9.1 Measures of Center and Spread


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1 Name Class Date 9.1 Measures of Center and Spread Essential Question: How can you describe and compare data sets? Explore Exploring Data Resource Locker Caleb and Kim have bowled three games. Their scores are shown in the chart below. Name Game 1 Game Game Average Score Caleb Kim Complete the table.. Image Credits: Mike Kemp/Rubberball/Corbis A B Find Caleb s average score over the three games and enter it in the table = _ = Find Kim s average score over the three games and enter it in the table = _ = How do their average scores compare? Caleb s / Kim s game is more consistent. Caleb s scores are farther from / closer to the average than Kim s. They bowl a fourth game, where Caleb scores 150 and Kim scores a 175. How does this affect their averages? Caleb s average. Kim s average. Does the Game 4 score affect the consistency of their scores? Explain. Module 9 77 Lesson 1
2 . Reflect 1. Discussion Is the average an accurate representation of Caleb s bowling?. Discussion Is the average an accurate representation of Kim s bowling? Explain 1 Measures of Center: Mean and Median Two commonly used measures of center for a set of numerical data are the mean and median. Measures of center represent a central or typical value of a data set. The mean is the sum of the values in the set divided by the number of values in the set. The median is the middle value in a set when the values are arranged in numerical order. Example 1 Find the mean and median of each data set. A The number of text messages that Isaac received each day for a week is shown. 47, 49, 54, 50, 48, 47, 55 Find the mean. Divide the sum by the numbers of data values. _ 50 = 50. The mean is 50 text messages a day. 7 Find the median. Rewrite the values in increasing order. 47, 47, 48, 49, 50, 54, 55. The median is 49 text messages a day. B The amount of money Elise earned in tips per day for 6 days is listed below. $75, $97, $60, $84, $119, $100 Find the mean. Divide the sum by the number of data values. _ 6 = The mean is. Find the median. Rewrite the values in increasing order. 75,,,,, 60 Find the mean of the middle two values. The median is. + = Module 9 78 Lesson 1
3 Reflect. Discussion For the data on tips, which measure of center is more accurate in describing the typical value? Explain. Your Turn Find the mean and median of each data set. 4. Niles scored 70, 74, 7, 71, 7, and 96 on his 6 geography tests. 5. Raul recorded the following golf scores in his last 7 games. 84, 94, 9, 89, 94, 81, 90 Explain Measures of Spread: Range and IQR Measures of spread are used to describe the consistency of data values. They show the distance between data values and their distance from the center of the data. Two commonly used measures of spread for a set of numerical data are the range and interquartile range (IQR). The range is the difference between the greatest and the least data values. Quartiles are values that divide a data set into four equal parts. The first quartile ( Q 1 ) is the median of the lower half of the set, the second quartile ( Q ) is the median of the whole set, and the third quartile ( Q ) is the median of the upper half of the set. The interquartile range (IQR) of a data set is the difference between the third and first quartiles. It represents the range of the middle half of the data. Range : 91 = 8 IQR: 7  = 4 1,,,,, 4, 4 5, 6, 6, 7, 7, 8, 8, 9 First quartile (Q 1 ): Third quartile (Q ): 7 Median (Q ): 5 Example Find the median, range, and interquartile range for the given data set. A The April high temperatures for 5 years in Boston are 77 F, 86 F, 84 F, 9 F, and 90 F Order the data values. Median: 77, 84, 86, 90, 9 Range: 977 = 16 median Interquartile range: lower half 77, 84, 86, 90, 9 Q 1 = median upper half IQR = Q  Q 1 = = 1 _ = 80.5 and Q = _ = 91.5 Module 9 79 Lesson 1
4 B The numbers of runs scored by a softball team in 0 games are given., 4, 8, 1, 7, 5, 4, 1,, 9, 11, 4, 14, 8,, 10,, 10, 9, 7 Order the data values.,,,, 4, 4, 4,,,,,,, 9, 10, 10, 11, 1, 1, 14 + Median = Range = 14  = median Interquartile range. Q and Q + IQR = Q  Q 1 =  = The median is. The range is. The IQR is. Reflect 6. Discussion Why is the IQR less than the range? Your Turn Find the median, range, and interquartile range for the given data set. 7. 1, 1, 6, 4, 8, 6 8. The high temperatures in degrees Fahrenheit on 11 days were 68, 71, 75, 74, 75, 71, 7, 71, 7, 74, and 79. Module 9 80 Lesson 1
5 Explain Measures of Spread: Standard Deviation Another measure of spread is the standard deviation, which represents the average of the distance between individual data values and the mean. The formula for finding the standard deviation of the data set { x 1, x, x, x, x n }, with n elements and mean x, is shown below. A standard deviation = ( x 1  x ) _ + ( x  x ) _ + + ( x n  x ) Find the standard deviation of 77, 86, 84, 9, 90. Find the mean mean = 5 = _ 40 5 = 86 Complete the table. Data Value, x Deviation from Mean, x  _ x Squared Deviation, (x  _ x ) = 9 (9 ) = 0 0 = =  ( ) = 7 7 = = 4 4 = 16 n B Find the mean of the squared deviations mean = 5 = _ = 0 Find the square root of the mean of the squared deviations, rounding to the nearest tenth The standard deviation is approximately 5.5. Find the standard deviation of, 4, 8, 1, 7, 5, 4, 1,, 9, 11, 4, 14, 8,, 10,, 10, 9, 7. Find the mean mean = 0 = _ 0 = Module 9 81 Lesson 1
