Employee Salaries The following table gives the annual

A number that is representative of a set of data is called a(n) .

Averages are referred to as measures of central .

The average that is found by summing the data and then dividing the sum by the number of pieces of data is called the .

The value in the middle of a set of ranked data is called the

The piece of data that occurs most frequently is called the .

The value halfway between the lowest and highest values in a set of data is called the .

The measures of position that divide a set of data into four equal parts are called .

Data that are listed from the lowest value to the highest value or from the highest value to the lowest value are called data.

a) The symbol for the sample mean is . b) The symbol for the population mean is .

a) Another name for the 25th percentile is the quartile. b) Another name for the 50th percentile is the quartile. c) Another name for the 75th percentile is the quartile.

In Exercises 11-20, determine the mean, median, mode, and midrange of the set of data. Where appropriate, round your answer to the nearest tenth. 8, 9, 9, 11, 13, 13, 13, 24, 26

In Exercises 1120, determine the mean, median, mode, and midrange of the set of data. Where appropriate, round your answer to the nearest tenth.7, 9, 10, 10, 10, 12, 12

In Exercises 11–20, determine the mean, median, mode, and midrange of the set of data. Where appropriate, round your answer to the nearest tenth. 76, 82, 94, 55, 100, 52, 96

In Exercises 1120, determine the mean, median, mode, and midrange of the set of data. Where appropriate, round your answer to the nearest tenth.4, 6, 10, 12, 10, 9, 365, 40, 37, 8

In Exercises 1120, determine the mean, median, mode, and midrange of the set of data. Where appropriate, round your answer to the nearest tenth.1, 3, 5, 7, 9, 11, 13, 15

In Exercises 1120, determine the mean, median, mode, and midrange of the set of data. Where appropriate, round your answer to the nearest tenth.40, 50, 30, 60, 90, 100, 140

In Exercises 1120, determine the mean, median, mode, and midrange of the set of data. Where appropriate, round your answer to the nearest tenth.1, 7, 11, 27, 36, 14, 12, 9, 1

In Exercises 1120, determine the mean, median, mode, and midrange of the set of data. Where appropriate, round your answer to the nearest tenth.1, 1, 1, 1, 4, 4, 4, 4, 6, 8, 10, 12, 15, 21

In Exercises 1120, determine the mean, median, mode, and midrange of the set of data. Where appropriate, round your answer to the nearest tenth.6, 8, 12, 13, 11, 13, 15, 17

In Exercises 1120, determine the mean, median, mode, and midrange of the set of data. Where appropriate, round your answer to the nearest tenth.5, 15, 5, 15, 5, 15

Cholesterol Level The total cholesterol level of 10 patients of Dr. Novak are 176, 202, 285, 153, 200, 182, 248, 132, 214, and 195. Determine the a) mean. b) median. c) mode. d) midrange.

Daily Commission The amount of money Steve Kline collected in sales commission in each of seven days is $48, $67, $51, $25, $102, $61, $80. Determine the mean, median, mode, and midrange.

Change in the Data The mean is the most sensitive average because it is affected by any change in the data. a) Determine the mean, median, mode, and midrange for 1, 2, 3, 5, 5, 7, 11. b) Change the 7 to a 10 in part (a). Determine the mean, median, mode, and midrange. c) Which averages were affected by changing the 7 to a 10? d) Which averages will be affected by changing the 11 to a 10 in part (a)?

Life Expectancy In 2010, the National Center for Health Statistics indicated a record average life expectancy of 78.2 years for the total U.S. population. The average life expectancy for men was 75.6 years, and for women it was 80.8 years. Which average do you think the National Center for Health is using?

A Grade of B To get a grade of B, a student must have a mean average of 80 or greater. Jim Condor has a mean average of 79 for 10 quizzes. He approaches his teacher and asks for a B, reasoning that he missed a B by only one point. What is wrong with Jims reasoning?

Employee Salaries The salaries of 10 employees of a small company follow. +29,000 +65,000 26,000 25,000 32,000 28,000 27,000 82,000 27,000 30,000 Determine the a) mean. b) median. c) mode. d) midrange. e) If the employees wanted to demonstrate the need for a raise, which average would they use to show they are being underpaid: the mean or the median? Explain. f) If the management did not want to give the employees a raise, which average would they use: the mean or the median? Explain.

Ice Cream Consumption The 10 countries with the highest per person consumption of ice cream in 2008 are listed below. 13.4 Measures of Central Tendency 803 Country Consumption (pints per person) Australia 31.7 USA 26.0 Nauru 25.3 New Zealand 23.4 Canada 22.7 Norway 22.3 Finland 20.9 Sweden 20.4 Iceland 18.5 Italy 18.5 Source: Euromonitor International Determine, to the nearest tenth, the a) mean. b) median. c) mode. d) midrange.

Living Expenses Bob Bennets monthly living expenses for 1 year are as follows: +1200 +1050 +1570 +1600 2000 1050 1550 1450 1800 1100 1310 1430 Where appropriate, round your answer to the nearest cent. Determine the a) mean. b) median. c) mode. d) midrange.

