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Understanding Measures of Variability in Data Sets

Learn about finding measures like quartiles and range, and how to create box-and-whisker plots to analyze data variability in mathematics.

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Understanding Measures of Variability in Data Sets

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  1. 9-4 Variability Course 3 Warm Up Problem of the Day Lesson Presentation

  2. 9-4 Variability Course 3 Warm Up 1. Order the test scores from least to greatest: 89, 93, 79, 87, 91, 88, 92. 2. Find the median of the test scores. Find the difference. 79, 87, 88, 89, 91, 92, 93 89 3. 17 – 0.9 4. 8.4 – 7. 6 16.1 0.8 3.4 166.9 5. 9.1 – 5.7 6. 190.3 – 23.4

  3. 9-4 Variability Course 3 Problem of the Day What are the possible values for x in the data set 22, 12, 33, 25, and x if the median is 25? any number greater than or equal to 25

  4. 9-4 Variability Course 3 Learn to find measures of variability.

  5. 9-4 Variability Course 3 Insert Lesson Title Here Vocabulary variability quartile box-and-whisker plot

  6. 9-4 Variability Course 3 The table below summarizes a veterinarian’s records for kitten litters born in a given year. While central tendency describes the middle of a data set, variability describes how spread out the data is. Quartiles divide a data set into four equal parts. The third quartile minus the first quartile is the range for the middle half of the data.

  7. 9-4 Variability Third quartile: 5 median of upper half First quartile: 3 median of lower half Median: 4 (second quartile) Course 3 The range of a data set is the largest value minus the smallest value. For the kitten data, the range is 6 — 2 = 4. Kitten Data Lower half Upper half 2 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 The range is affected by outliers, so another measure is often used. Quartiles divide a data set into four equal parts. The third quartile minus the first quartile is the range for the middle half of the data.

  8. 9-4 Variability Course 3 Additional Example 1A: Finding Measures of Variability Find the first and third quartiles for the data set. 15, 83, 75, 12, 19, 74, 21 Order the values. 12 15 19 21 74 75 83 first quartile: 15 third quartile: 75

  9. 9-4 Variability 63 + 75 2 first quartile: = 69 79 + 88 2 third quartile: = 83.5 Course 3 Additional Example 1B: Finding Measures of Variability Find the first and third quartiles for the data set. 75, 61, 88, 79, 79, 99, 63, 77 Order the values. 61 63 75 77 79 79 88 99

  10. 9-4 Variability Course 3 Check It Out: Example 1A Find the first and third quartiles for the data set. 25, 38, 66, 19, 91, 47, 13 Order the values. 13 19 25 38 47 66 91 first quartile: 19 third quartile: 66

  11. 9-4 Variability 33 + 45 2 first quartile: = 39 49 + 59 2 third quartile: = 54 Course 3 Check It Out: Example 1B Find the first and third quartiles for the data set. 45, 31, 59, 49, 49, 69, 33, 47 Order the values. 31 33 45 47 49 49 59 69

  12. 9-4 Variability 1 2 3 4 5 6 7 8 9 Course 3 A box-and-whiskerplot shows the distribution of data. The middle half of the data is represented by a “box” with a vertical line at the median. The lower fourth and upper fourth quarters are represented by “whiskers” that extend to the smallest and largest values. Median First quartile Third quartile

  13. 9-4 Variability 15 + 17 2 21 + 21 2 third quartile: = 21 first quartile: = 16 19 + 19 2 median: = 19 Course 3 Additional Example 2: Making a Box-and-Whisker Plot Use the given data to make a box-and-whisker plot: 21, 25, 15, 13, 17, 19, 19, 21 Step 1. Order the data and find the smallest value, first quartile, median, third quartile, and largest value. 13 15 17 19 19 21 21 25 smallest value: 13 largest value: 25

