4-4 Variability Warm Up Problem of the Day Lesson Presentation Pre-Algebra Warm Up Problem of the Day Lesson Presentation

Pre-Algebra 4-4 Variability 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 5. 9.1 – 5.7 6. 190.3 – 23.4 3.4 166.9

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

Learn to find measures of variability.

Vocabulary variability range quartile box-and-whisker plot

Litter Size Number of Litters The table below summarizes a veterinarian’s records for kitten litters born in a given year. The range of a data set is the largest value minus the smallest value. Litter Size 2 3 4 5 6 Number of Litters 1 8 11 While central tendency describes the middle of a data set, variability describes how spread out the data is. 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.

The range of a data set is the largest value minus the smallest value 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 First quartile: 3 median of lower half Median: 4 (second quartile) Third quartile: 5 median of upper half Litter Size 2 3 4 5 6 Number of Litters 1 8 11 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.

Additional Example 1A: Finding Measures of Variability Find the range and the first and third quartiles for the data set. A. 15, 83, 75, 12, 19, 74, 21 12 15 19 21 74 75 83 Order the values. range: 83 – 12 = 71 first quartile: 15 third quartile: 75

Additional Example 1B: Finding Measures of Variability Find the range and the first and third quartiles for the data set. B. 75, 61, 88, 79, 79, 99, 63, 77 61 63 75 77 79 79 88 99 Order the values. range: 99 – 61 = 38 first quartile: = 69 63 + 75 2 third quartile: = 83.5 79 + 88 2

Try This: Example 1A Find the range and the first and third quartiles for the data set. A. 25, 38, 66, 19, 91, 47, 13 13 19 25 38 47 66 91 Order the values. range: 91 – 13 = 78 first quartile: 19 third quartile: 66

Try This: Example 1B Find the range and the first and third quartiles for the data set. B. 45, 31, 59, 49, 49, 69, 33, 47 31 33 45 47 49 49 59 69 Order the values. range: 69 – 31 = 38 first quartile: = 39 33 + 45 2 third quartile: = 54 49 + 59 2

A box-and-whisker plot shows the distribution of data A box-and-whisker plot 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. 1 2 3 4 5 6 7 8 9 Median First quartile Third quartile

Use the given data to make a box-and-whisker plot: 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 first quartile: = 16 15 + 17 2 third quartile: = 21 21 + 21 2 median: = 19 19 + 19 2

Use the given data to make a box-and-whisker plot. 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. 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 first quartile 16 median 19 third quartile 21 largest value 25 12 14 16 18 20 22 24 26 28

Use the given data to make a box-and-whisker plot. Additional 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. 13 15 17 19 19 21 21 25 12 14 16 18 20 22 24 26 28

Use the given data to make a box-and-whisker plot. Try This: 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 first quartile: = 25 24 + 26 2 third quartile: = 32 31 + 33 2 median: = 30 29 + 31 2

Use the given data to make a box-and-whisker plot. Try This: 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 22 24 26 28 30 32 34 36 38

Use the given data to make a box-and-whisker plot. Try This: 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 22 24 26 28 30 32 34 36 38

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.

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.

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.

Try This: Example 3 Oakland Final 1 2 3 4 T Oakland 6 12 21 Tampa Bay 17 14 48 Oakland 0 3 6 9 12 15 18 Tampa Bay 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.

Try This: Example 3 Continued A. Compare the medians and ranges. Oakland 0 3 6 9 12 15 18 Tampa Bay The median for Tampa Bay is significantly greater, however the range for Tampa Bay is slightly greater.

Try This: Example 3 Continued B. Compare the differences between the third quartile and first quartile for each. Oakland 0 3 6 9 12 15 18 Tampa Bay The difference between the third quartile and first quartile is the length of the box, which is slightly greater for Oakland.

Lesson Quiz: Part 1 Find the range and 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 range = 36; Q1 = 47; Q3 = 53 range = 18; Q1 = 2.5; Q3 = 12

Lesson Quiz: Part 2 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? 78 87 91 94 98 90