8.1 Means, Medians, and Modes - McGraw Hill …

1 2001 McGraw - hill CompaniesMeans, Medians, and the the the the a mode615A very useful concept is the average of a group of numbers. An average is a number that istypical of a larger group of numbers. In mathematics we have several different kinds of av-erages that we can use to represent a larger group of numbers. The first of these is the find the mean for a group of numbers, follow these two steps:Step 1 Add all of the numbers in the 2 Divide that sum by the number of items in the by Step:Finding the MeanExample 1 Finding the MeanFind the mean of the group of numbers 12, 19, 15, and 1 Add all of the 19 15 14 60 Step 2 Divide that sum by the number of 4 15 There are four items in this mean of this group of numbers is YOURSELF 1 Find the mean of the group of numbers 17, 24, 19,and s apply the mean to a word 2 Finding the MeanThe ticket prices (in dollars) for the nine concerts held at the Civic Arena this school yearwere33, 31, 30, 59, 30, 35, 32, 36, 56 What was the mean price for these tickets?

2 Step 1 Add all the 31 30 59 30 35 32 36 56 342616 CHAPTER8 DATAANALYSIS ANDSTATISTICSStep 2 Divide by 9 38 The mean ticket price was $38. 2001 McGraw - hill CompaniesCHECK YOURSELF 2 The costs (in dollars) of the six textbooks that Aaron needs for the fall quarter are75, 69, 57, 87, 76, 80 Find the mean cost of these books. Finding the MedianFind the median for the following groups of numbers.(a)35, 18, 27, 38, 19, 63, 22 Step 1 Rewrite the numbers in order from smallest to , 19, 22, 27, 35, 38, 63 Step 2 Count from both ends to find the number in the from both ends, we find that 27 is the median . There are three numbers above27 and three numbers below it.(b)29, 88, 74, 81, 62, 37 Step 1 Rewrite the numbers in order from smallest to , 37, 62, 74, 81, 88 Step 2 Count from both ends to find the number in the from both ends, we find that there are two numbers in the middle, 62 and go on to step the mean is probably the most common way to find an average for a groupof numbers, it is not always the most representative.

3 Another kind of average is called medianis the number for which there are as many instances that are abovethat number as there are instances below it. To find the median , follow thesesteps:Step 1 Rewrite the numbers in order from smallest to 2 Count from both ends to find the number in the 3If there are two numbers in the middle, add them together and findtheir by Step:Finding the MedianExample 3 NOTED ivide by 9 becausethere are 9 ticket , MEDIANS, 3If there are two numbers in the middle, find their mean.(62 74) 2 136 2 68 2001 McGraw - hill CompaniesCHECK YOURSELF 3 Find the median for each group of numbers:(a)8, 6, 19, 4, 21, 5, 27(b)43, 29, 13, 37, 29, 53 CHECK YOURSELF 4 The following are Jessica s phone bills for each month of , 67, 31, 24, 15, 17, 41, 27, 17, 22, 26, 47(a)Find the mean amount of her phone bills.

4 (b)Find the median amount of her phone are times in which the median is a better representative of a group of numbersthan the mean is. Example 4 illustrates such a 4 Comparing the Mean and the MedianThe following numbers represent the hourly wage of seven employees of a local chip manu-facturing , 11, 14, 16, 32, 13, 14(a)Find the mean hourly 1 Add all of the numbers in the 11 14 16 32 13 14 112 Step 2 Divide that sum by the number of items in the 7 16 The mean wage is $16 an hour.(b)Find the median wage for the seven 1 Rewrite the numbers in order from smallest to , 12, 13, 14, 14, 16, 32 Step 2 Count from both ends to find the number in the middle number is 14.

5 There are three numbers above it and three numbers belowit. The median salary is $14 per hour. Which salary do you think is more typical of theworkers? Why?618 CHAPTER8 DATAANALYSIS ANDSTATISTICS 2001 McGraw - hill CompaniesExample 5 Finding a ModeFind the mode for the set of numbers , 24, 24, 24, 24, 27, 28, 32, 32 The mode, 24, is the number that appears most YOURSELF 5 Find the mode for the set of numbers , 7, 7, 9, 11, 13, 13, 15, 15, 15, 15, 21 CHECK YOURSELF 6 The following types of computers were available in the lab. Which type was themode?Apple, IBM, Compaq, Dell, Apple, IBM, Apple, Compaq, Dell, Apple, IBM, Apple, Dell, Apple, Compaq One advantage of the mode is that it can be used with data that are not a set of 6 Finding a ModeFollowing are the eye colors from a class of 12 students.