6 Complete the table. Data Value, x Deviation from Mean, x  _ x Squared Deviation, (x  _ x )  = ( )  = ( ) = ( ) = ( ) 7  = ( ) = ( ) = ( ) = ( ) 11  = ( ) 1  = ( ) = ( ) Module 9 8 Lesson 1
7 Find the mean of the squared deviations. mean = 5 = _ 0 = Find the square root of the mean of the squared deviations, rounding to the nearest tenth. The standard deviation is approximately. Reflect 9. In terms of data values used, what makes calculating the standard deviation different from calculating the range? Your Turn 10. Find the standard deviation of 1, 1, 6, 4, 8, 6. Data Value, x Deviation from Mean, x  x _ Squared Deviation, (x  x ) _ Find the standard deviation of 68, 71, 75, 74, 75, 71, 7, 71, 7, 74, and 79. Module 9 8 Lesson 1
8 Elaborate 1. In Your Turn 11, what is the mean of the deviations before squaring? Use your answer to explain why squaring the deviations is reasonable. 1. How can you determine the first and third quartiles of a data set? 14. How can you determine the standard deviation of a data set? 15. Essential Question CheckIn What does the measure of center of a data set indicate? Evaluate: Homework and Practice 1. The data set {1, 4, 14, 15, 14} gives the times of Tara s oneway ride to school (in minutes) for one week. Is the average (mean) of the times a good description of Tara s ride time? Explain. Online Homework Hints and Help Extra Practice Find the mean and median of each data set.. The numbers of hours Cheri works each day are, 7, 4, 6, and 5.. The weights in pounds of 6 members of a basketball team are 15, 16, 150, 119, 150, and , 18, 1, 10, 9 Module 9 84 Lesson 1
9 5. The average yearly gold price for the period from : $79.11, $71.04, $09.7, $6.8, $409.7, $444.74, $60.46, $695.9, $871.96, $ There are 8, 0, 9, 6, 1, and 0 students in a school s 6 Algebra 1 classes. 7. 1, 14, 18, 1, 1, 17, 15, 1 8. The numbers of members in 5 karate classes are 1, 1, 10, 16, and Find the range and interquartile range for, 7, 4, 6, and Find the range and interquartile range for 15, 16, 150, 119, 150, and Find the range and interquartile range for 6, 18, 1, 10, and Find the range and interquartile range for $79.11, $71.04, $09.7, $6.8, $409.7, $444.74, $60.46, $695.9, $871.96, and $ Find the range and interquartile range for 8, 0, 9, 6, 1, and 0. Module 9 85 Lesson 1
10 14. Find the range and interquartile range for 1, 14, 18, 1, 1, 17, 15, and Find the range and interquartile range for 1, 1, 15, 17, and Find the standard deviation of, 7, 4, 6, and Find the standard deviation of 15, 16, 150, 119, 150, and Find the standard deviation of 6, 18, 1, 10, and Find the standard deviation of $79.11, $71.04, $09.7, $6.8, $409.7, $444.74, $60.46, $695.9, $871.96, and $ Find the standard deviation of 8, 0, 9, 6, 1, and Find the standard deviation of 1, 14, 18, 1, 1, 17, 15, and 1. Module 9 86 Lesson 1
11 . Determine whether or not the third quartile will be a member of the data set. Select the correct answer for each lettered part. A. {79, 91, 90, 99, 91, 80, 80, 90} Yes No B. {98, 96, 96, 95, 91, 81, 87} Yes No C. {88, 95, 89, 9, 88, 9, 84, 9, 85, 9} Yes No D. {97, 84, 96, 8, 9, 88, 8, 91, 94} Yes No E. {94, 85, 95, 80, 97} Yes No F. {85, 89, 81, 89, 85, 84} Yes No Use this data for Exercises and 4. The numbers of members in 6 yoga clubs are 80, 74, 77, 71, 75, and 91.. Find the standard deviation of the numbers of members to the nearest tenth. H.O.T. Focus on Higher Order Thinking. Image Credits: Kristian Sekulic/Getty Images 4. Explain the Error Suppose a person in the club with 91 members transfers to the club with 71 members. A student claims that the measures of center and the measures of spread will all change. Correct the student s error. 5. What If? If all the values in a set are increased by 10, does the range also increase by 10? Explain. 6. Communicate Mathematical Ideas Jorge has a data set with the following values: 9, 80, 88, 95, and x. If the median value for this set is 88, what must be true about x? Explain. 7. Critical Thinking If the value for the median of a set is not found in the data set, what must be true about the data set? Explain. Module 9 87 Lesson 1
12 Lesson Performance Task The table lists the ages of randomly selected members in the soprano and bass sections of a town choir. Find the mean, median, range, interquartile range, and standard deviation for each type of singer in the data set. Interpret each result. What can you conclude about the ages of the different types of singers? Age of Soprano Singers Age of Bass Singers Module 9 88 Lesson 1
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