Section 6.2 : Measures of Central Tendency and Dispersion

Learning Targets: S.ID.2, N.Q.2, S.ID.1, S.ID.3

Important Terms and Definitions

Measure of Central Tendency: A single number that describes a group of data. (i.e. mean, median, mode, etc)

Mean (average): Found by dividing the sum of the data by the number of data items

Median: The middle number when the data are arranged in either increasing or decreasing order

*Note: If there is an even number of data, the median is the average of the two middle scores.

Mode: The number that appears most frequently. A group of numbers may have no mode or more than one mode.

Outliers: a data value that is much higher or lower than the other values in the set

Quick Info

1. The mode is useful when dealing with data that are not numerical, such as color of eyes 2. The mean is influenced by very large or very small numbers, while median is not.

Therefore, if there are outliers, use median. If there aren’t outliers, find the mean.

Measure of Dispersion: Describes how dispersed/spread out the values in the data set are

Range: Highest minus lowest. Gives you the measure of the spread of the data

Line Plot: A data display in which each mark above a number line corresponds to each data value. These can also be called dot plots when the mark used is a dot.

Finding Measures of Central Tendency and Dispersion

Example: Find the mean, median, mode and range of the data set.

1, 18, 25, 32, 32, 35, 43, 50, 64, 70

Mean = 1+ 18+25+32+32+35+ 43+50+64+7010

= 37010

= 37

Median = 32+352

= 33.5

Mode = 32

Range = 70 − 1 = 69

(ex 1) The data set below shows the scores from when you and your friends played putt-putt this past summer at Sports World. Find the mean, median, mode and range of the data set. Which measure of central tendency best describes the data?

39, 43, 42, 38, 67, 41, 45, 43, 40

(ex 2) Find the mean, median, mode and range of the data set. Which measure of central tendency best describes the data?

8.2, 9.3, 8.5, 8.8, 9.0

(ex 3) Over the past 6 seasons, one baseball player’s batting averages were .265, .327, .294,.316, .281, and .318. A second player’s batting averages were . 304, .285, .312, .291, .303,and .314. What are the range and mean of each player’s batting averages? Use your results to compare the players’ batting skills.

Using Mean to Find a Missing Data Value

Example: Suppose your grades on three science exams are 82, 94, and 89. What grade do you need on your next exam to have an average of 90?

Mean = 82+94+89+𝑥4

90 = 82+94+89+𝑥4

360 = 82 + 94 + 89 + 𝑥

360 = 265 + 𝑥

𝑥 = 95

(ex 3) Write and solve an equation to find the value of x.

31.7, 42.8, 26.4, 𝑥 ; mean: 35

Line Plots

Example: Your English teacher asks everyone in the class to write an autobiography, but doesn’t give any instructions in regards to how long the paper must be. After everyone turned in their papers, your teacher counted how many pages each one was. Create a line plot to represent the data. 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 1 2 3 4 5 6 7 8 9 10

(ex 4) The data set below shows the ages of the members of the Speech team. Make a line plot of the data. Then calculate the mean, median, mode and range of the data.

14, 14, 15, 15, 16, 15, 15, 16, 17, 17

(ex 5) The line plots below show the monthly earnings in thousands of dollars for two different ice cream companies in Boardman. Use the information in the line plots to determine which company has a greater mean and a greater median.

Company C

x x x x x x x x x x x x 30 31 32 33 34 35

Company H

x x x x x x x x x x x x 30 31 32 33 34 35

Homework – Section 6.2 : Measures of Central Tendency and Dispersion

Find the mean, median, mode, and range of each data set. Which measure of central tendency best describes the data?

1. Number of text messages sent per hour: 15, 30, 32, 48, 2, 35, 40, 19, 33, 30 2. Points scored per game: 15, 8, 12, 19, 22, 10, 12, 15, 21, 17, 11 3. 42.1, 46.4, 58.2, 67.3, 49.1, 40.2, 22.3, 46.6 4. 49, 52, 53, 56, 62, 61, 55, 52

Write and solve an equation to find the value of x.

5. 100, 121, 105, 113, 108, 𝑥; mean: 112 6. 99, 86, 76, 95, 𝑥; mean: 91 7. Ryan recently launched a new website for his band. In the past six days, he has recorded

the following number of daily hits: 37, 29, 37, 56, 45, 38. He is really hoping that by the end of the week, it will have averaged 40 hits per day for the week. For this to happen, how many hits must it have on the final day of the week?

8. The closing prices over a period of time, in dollars, of two different stocks are given below. What are the range and mean of each set of data? Use your results to compare Stock A and Stock B.

Stock A : 7, 4, 3, 6, 1 Stock B : 24, 15, 2, 10, 5

9. The list below shoes the number of students in the freshman homerooms here at Boardman High School. Find the mean, median, mode, and range of the data, and then make a line plot of the data. How can you use the line plot to determine whether the mean and median are equal?

21, 22, 19, 20, 23, 21, 20, 24, 20, 19, 20, 21, 21, 23, 21, 22, 19, 21, 19, 20

10. The line plots below show the scores on 10 point Notebook Quizzes for two classes of students. Use the information in the line plots to determine which class has a greater mean and a greater median.

1st Period

x x x x x x x x x x x x x x x x x x x x 5 6 7 8 9 10

2nd Period

x x x x x x x x x x x x x x x x x x x x 5 6 7 8 9 10