Learn calculate, interpret, and compare measures of variation in a data set.
Vocabulary box-and-whisker plot quartiles variation inter-quartile range (IQR) 2
Ms. Snow asks some of her students how many pets they have Ms. Snow asks some of her students how many pets they have. The responses are 9, 0, 4, 1, 1, 2, 3, 5, and 2 pets. You can display this data using a box-and-whisker plot. A box-and-whisker plot or box plot is a data display that shows how data are distributed by using the median, quartiles, least value, and greatest value. Quartiles are three values, one of which is the median, that divide a data set into fourths. Each quartile contains one-fourth, or 25%, of the data. 3
A box-and-whisker plot can be used to show how the values in a data set are distributed. Variation (variability) is the spread of the values. The interquartile range (IQR) is the difference between the first and third quartiles. It is a measure of the spread of the middle 50% of the data. A small interquartile range means that the data in the middle of the set are close in value. A large interquartile range means that the data in the middle are spread out. 4
Additional Example 1: Making a Box-and-Whisker Plot The average number of hours that several students watch television in a day is given. Use the data to make a box-and-whisker plot: 2, 1, 5, 2, 1, 2, 3, 2, 2. Step 1 Order the data from least to greatest. 1, 1, 2, 2, 2, 2, 2, 3, 5 Step 2 Find the least and greatest values, the median, and the first and third quartiles. 2 5 1 3 Least value Greatest Value Third Quartile Median First Quartile 1.5 2.5 5
The first quartile is the mean of 1 and 2 The first quartile is the mean of 1 and 2. The third quartile is the mean of 2 and 3. Step 3 Draw a number line, and plot a point above each of the five values you just identified. Draw a box through the first and third quartiles and a vertical line through the median. Draw lines from the box to the least value and the greatest value. (These are the whiskers.) Continued: Example 1 0 1 2 3 4 5 6 6
Step 1 Order the data from least to greatest. Check It Out: Example 1 The next 9 customers in line are waiting to purchase the following number of items: 6, 10, 8, 5, 9, 4, 10, 7, 5. Use the data to make a box-and-whisker plot. Step 1 Order the data from least to greatest. 4, 5, 5, 6, 7, 8, 9, 10, 10 Step 2 Find the least and greatest values, the median, and the first and third quartiles. 6 4 8 10 5 9 Least value First Quartile Median Third Quartile Greatest Value 7 9.5 7
Continued: Check It Out Example 1 The first quartile is the mean of 5 and 5. The quartile is the mean of 9 and 10. Step 3 Draw a number line, and plot a point above each of the five values you just identified. Draw a box through the first and third quartiles and a vertical line through the median. Draw lines from the box to the least value and the greatest value. (These are the whiskers.) 0 1 2 3 4 5 6 7 8 9 10 11 8
Additional Example 2: Finding the Interquartile Range Find the inter-quartile range for the data set 87, 71, 72, 73, 84, 92, 73. 71, 72, 73, 73, 84, 87, 82 Order the data from least to greatest. 84, 73, 71, 72, 82 87, Find the median and quartiles IQR=87-72=15 Find the difference between the first quartile (72) and the third quartile (87). The inter-quartile range is 15. 9
Order the data from least to greatest. Check It Out: Example 2 Find the inter-quartile range for the data set 17, 39, 38, 9, 29, 40, 27 9, 17, 27, 29, 38, 39, 40 Order the data from least to greatest. 38, 29, 9, 17, 27, 40 39, Find the median and quartiles IQR=39-17=22 Find the difference between the first quartile (17) and the third quartile (39). The inter-quartile range is 22. 10
Additional Example 3: Finding Mean Absolute Deviation A scientist is studying temperature variation. She determines that the temperature at noon on four days is 75F, 82F, 78F, and 67F. What is the mean absolute deviation of the temperatures? Find the mean. . 75+82+78+67 = 75.5 4 75 is 0.5 unit from 75.5. 82 is 6.5 units from 75.5. 78 is 2.5 units from 75.5. 67 is 8.5 units from 75.5. Find the mean Find the distance on a number line each data value is from the mean. each data value is from the mean. 11
Find the mean of the distances Continued: Example 3 0.5+6.5+2.5+8.5 = 4.5. 4 Find the mean of the distances The mean absolute deviation of the temperatures is 4.5F. So, on average, the temperatures were within is 4.5F of the mean, 75.5F. 12
Find the distance on a number line each data value is from the mean. Check It Out: Example 3 A scientist is studying temperature variation. She determines that the temperature at noon on four days is 64F, 75F, 80F, and 78F. What is the mean absolute deviation of the temperatures? Find the mean. 64+75+80+78 = 74.25 4 64 is 10.25 unit from 74.25. 75 is 0.75 units from 74.25. 80 is 5.75 units from 74.25. 78 is 3.75 units from 74.25. Find the mean Find the distance on a number line each data value is from the mean. 13
Continued: Check It Out Example 3 Find the mean of the distances 10.25+0.75+5.75+3.75 = 5.125. 4 The mean absolute deviation of the temperatures is 5.125F. So, on average, the temperatures were within is 5.125F of the mean, 74.25F.