Mean, Median, Mode and Range Additional Data andOutliers Chapter 6 – Lessons 6.2 and 6.3 Mean, Median, Mode and Range Additional Data andOutliers

Learn to find the mean, median, mode and range of a data set. Learn the effect of additional data and outliers.

Vocabulary mean median mode range outlier

Example 1: Finding the Mean of a Data Set Find the mean of each data set. 1 2 4 5 3 8 Depths of Puddles (in.)‏ mean: 5 + 8 + 3 + 5 + 4 + 2 + 1 = 28 28 ÷ 7 = 4 Add all values. Divide the sum by the number of items. The mean is 4 inches.

White board (or mental) practice: Find the mean of each data set. 9 6 5 2 10 1 Rainfall per month (in.)‏ mean: 1 + 2 + 10 + 2 + 5 + 6 + 9 = 35 35 ÷ 7 = 5 Add all values. Divide the sum by the number of items. The mean is 5 inches.

Some other descriptions of a set of data are called the median, mode, and range. The median is the middle value when the data are in numerical order, or the mean of the two middle values if there are an even number of items. The mode is the value or values that occur most often. There may be more than one mode for a data set. When all values occur an equal number of times, the data set has no mode. The range is the difference between the least and greatest values in the set.

Example 2: Finding the Mean, Median, Mode, and Range of a Data Set Find the mean, median, mode, and range of the data set. 9th Grade 15 8th Grade 14 7th Grade 11 6th Grade 12 Car Wash Totals mean: 12 + 11 + 14 + 15 4 = 13 Write the data in numerical order. 11, 12, 14, 15 median: 11, 12, 14, 15 There are an even number of items, so find the mean of the two middle values. 12 + 14 2 = 13 mode: none range: 15 – 11 = 4 The mean is 13, the median is 13, there is no mode, and the range is 4.

White board practice: Find the mean, median, mode, and range of the data set. 9th Grade 14 8th Grade 22 7th Grade 11 6th Grade 17 Bake Sale Totals mean: 17 + 11 + 22 + 14 4 = 16 Write the data in numerical order. 11, 14, 17, 22 median: 11, 14, 17, 22 There are an even number of items, so find the mean of the two middle values. 14 + 17 2 = 15.5 mode: none range: 22 – 11 = 11 The mean is 16, the median is 15.5, there is no mode, and the range is 11.

The mean, median, and mode may change when you add data to a data set.

Example 3: Sports Application A. Find the mean, median, and mode of the data in the table. 7 5 11 Games 2002 2001 2000 1999 1998 Year EMS Football Games Won mean = 7 modes = 5, 7 median = 7 B. EMS also won 13 games in 1997 and 8 games in 1996. Add this data to the data in the table and find the mean, median, and mode. mean = 8 modes = 5, 7 median = 7 The mean increased by 1, the modes remained the same, and the median remained the same.

MA Basketball Games Won White board practice: A. Find the mean, median, and mode of the data in the table. 11 6 4 13 Games 2002 2001 2000 1999 1998 Year MA Basketball Games Won B. MA also won 15 games in 1997 and 8 games in 1996. Add this data to the data in the table and find the mean, median, and mode.

MA Basketball Games Won White board practice: Solution A. Find the mean, median, and mode of the data in the table. 11 6 4 13 Games 2002 2001 2000 1999 1998 Year MA Basketball Games Won mean = 8 mode = 6 median = 6 B. MA also won 15 games in 1997 and 8 games in 1996. Add this data to the data in the table and find the mean, median, and mode. mean = 9 mode = 6 median = 8 The mean increased by 1, the mode remained the same, and the median increased by 2.

An outlier is a value in a set that is very different from the other values.

Example 4: Application Ms. Gray is 25 years old. She took a class with students who were 55, 52, 59, 61, 63, and 58 years old. Find the mean, median, and mode with and without Ms. Gray’s age. Data with Ms. Gray’s age: mean ≈ 53.3 no mode median = 58 Data without Ms. Gray’s age: mean = 58 no mode median = 58.5 When you add Ms. Gray’s age, the mean decreases by about 4.7, the mode stays the same, and the median decreases by 0.5. The mean is the most affected by the outlier. The median t is closer to most of the students’ ages. Ms. Grey’s age is an outlier because she is much younger than the others in the group. Helpful Hint