Internet Retailers The ten U.S. retailers with the highest Internet sales in 2008 are listed below.Determine to the nearest tenth the a) mean. b) median. c) mode. d) midrange.

Exam Average Malcolm Sanders mean average on five exams is 86. Determine the sum of his scores.

Exam Average Jeremy Urbans mean average on six exams is 92. Determine the sum of his scores.

Creating a Data Set Construct a set of five pieces of data in which the mode has a lower value than the median and the median has a lower value than the mean.

Creating a Data Set Construct a set of six pieces of data with a mean, median, and midrange of 75 and where no two pieces of data are the same.

Creating a Data Set Construct a set of six pieces of data with a mean of 84 and where no two pieces of data are the same.

Water Park For the 2011 season, 27,000 people visited the Blue Lagoon Water Park. The park was open 120 days for water activities. The highest number of visitors on a single day was 500. The lowest number of visitors on a single day was 50. Determine whether it is possible to find the following with the given information. a) the mean number of visitors per day b) the median number of visitors per day c) the mode number of visitors per day d) the midrange number of visitors per day

Determine a Necessary Grade A mean average of 80 or greater for five exams is needed for a final grade of B in a course. Jorge Riveras first four exam grades are 73, 69, 85, and 80. What grade does Jorge need on the fifth exam to get a B in the course?

Grading Methods A mean average of 60 on seven exams is needed to pass a course. On her first six exams, Sheryl Ward received grades of 51, 72, 80, 62, 57, and 69. a) What grade must she receive on her last exam to pass the course? b) An average of 70 is needed to get a C in the course. Is it possible for Sheryl to get a C? If so, what grade must she receive on the seventh exam? c) If her lowest grade of the exams already taken is to be dropped, what grade must she receive on her last exam to pass the course? d) If her lowest grade of the exams already taken is to be dropped, what grade must she receive on her last exam to get a C in the course?

Central Tendencies Which of the measures of central tendency must be an actual piece of data in the distribution?

Creating a Data Set Construct a set of six pieces of data such that if only one piece of data is changed, the mean, median, and mode will all change.

Changing One Piece of Data Consider the set of data 1, 1, 1, 2, 2, 2. If one 2 is changed to a 3, which of the following will change: mean, median, mode, midrange?

Changing One Piece of Data Is it possible to construct a set of six different pieces of data such that by changing only one piece of data you cause the mean, median, mode, and midrange to change? Explain.

Grocery Expenses The Taylors have recorded their weekly grocery expenses for the past 12 weeks and determined that the mean weekly expense was $85.20. Later Mrs. Taylor discovered that 1 weeks expense of $74 was incorrectly recorded as $47. What is the correct mean?

Percentiles For any set of data, what must be done to the data before percentiles can be determined?

Percentiles Josie Waverly scored in the 73rd percentile on the verbal part of her College Board test. What does that mean?

Percentiles When a national sample of heights of kindergarten children was taken, Kevin Geis was told that he was in the 35th percentile. Explain what that means.

Percentiles A union leader is told that, when all workers salaries are considered, the first quartile is $22,750. Explain what that means.

Quartiles The prices of a gallon of the 21 top-rated interior paints, as rated in the March 2010 issue of Consumer Reports, are as follows: +18 +19 +19 +19 +20 +20 +21 +21 +22 +22 +23 +23 +25 +27 +30 +33 $35 +35 +35 +58 +61 Determine a) Q2. b) Q1. c) Q3.

Quartiles The prices of the 20 top-rated 16-inch laptops, as rated in the June 2010 issue of Consumer Reports, are as follows: $450 $460 $500 $530 $550 $550 $650 $650 $680 $700 $720 $730 $800 $850 $900 $1000 $1200 $1350 $1700 $1900 Determine a) Q2. b) Q1. c) Q3.

The 50th Percentile Give the names of two other statistics that have the same value as the 50th percentile.

College Admissions Jonathan Burd took an admission test for the University of California and scored in the 85th percentile. The following year, Jonathans sister Kendra took a similar admission test for the University of California and scored in the 90th percentile. a) Is it possible to determine which of the two answered the higher percent of questions correctly on their respective exams? b) Is it possible to determine which of the two was in a better relative position with regard to their respective populations? Explain.

Employee Salaries The following statistics represent weekly salaries at the Midtown Construction Company: Mean $600 First quartile $560 Median $590 Third quartile $625 Mode $580 83rd percentile $665 a) What is the most common salary? b) What salary did half the employees salaries surpass? c) About what percent of employees salaries surpassed $625? d) About what percent of employees salaries were less than $560? e) About what percent of employees salaries surpassed $665? f) If the company has 100 employees, what is the total weekly salary of all employees?

The Mean of the Means Consider the following five sets of values. i) 5 6 7 7 8 9 14 ii) 3 6 8 9 iii) 1 1 1 2 5 iv) 6 8 9 12 15 v) 50 51 55 60 80 100 a) Compute the mean of each of the five sets of data. b) Compute the mean of the five means in part (a). c) Find the mean of the 27 pieces of data. d) Compare your answer in part (b) to your answer in part (c). Are the values the same? Does your answer make sense?