  14. 9-4 Variability 12 14 16 18 20 22 24 26 28 Course 3 Additional Example 2 Continued Use the given data to make a box-and-whisker plot. Step 2. Draw a number line and plot a point above each value from Step 1. 13 15 17 19 19 21 21 25 smallest value 13 first quartile 16 third quartile 21 largest value 25 median 19

  15. 9-4 Variability 12 14 16 18 20 22 24 26 28 Course 3 Additional Example 2 Continued Use the given data to make a box-and-whisker plot. Step 3. Draw the box and whiskers. 13 15 17 19 19 21 21 25

  16. 9-4 Variability 31 + 33 2 24 + 26 2 third quartile: = 32 first quartile: = 25 29 + 31 2 median: = 30 Course 3 Check It Out: Example 2 Use the given data to make a box-and-whisker plot. 31, 23, 33, 35, 26, 24, 31, 29 Step 1. Order the data and find the smallest value, first quartile, median, third quartile, and largest value. 23 24 26 29 31 31 33 35 smallest value: 23 largest value: 35

  17. 9-4 Variability 22 24 26 28 30 32 34 36 38 Course 3 Check It Out: Example 2 Continued Use the given data to make a box-and-whisker plot. Step 2. Draw a number line and plot a point above each value. 23 24 26 29 31 31 33 35

  18. 9-4 Variability 22 24 26 28 30 32 34 36 38 Course 3 Check It Out: Example 2 Continued Use the given data to make a box-and-whisker plot. Step 3. Draw the box and whiskers. Step 2. Draw a number line and plot a point above each value. 23 24 26 29 31 31 33 35

  19. 9-4 Variability Course 3 Additional Example 3: Comparing Data Sets Using Box-and-Whisker Plots Note: 57 is the first quartile and the median. These box-and-whisker plots compare the ages of the first ten U.S. presidents with the ages of the last ten presidents (through George W. Bush) when they took office.

  20. 9-4 Variability Course 3 Additional Example 3 Continued Note: 57 is the first quartile and the median. A. Compare the medians and ranges. The median for the first ten presidents is slightly greater. The range for the last ten presidents is greater.

  21. 9-4 Variability Course 3 Additional Example 3 Continued Note: 57 is the first quartile and the median. B. Compare the differences between the third quartile and first quartile for each. The difference between the third quartile and first quartile is the length of the box, which is greater for the last ten presidents.

  22. 9-4 Variability Oakland 0 3 6 9 12 15 18 Tampa Bay 0 3 6 9 12 15 18 Course 3 Check It Out: Example 3 These box-and-whisker plots compare the scores per quarter at Super Bowl XXXVII. The data in the T column is left out because it is a total of all the quarters.

  23. 9-4 Variability Oakland 0 3 6 9 12 15 18 Tampa Bay 0 3 6 9 12 15 18 Course 3 Check It Out: Example 3A Compare the medians and ranges. The median for Tampa Bay is significantly greater, however the range for Tampa Bay is slightly greater.

  24. 9-4 Variability Oakland 0 3 6 9 12 15 18 Tampa Bay 0 3 6 9 12 15 18 Course 3 Check It Out: Example 3B Compare the differences between the third quartile and first quartile for each. The difference between the third quartile and first quartile is the length of the box, which is slightly greater for Oakland.

  25. 9-4 Variability Course 3 Insert Lesson Title Here Lesson Quiz: Part I Find the first and third quartile for each data set. 1. 48, 52, 68, 32, 53, 47, 51 2. 3, 18, 11, 2, 7, 5, 9, 6, 13, 1, 17, 8, 0 Q1 = 47; Q3 = 53 Q1 = 2.5; Q3 = 12

  26. 9-4 Variability 78 87 91 94 98 Course 3 Insert Lesson Title Here Lesson Quiz: Part II Use the following data for problems 3 and 4. 91, 87, 98, 93, 89, 78, 94 3. Make a box-and-whisker plot 4. What is the mean? 90

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