9-1 Measures of Center and Spread COMMON CORE

UNIT 9: Data Analysis

Name_____________________ Class_____________ Date__________

Measures of Center and Spread

Essential question: What statistics can you use to characterize and compare the center and spread of data sets?

9-1

COMMON CORE

CC.9-12.S.ID.2*

Two commonly used measures of the center of a set of numerical data are the mean and median. Let n be the number of data values. The mean is the sum of the data values divided by n. When the data values are ordered from least to greatest, the median is either the middle value if n is odd or the average of the two middle values if n is even. The median divides the data set into two halves. The first quartile (Q1) of a data set is the median of the lower half of the data. The third quartile (Q3) is the median of the upper half.

Two commonly used measures of the spread of a set of numerical data are the range and interquartile range. The range is the difference between the greatest data value and the least data value. The interquartile range (IQR) is the difference between the third quartile and first quartile: IQR = Q3 - Q1.

1 EXAMPLE Finding Mean, Median, Range, and Interquartile Range

The April high temperatures (in degrees Fahrenheit) for five consecutive years in Boston are listed below. Find the mean, median, range, and interquartile range for this data set

77 86 84 93 90

A Find the mean.

Mean =

=

B Find the median.

Write the data values from least to greatest:

Identify the middle value: C Find the range.

Range = 93 -

=

D Find the interquartile range.

Find the first and third quartiles. Do not include the median as part of either the lower half or the upper half of the data.

Q1 =

=

and Q3 =

=

Find the difference between Q3 and Q1 : IQR =

-

=

REFLECT

1a. If 90?F is replaced with 92?F, will the median or mean change? Explain.

1b. Why is the IQR less than the range?

Standard Deviation Another measure of spread is standard deviation. It is found by squaring the deviations of the data values from themean of the data values, then finding the mean of those squared deviations, and finally taking the square root of the mean of the squared deviations. The steps for calculating standard deviation are listed below. 1. Calculate the mean, x. 2. Calculate each data value's deviation from the mean by finding x - x for each data value x. 3. Find (x - x)2, the square of each deviation. 4. Find the mean of the squared deviations. 5. Take the square root of the mean of the squared deviations.

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UNIT 9: Data Analysis

2 EXAMPLE Calculating the Standard Deviation

Calculate the standard deviation for the data from the previous example. A Complete the table using the fact that the mean of the data is x = 86.

Data value, x

77 86 84 93 90

Deviation from mean, x - x

77 - 86 = -9

Squared deviation, (x - x)2

(-9)2 = 81

B Find the mean of the squared deviations.

Mean =

=

C Take the square root of the mean of the squared deviations. Use a calculator, and round to the nearest tenth.

Square root of mean =

REFLECT 2a. What is the mean of the deviations before squaring? Use your answer to explain why squaring the deviations is reasonable.

2b. In terms of the data values used, what makes calculating the standard deviation different from calculating the range? 2c. What must be true about a data set if the standard deviation is 0? Explain.

Numbers that characterize a data set, such as measures of center and spread, are called statistics. They are useful when comparing large sets of data.

3 EXAMPLE Comparing Statistics for Related Data Sets

The tables below list the average ages of players on 15 teams randomly selected from the 2010 teams in the National Football League (NFL) and Major League Baseball (MLB).Compare the average ages of NFL players to the average ages of MLB players.

NFL Players' Average Ages

Team

Average Age

Bears

25.8

Bengals

26.0

Broncos

26.3

Chiefs

25.7

Colts

25.1

Eagles

25.2

Jets

26.1

Lions

26.4

Packers

25.9

Patriots

26.6

Saints

26.3

Seahawks

26.2

Steelers

26.8

MLB Players' Average Ages

Team

Average Age

Astros

28.5

Cardinals

29.0

Cubs

28.0

Diamondbacks

27.8

Dodgers

29.5

Giants

29.1

Marlins

26.9

Mets

28.9

Nationals

28.6

Padres

28.7

Pirates

26.9

Phillies

30.5

Reds

28.7

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UNIT 9: Data Analysis

Texans

25.6

Titans

25.7

A On a graphing calculator, enter the two sets of data into two lists, L1 and L2. Examine the data as you enter the values, and record your general impressions about how the data sets compare before calculating any statistics.

Rockies

28.9

Yankees

29.3

B Calculate the statistics for the NFL data in list L1. Then do the same for the MLB data in L2. Record the results in the table below. Your calculator may use the following notations and abbreviations for the statistics you're interested in.

Mean: x

Median: Med

IQR: May not be reported directly, but can be obtained by subtracting Q1 from Q3

Standard deviation: x

NFL MLB

Center

Mean

Median

Spread

IQR (Q3 - Q1)

Standard Deviation

C Compare the corresponding statistics for the NFL data and the MLB data. Are your comparisons consistent for the two measures of center and the two measures of spread? Do your comparisons agree with your general impressions from Part A?

REFLECT 3a. Based on a comparison of the measures of center, what conclusion can you draw about the typical age of an NFL player and of an MLB player?

3b. Based on a comparison of the measures of spread, what conclusion can you draw about variation in the ages of NFL players and of MLB players?

3c. What do you notice about the mean and median for the NFL? For the MLB?

3d. What do you notice about the IQR and standard deviation for the NFL? For the MLB?

PRACTICE

The numbers of students in each of a school's six Algebra 1 classes are listed below. Find each statistic for this data set. 28 30 29 26 31 30

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UNIT 9: Data Analysis

1. Mean = 3. Range =

2. Median = 4. IQR =

5. Find the standard deviation of the Algebra 1 class data by completing the table and doing the calculations below it.

Data value, x

28 30 29 26 31 30

Deviation from mean, x - x

Squared deviation, (x - x)2

Mean of squared deviations =

Standard deviation 6. Error Analysis Suppose a student in the Algebra 1 class with 31 students transfers to the class with 26 students. The student claims that the measures of center

and the measures of spread will all change. Correct the student's error.

7. The table lists the heights (in centimeters) of 8 males and 8 females on the U.S. Olympic swim team, all randomly selected from swimmers on the team who participated in the 2008 Olympic Games held in Beijing, China.

Heights of Olympic male swimmers

Heights of Olympic female swimmers

196 188 196 185 203 183 183 196 173 170 178 175 173 180 180 175

a. Use a graphing calculator to complete the table below.

Olympic male swimmers

Olympic female swimmers

Center

Mean

Median

Spread

IQR (Q3 - Q1)

Standard Deviation

b. Discuss the consistency of the measures of center for male swimmers and the measures of center for female swimmers, and then compare the measures of center for male and female swimmers.

c. What do the measures of spread tell you about the variation in the heights of the male and female swimmers?

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