6 Which color is the mode?blue, brown, hazel, blue, brown, brown, brown, brown, blue, brown, hazel, greenBecause brown occurs most frequently, it is the YOURSELF $743. (a)8; (b)334. (a)$30; (b)$ The modeof a set of data is the item or number that appears most :ModeAnother measure used as an average is called the set with two differentmodes is called each eye colorappeared three times, therewould be no mode! Not everydata set has a mode. 2001 McGraw - hill CompaniesExercisesFind the mean for each set of , 9, 10, 8, , 15, 17, 17, , 15, 17, 19, 24, , 43, 56, 67, 69, , 14, 15, 16, 16, 16, 17, 22, 25, , 25, 27, 32, 36, 37, 43, 44, 44, , 8, 9, 11, , 18, 11, 7, , 8, 11, 14, , 23, 25, 27, 22, 20 Find the median for each set of , 3, 5, 6, , 13, 15, 17, , 24, 27, 31, 36, 38, , 4, 9, 16, 25, 36, , 13, 47, 25, 68, 51, , 71, 33, 69, 71, 25, 75 Find the mode for each set of , 13, 16, 18, , 43, 56, 67, 69, , 44, 25, 27, 32, 36, 37, , 8, 10, 9, 9, 10, , 13, 7, 14, 4, 11, , 2, 3, 3, 4, 9, 9, 3 Solve the following temperatures of 86 , 91 , 92 , 103.

7 And 98 were recorded forthe first 5 days of July. What was the mean high temperature? Section Date 2001 McGraw - hill salesperson drove 238, 159, 87, 163, and 198 miles (mi) on a 5-day was the mean number of miles driven per day? mileage ratings for seven new diesel cars were 43, 29, 51,36, 33, 42, and 32 miles per gallon (mi/gal). What was the mean rating? enrollments in the four elementary schools of a district are 278,153, 215, and 198 students. What is the mean enrollment? get an A in history, you must have a mean of 90 on five tests. Yourscores thus far are 83, 93, 88, and 91. How many points must you have on thefinal test to receive an A?

8 (Hint:First find the total number of points you need toget an A.) pass biology, you must have a mean of 70 on six quizzes. So far yourscores have been 65, 78, 72, 66, and 71. How many points must you have on the finalquiz to pass biology? had scores of 87, 82, 93, 89, and 84 on five tests. Tamika hadscores of 92, 83, 89, 94, and 87 on the same five tests. Who had the higher meanscore? By how much? Wong family had heating bills of $105, $110, $90, and $67 in thefirst 4 months of 1999. The bills for the same months of 2000 were $110, $95, $75,and $76. In which year was the mean monthly bill higher? By how much? 2001 McGraw - hill CompaniesMonthly energy use, in kilowatt-hours (kWh), by appliance type for four typical is shown range971158096 Electric heat120010861103975 Water heater407386368423 Refrigerator12715498121 Lights759910894 Air conditioner12311796120 Color is the mean number of kilowatt-hours used each month by the fourfamilies for heating their homes?

9 Is the mean number of kilowatt-hours used each month by the fourfamilies for hot water? is the mean number of kilowatt-hours used per appliance by theMcCarthy family? is the mean number of kilowatt-hours used per appliance by theGregg family?Use your calculator for exercises 35 and kept the following records of his utility bills for 12 months: $53,$51, $43, $37, $32, $29, $34, $41, $58, $55, $49, and $58. What was the meanmonthly bill? following scores were recorded on a 200-point final examination:193, 185, 163, 186, 192, 135, 158, 174, 188, 172, 168, 183, 195, 165, 183. What wasthe mean of the scores? following are eye colors from a class of eight students. Which color is the mode?

10 Hazel, green, brown, brown, blue, green, hazel, weather in Philadelphia over the last seven days was as follows:Rain, sunny, cloudy, rain, sunny, rain, rain. What type of weather was the mode? the advantages and disadvantages of the mean, median , and mode. 2001 McGraw - hill a certain math class, you take four tests and the final, which counts as two grade is the average of the six tests. At the end of the course, you compute boththe mean and the median .(a)You want to convince the teacher to use the mean to compute your average. Writea note to your teacher explaining why this is a better choice. Choose numbersthat make a convincing argument.(b)You want to convince the teacher to use the median to compute your a note to your teacher explaining why this is a better choice.