Ruth Versus Mantle The tables below compare the batting performances for selected years for two well-known former baseball players, Babe Ruth and Mickey Mantle. Babe Ruth Boston Red Sox 19141919 New York Yankees 19201934 a) For each player, compute the batting average percent (pct.) for each year by dividing the number of hits by the number of at bats. Round to the nearest thousandth. Place the answers in the pct. column. b) Going across each of the five horizontal lines (for example Ruth, 1925, vs. Mantle, 1954), compare the percents (pct.) and determine which is greater in each case. c) For each player, compute the mean batting average percent for the 5 given years by dividing the total hits by the total at bats. Which is greater, Ruths or Mantles? d) Based on your answer in part (b), does your answer in part (c) make sense? Explain. e) Find the mean percent for each player by adding the five pcts. and dividing by 5. Which is greater, Ruths or Mantles? f ) Why do the answers obtained in parts (c) and (e) differ? Explain. g) Who would you say has the better batting average percent for the 5 years selected? Explain.

Employee Salaries The following table gives the annual salary distribution for employees at Kulzers Home Improvement. Using the information provided in the table, determine the a) mean annual salary. b) median annual salary. c) mode annual salary. d) midrange annual salary. e) Which is the best measure of central tendency for this set of data? Explain your answer.

Weighted Average Sometimes when we wish to find an average, we may wish to assign more importance, or weight, to some of the pieces of data. To calculate a weighted average, we use the formula: weighted average = xw w , where w is the weight of the piece of data, x; xw is the sum of the products of each piece of data multiplied by its weight; and w is the sum of the weights. For example, suppose that students in a class need to submit a report that counts for 20% of their grade, they need to take a midterm exam that counts for 30% of their grade, and they need to take a final exam that counts for 50% of their grade. Suppose that a student got a 72 on the report, an 85 on the midterm exam, and a 93 on the final exam. To determine this students weighted average, first find xw: xw = 7210.202 + 8510.302 + 9310.502 = 86.4. Next find w, the sum of the weights: w = 0.20 + 0.30 + 0.50 = 1.00. Now determine the weighted average as follows. Weighted average = xw w = 86.4 1.00 = 86.4 Thus, the weighted average is 86.4. Note that w does not always have to be 1.00. In Exercises 55 and 56, use the weighted average formula. Course Average Suppose that your final grade for a course is determined by a midterm exam and a final exam. The midterm exam is worth 40% of your grade, and the final exam is worth 60%. If your midterm exam grade is 84 and your final exam grade is 94, calculate your final average.

Weighted Average Sometimes when we wish to find an average, we may wish to assign more importance, or weight, to some of the pieces of data. To calculate a weighted average, we use the formula: weighted average = xw w , where w is the weight of the piece of data, x; xw is the sum of the products of each piece of data multiplied by its weight; and w is the sum of the weights. For example, suppose that students in a class need to submit a report that counts for 20% of their grade, they need to take a midterm exam that counts for 30% of their grade, and they need to take a final exam that counts for 50% of their grade. Suppose that a student got a 72 on the report, an 85 on the midterm exam, and a 93 on the final exam. To determine this students weighted average, first find xw: xw = 7210.202 + 8510.302 + 9310.502 = 86.4. Next find w, the sum of the weights: w = 0.20 + 0.30 + 0.50 = 1.00. Now determine the weighted average as follows. Weighted average = xw w = 86.4 1.00 = 86.4 Thus, the weighted average is 86.4. Note that w does not always have to be 1.00. In Exercises 55 and 56, use the weighted average formula. Grade Point Average In a four-point grade system, an A corresponds to 4.0 points, a B corresponds to 3.0 points, a C corresponds to 2.0 points, and a D corresponds to 1.0 points. No points are awarded for an F. Last semester, Tanya Reeves received a B in a four-credit hour course, an A in a three-credit hour course, a C in a three-credit hour course, and an A in another three-credit hour course. Grade point average (GPA) is calculated as a weighted average using the credit hours as weights and the number of points corresponding to the grade as pieces of data. Calculate Tanyas GPA for the previous semester. (Round your answer to the nearest hundredth.)

Your Exam Average a) Calculate the mean, median, mode, and midrange of your exam grades in your mathematics course. b) Which measure of central tendency best represents your average grade? c) Which measure of central tendency would you rather use as your average grade?

Purchases Matthew Riveria purchased some items at Staples each day for five days. The mode of the number of items Matthew purchased is higher than the median of the number of items he purchased. The median of the number of items Matthew purchased is higher than the mean of the number of items he purchased. He purchased at least two items but no more than seven items each day. a) How many items did Matthew purchase each day? (Note: There is more than one correct answer.) b) Determine the mean, median, and mode for your answer to part (a).

Two other measures of location that we did not mention in this section are stanines and deciles. Use statistics books, books on educational testing and measurements, and Internet Web sites to write a report on what stanines and deciles are and when percentiles, quartiles, stanines, and deciles are used.