Unit 5: Probability & Statistics SMART Packet #1 Measures of Central Tendency: Mean, Median, Mode Student: Teacher: Standards A.S.4 A.A.6 A.S.16 Compare and contrast the appropriateness of different measures of central tendency for a given data set Analyze and solve verbal problems whose solution requires solving a linear equation in one variable or linear inequality in one variable Recognize how linear transformations of one-variable data affect the data s mean, median, mode, and range
Measures of Central Tendency Measure of Central Tendency MEAN MEDIAN MODE Definition The average. The middle value of the data, when it is in numerical order. The number that appears the most. Steps to Find the Measure of Central Tendency 1. Add up all of the numbers in the data set. 2. Divide the total found in step 1 by the amount of numbers in the set. 1. Put the numbers in order from low to high. The number in the middle is the median. 2. If there are two numbers in the middle, add them together and divide by 2. 1. Find the number that appears the most. 2. If no number appears the most, write no mode. Example 1: Find the mean, median and mode for the following data: 5, 15, 10, 16, 8, 10, 20, 12 Mean: sum of all numbers in the data amount of numbers in the data 96 = = 12 8 The mean is 12. Median: 5 8 10 10 12 15 16 20 10 and 12 are both in the middle 10 +12 = 11 2 The median is 11. Mode: 10 appears more than any other number. The mode is 10.
Example 2: The prices of seven race cars sold last week are listed in the table below. (a) What is the mean value of these race cars, in dollars? mean = = = sum of all numbers in the data amount of numbers in the data 126,000 + 2(140,000) + 180,000 + 2(400,000) + 819,000 7 2,205,000 7 = 315,000 The mean value of these races cars is 315,000. (b) What is the median value of these race cars, in dollars? 126,000 140,000 140,000 180,000 400,000 400,000 819,000 180,000 is the median (c) State which of these measures of central tendency best represents the value of the seven race cars. Justify your answer. The median best represents the value of the seven race cars because the data set includes an outlier, a number that is much larger than the rest data in the set.
PRACTICE, PART I 1. Sara's test scores in science were 70, 78, 80, 92, 84, 76, and 80. Determine the mean, the median, and the mode of Sara's test scores. Mean: Median: Mode: 2. The weekly salaries of six employees at McDonalds are $140, $220, $90, $180, $140, $200. Find the mean, median, and mode for these six salaries. Mean: Median: Mode:
3. The ages of five children in a family are 3, 3, 5, 8, and 18. Which statement is true for this group of data? (1) mode > mean (3) median = mode HINT: Find the mean, (2) mean > median (4) median > mean median, and mode. Then compare your results. 4. Which statement is true about the data set 3, 4, 5, 6, 7, 7, 10? (1) mean = mode (3) mean = median (2) mean > mode (4) mean < median 5. Seth bought a used car that had been driven 20,000 miles. After he owned the car for 2 years, the total mileage of the car was 49,400. Find the average number of miles he drove each month during those 2 years.
6. The values of 11 houses on Washington St. are shown in the table below. (a) Find the mean value of these houses in dollars. (b) Find the median value of these houses in dollars. (c) State which measure of central tendency, the mean or the median, best represents the values of these 11 houses. Justify your answer
Example 3: On his first 5 biology tests, Bob received the following scores: 72, 86, 92, 63, and 77. What test score must Bob earn on his sixth test so that his average (mean score) for all six tests will be 80? Show how you arrived at your answer. Step 1: Set up an equation to represent the situation using all 6 test scores. Let x represent the unknown test score 72 + 86 + 92 + 63 + 77 + x = 80 6 Step 2: Solve for x. 72 + 86 + 92 + 63 + 77 + x = 80 6 390 + x = 80 6 Combine like terms. 390 + x = 480 Multiply both sides of the equation by 6. x = 90 Bob needs to earn a 90 on his sixth test. Subtract 390 from both sides. PRACTICE, PART II 7. TOP Electronics is a small business with five employees. The mean (average) weekly salary for the five employees is $360. If the weekly salaries of four of the employees are $340, $340, $345, and $425, what is the salary of the fifth employee? + + + + =
8. The students in Woodland High School's meteorology class measured the noon temperature every schoolday for a week. Their readings for the first 4 days were Monday, 56 ; Tuesday, 72 ; Wednesday, 67 ; and Thursday, 61. If the mean (average) temperature for the 5 days was exactly 63, what was the temperature on Friday? 9. In his first three years coaching baseball at High Ridge High School, Coach Batty's team won 7 games the first year, 16 games the second year, and 4 games the third year. How many games does the team need to win in the fourth year so that the coach's average will be 10 wins per year? (1) 13 (3) 3 (2) 10 (4) 9
10. The exact average of a set of six test scores is 92. Five of the scores are 90, 98, 96, 94, and 85. What is the other test score? (1) 92 (3) 89 (2) 91 (4) 86 11. For five algebra examinations, Maria has an average of 88. What must she score on the sixth test to bring her average up to exactly 90? (1) 98 (3) 94 (2) 92 (4) 100 12. If 6 and x have the same mean (average) as 2, 4, and 24, what is the value of x? (1) 5 (3) 10 (2) 36 (4) 14