White board practice: Ms. Pink is 56 years old. She volunteered to work with people who were 25, 22, 27, 24, 26, and 23 years old. Find the mean, median, and mode with and without Ms. Pink’s age. Data with Ms. Pink’s age: mean = mode = median = Data without Ms. Pink’s age: mean = mode = median =

White board practice: Solution Ms. Pink is 56 years old. She volunteered to work with people who were 25, 22, 27, 24, 26, and 23 years old. Find the mean, median, and mode with and without Ms. Pink’s age. Data with Ms. Pink’s age: mean = 29 no mode median = 25 Data without Ms. Pink’s age: mean = 24.5 no mode median = 24.5 When you add Ms. Pink’s age, the mean increases by 4.5, the mode stays the same, and the median increases by 0.5. The mean is the most affected by the outlier. The median is closer to most of the students’ ages.

Example 5: Describing a Data Set The Yorks are shopping for skates. They found 8 pairs of skates with the following prices: $35, $42, $75, $40, $47, $34, $45, $40 What are the mean, median, and mode of this data set? Which statistic best describes the data set? Mean: 35 + 42 + 75 + 40 + 47 + 34 + 45 + 40 358 8 = = 44.75 8 The mean is $44.75. The mean is higher than most of the prices because of the $75 skates, and the mode doesn’t consider all of the data.

Example 5 Continued The Yorks are shopping for skates. They found 8 pairs of skates with the following prices: $35, $42, $75, $40, $47, $34, $45, $40 What are the mean, median, and mode of this data set? Which statistic best describes the data set? Median: 34, 35, 40, 40, 42, 45, 47, 75 40 + 42 2 82 2 = = 41 The median is $41. The median price is the best description of the prices. Most of the skates cost about $41.

Example 5 Continued The Yorks are shopping for skates. They found 8 pairs of skates with the following prices: $35, $42, $75, $40, $47, $34, $45, $40 What are the mean, median, and mode of this data set? Which statistic best describes the data set? mode: The value $40 occurs 2 times, and is more than any other value. The mode is $40. The mode represents only 2 of the 8 values. The mode does not describe the entire data set.

Some data sets, such as {red, blue, red}, do not contain numbers. In this case, the only way to describe the data set is with the mode.

Lesson Quiz Use the following data set: 18, 20, 56, 47, 30, 18, 21. 1. Find the range. 2. Find the mean. 3. Find the median. 4. Find the mode. 5. Bonnie ran a mile in 8 minutes, 8 minutes, 7 minutes, 9 minutes, and 8 minutes. What was her mean time? 38 30 21 18 8 minutes

6. Identify the range of the following data set. 20, 22, 58, 48, 32, 20, 23 A. 38 B. 28 C. 18 D. 8

7. Identify the mean of the following data set. 20, 25, 60, 42, 30, 20, 20 A. 31 B. 34 C. 38 D. 42

8. Identify the median of the following data set. 20, 22, 58, 48, 32, 20, 23 A. 48 B. 32 C. 23 D. 20

9. Identify the mode of the following data set. 20, 28, 55, 48, 30, 20, 25 A. 20 B. 23 C. 25 D. 32

10. Rebecca spent $7, $12, $8, and $13 over the past 4 days buying vegetables. What was the mean amount spent on vegetables? A. $10 B. $9 C. $8 D. $7

At the college bookstore, your brother buys 6 textbooks at the following prices: $21, $58, $68, $125, $36, and $140. 11. Find the mean. 12. Find the median. 13. Find the mode. 14. Your brother signs up for an additional class, and the textbook costs $225. Recalculate the mean, including the extra book. $74.67 $63 none $96.14

15. The weights of 7 members of a family are 48 kg, 52 kg, 63 kg, 75 kg, 52 kg, 64 kg, and 67 kg. Identify the median. A. 48 kg B. 52 kg C. 63 kg D. 75 kg

16. The heights of seven dogs at a vet are 17 inches, 14 inches, 13 inches, 21 inches, 17 inches, 15 inches, and 22 inches. Identify the mode. A. 17 in. B. 16 in. C. 15 in. D. none

17. Lopez buys 5 collectibles at the following prices: $15, $12, $15, $13 and $16. He then buys another collectible at $75. Identify the mean with and without the sixth collectible. A. $24.33; $14.20 B. $14.20; $13.83 C. $14.20; $12.64 D. $24.33; $29.20