which class do you think is better? Justify your answer. (iv) Ms Y’s Class have better results on average but the higher interquartile range means their results are more varied than Ms X’s class.

  28. 6. The time spent (in minutes) by 20 people waiting at a bus stop has been recorded as follows: 3∙2, 2∙4, 3∙2, 1∙3, 1∙6, 2∙8, 1∙4, 2∙9, 3∙2, 4∙8, 1∙7, 3, 0∙9, 3∙7, 5∙6, 1∙4, 2∙6, 3∙1, 1∙6, 1∙1 Find the median waiting time and the upper and lower quartiles. (i) List in order: 0·9, 1·1, 1·3, 1·4, 1·4, | 1·6, 1·6, 1·7, 2·4, 2·6, | 2·8, 2·9, 3, 3·1, 3·2, | 3·2, 3·2, 3·7, 4·8, 5·6. Median =  Between 10th and 11th position

  29. 6. The time spent (in minutes) by 20 people waiting at a bus stop has been recorded as follows: 3∙2, 2∙4, 3∙2, 1∙3, 1∙6, 2∙8, 1∙4, 2∙9, 3∙2, 4∙8, 1∙7, 3, 0∙9, 3∙7, 5∙6, 1∙4, 2∙6, 3∙1, 1∙6, 1∙1 Find the median waiting time and the upper and lower quartiles. (i) List in order: 0·9, 1·1, 1·3, 1·4, 1·4, | 1·6, 1·6, 1·7, 2·4, 2·6, | 2·8, 2·9, 3, 3·1, 3·2, | 3·2, 3·2, 3·7, 4·8, 5·6. Lower quartileQ1 Upper quartile Q3

  30. 6. The time spent (in minutes) by 20 people waiting at a bus stop has been recorded as follows: 3∙2, 2∙4, 3∙2, 1∙3, 1∙6, 2∙8, 1∙4, 2∙9, 3∙2, 4∙8, 1∙7, 3, 0∙9, 3∙7, 5∙6, 1∙4, 2∙6, 3∙1, 1∙6, 1∙1 Find the range and interquartile range of the waiting time. (ii) Range: 5·6 – 0·9 = 4·7 minutes IQR: Q3 – Q1 = 3·2 – 1·5 = 1·7 minutes

  31. 6. The time spent (in minutes) by 20 people waiting at a bus stop has been recorded as follows: 3∙2, 2∙4, 3∙2, 1∙3, 1∙6, 2∙8, 1∙4, 2∙9, 3∙2, 4∙8, 1∙7, 3, 0∙9, 3∙7, 5∙6, 1∙4, 2∙6, 3∙1, 1∙6, 1∙1 Copy and complete the following statements: (ii) (a) ‘50% of the waiting times were greater than ......... minutes.’ 50% of the waiting times were greater than 2·7 minutes. (b) ‘75% of the waiting times were less than ...... minutes.’ 75% of the waiting times were less than 3·2 minutes. (c) ‘The minimum waiting time was ........ minutes and the maximum waiting time was ..... minutes. The waiting times were spread over ...... minutes.’ The minimum waiting time was 0·9 minutes and the maximum waiting time was 5·6 minutes. The waiting times were spread over 4·7 minutes.

  32. 7. Rory’s golf scores for his last 10 rounds were: 90, 106, 84, 103, 112, 100, 105, 81, 104, 98 • Use this data to calculate: (a) • (i) The median List in order: 81, 84, 90, 98, 100, | 103, 104, 105, 106, 112 Median:  Between 5th and 6th position

  33. 7. Rory’s golf scores for his last 10 rounds were: 90, 106, 84, 103, 112, 100, 105, 81, 104, 98 • Use this data to calculate: (a) (ii) The lower quartile List in order: 81, 84, 90, 98, 100, | 103, 104, 105, 106, 112 The lower quartile Q1 = 90

  34. 7. Rory’s golf scores for his last 10 rounds were: 90, 106, 84, 103, 112, 100, 105, 81, 104, 98 • Use this data to calculate: (a) (iii) The upper quartile List in order: 81, 84, 90, 98, 100, | 103, 104, 105 , 106, 112 The upper quartile Q3 = 105

  35. 7. Rory’s golf scores for his last 10 rounds were: 90, 106, 84, 103, 112, 100, 105, 81, 104, 98 • Use this data to calculate: (a) (iv) The interquartile range IQR = Q3 – Q1 = 105 – 90 = 15

  36. 7. Rory’s golf scores for his last 10 rounds were: 90, 106, 84, 103, 112, 100, 105, 81, 104, 98 • Shane is a member in Rory’s club. He has a median = 100, IQR = 7 and range = 19. Who should be picked for the last place on their club’s team. Justify your answer. (b) Rory has a range of 112 – 81 = 31 and Shane has a range of 19. This means that Rory’s scores are more widely spread than Shane’s. Rory has an IQR of 15 and Shane has an IQR of 7. This means that when outliers are removed, Rory’s scores are still more widely spread than Shane’s. Rory has a median of 101·5 and Shane has a median of 100. This means that Rory’s scores are slightly higher than Shane’s.

  37. 7. Rory’s golf scores for his last 10 rounds were: 90, 106, 84, 103, 112, 100, 105, 81, 104, 98 • Shane is a member in Rory’s club. He has a median = 100, IQR = 7 and range = 19. Who should be picked for the last place on their club’s team. Justify your answer. (b) So, while Shane is more consistent, Rory scores higher points. Therefore, Rory might be the better pick for the team. Either answer is acceptable, so long as you give valid reasons for your selection.

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