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50 Lesson 2-K ~ Measures Of Center And Variability In Two Data Sets

MEASURES OF CENTER AND VARIABILITY IN TWO DATA SETS

LESSON 2-K

It is helpful to have one number that describes a data set when comparing it with another data set. Which number should be used?

You might choose to compare two data sets using measures of center. Measures of center include mean, median and mode. The mean is the average of the numbers. The median is the number in the middle of the ordered data set. The mode is the number or numbers that occur most often. Depending on the numbers in the data set, there can be one mode, multiple modes or no mode.

Carl and Anna wanted to compare their test scores. Carl has taken 9 tests and Anna has taken 10 tests. Compare the means, medians and modes of their two sets of test scores.

Carl 85, 81, 93, 60, 75, 86, 95, 87, 85

Anna 78, 96, 96, 84, 84, 73, 98, 100, 76, 85

Find each mean. Carl: 85+81+93+60+75+86+95+87+85 _______________________ 9 = 747 ___ 9 = 83

Anna: 78+96+96+84+84+73+98+100+76+85 ___________________________ 10 = 870 ___ 10 = 87

The mean of Carl’s test scores is 83. The mean of Anna’s test scores is 87. They differ by 4 points.

EXAMPLE 1

solution

Lesson 2-K ~ Measures Of Center And Variability In Two Data Sets 51

Find each median. Carl: 60, 75, 81, 85, 85, 86, 87, 93, 95Order the numbers from → → → → ← ← ← ←least to greatest. Identifythe middle number(s). Anna: 73, 76, 78, 84, 84, 85, 96, 96, 98, 100 → → → → ← ← ← ←

84 and 85 are the middle 84 85 84.52+

=numbers. Find the mean of these numbers.

The median of Carl’s test scores is 85. The median of Anna’s test scores is 84.5. They differ by a score of 0.5.

Find each mode. Carl: 85 appears twice Anna: 84 and 96 both appear twice

The mode of Carl’s test scores is 85. The modes of Anna’s test scores are 84 and 96. Carl’s mode is similar to his mean and median. Anna has multiple modes and 96 is not similar to her mean or median. The mode is not often the best number to use to summarize data.

Carl’s test score of 60 is not typical of his other scores. It is much lower than his other scores and spreads his data out further from the mean. Because of this, the median is a better number to use than the mean as a single number to describe his scores. It tells you that 50% of his scores were above the median score and 50% were below.

At times the measures of center do not best compare two data sets. You might compare two data sets using measures of spread such as range, interquartile range or mean absolute deviation. These are also called measures of variability. Measures of variability help determine how spread apart the numbers in a data set are and give additional meaning to the measures of center (mean, median and mode).

It is helpful to first find the five-number summary for a set of data before finding its measures of variability.

The word “quartile” refers to how the data is separated into quarters. Twenty-five percent of the data falls between each pair of values.

Minimum MaximumMedianQ1 Q3

25% 25%25% 25%

EXAMPLE 1(CONTINUED)


Carl wanted to find the five-number summary for his test scores given in Example 1. Find the five-number summary of the test scores following the steps below. Carl: 85, 81, 93, 60, 75, 86, 95, 87, 85

1. Put the numbers in order and find the median.

60, 75, 81, 85, 85, 86, 87, 93, 95median

2. Find the median of the lower half of the data (this is called the 1st Quartile). If there are two numbers in the middle, include one in each half of the data.

60, 75,|81, 85, 85, 86, 87, 93, 95 78 Q1

3. Find the median of the upper half of the data (this is called the 3rd Quartile).

60, 75,|81, 85, 85, 86, 87,|93, 95 90 Q3

4. Identify the minimum and maximum values.

60, 75,|81, 85, 85, 86, 87,|93, 95 min max

The five-number summary of Carl’s test scores is 60 ~ 78 ~ 85 ~ 90 ~ 95. This gives a picture of the spread of his data.

Find the five-number summary for Anna’s test scores: 78, 96, 96, 84, 84, 73, 98, 100, 76, 85

Put the numbers in order and 73, 76, 78, 84, 84,|85, 96, 96, 98, 100find the median. 84.5 median

Find the 1st Quartile (Q1). 73, 76, 78, 84, 84,|85, 96, 96, 98, 100 If there are two numbers Q1in the middle, include one in each half of the data.

Find the 3rd Quartile (Q3). 73, 76, 78, 84, 84,|85, 96, 96, 98, 100 Q3

Find the minimum and maximum. 73, 76, 78, 84, 84,|85, 96, 96, 98, 100 min max

The five number summary is 73 ~ 78 ~ 84.5 ~ 96 ~ 100.

EXAMPLE 2

solution


The information already found to compare Carl and Anna’s test scores is summarized below.

Carl AnnaTest Scores (in order)

60, 75, 81, 85, 85, 86, 87, 93, 95 73, 76, 78, 84, 84, 85, 96, 96, 98, 100

Mean 83 87

Median 85 84.5

Mode(s) 85 84 and 96

Five-Number Summary

60 ~ 78 ~ 85 ~ 90 ~ 95 73 ~ 78 ~ 84.5 ~ 96 ~ 100

Measures of variability show other ways to compare the test scores. The range, interquartile range and mean absolute deviation for Carl’s and Anna’s test scores can be found using the steps below.

Step 1: Copy the chart below.

Carl Anna

Range

Interquartile Range

Mean Absolute Deviation

Step 2: Carl and Anna want to know the range of their data sets. The range is the difference between the maximum value in the data set and the minimum value in the data set. a. Find the minimum value and the maximum value in each data set. b. Subtract the minimum value from the maximum value for Carl and again for Anna. These values represent the range of each data set. Write the values in the chart.

Step 3: Another way to see spread in a data set is to look at its interquartile range. The interquartile range is the range of the middle half of the data. It excludes outliers in the data. a. Find the value for the 1st Quartile (Q1) and the 3rd Quartile (Q3) in the five-number summaries for Carl and Anna. b. Find the difference between the 3rd Quartile (Q3) and the 1st Quartile (Q1) for Carl and again for Anna. These are the interquartile ranges for each data set. Write these values in the chart.

Step 4: Another way to measure the spread of data is to see how far each number is from the mean of the data set. The average of these differences is the mean absolute deviation. a. A chart helps organize the work when finding the mean absolute deviation. There is a chart on the next page for Carl and Anna. Carl’s chart is filled in. Copy and complete the chart for Anna’s data.

EXPLORE! COMPARING TESTS


Carl

Test Score Deviation from Mean

Absolute Deviation

60 60 − 83 = −23 2375 75 − 83 = −8 881 81 − 83 = −2 285 85 − 83 = 2 285 85 − 83 = 2 286 86 − 83 = 3 387 87 − 83 = 4 493 93 − 83 = 10 1095 95 − 83 = 12 12

Anna

Test Score Deviation from Mean

Absolute Deviation

737678848485969698

100

b. Find the sum of the absolute deviations for Carl. c. Divide the sum of the absolute deviations for Carl by the number of tests taken (9). Round your answer to the nearest hundredth. This is the mean absolute deviation for Carl. Write it in the chart from Step 1. d. Complete the chart for Anna. Repeat parts b-c to find the mean absolute deviation for Anna.

Step 5: Write at least two complete sentences comparing Carl’s and Anna’s test scores.

A small mean absolute deviation tells you there is less spread in the data set. It means most or all of the numbers are close to the mean. When the mean absolute deviation is larger, the numbers are more spread out from the mean and there is greater variability in the numbers in the data set.

EXPLORE! (CONTINUED)


Find the mean absolute deviation for the data set 5, 6, 8, 10, 11

Find the mean. 5 + 6 + 8 + 10 + 11 _____________ 5 = 40 __ 5 = 8

Organize the data in a chart to Value

Deviation from Mean

Absolute Deviation

5 5 – 8 = –3 36 6 – 8 = –2 28 8 – 8 = 0 0

10 10 – 8 = 2 211 11 – 8 = 3 3

find the deviation from the meanand the absolute deviation.

Find the mean of the absolute deviations. 3 2 0 2 3 105 5

+ + + += = 2

The mean absolute deviation is 2. There is some variability from the mean.

EXERCISES

Find the mean, median, and mode(s) for each data set. If there is no mode, state “no mode.” 1. 7, 17, 35, 19, 14, 24, 17 2. 45, 31, 55, 31, 51, 31, 55, 23, 31, 27

3. 82, 92, 78, 76, 80, 86, 90, 80 4. 9, 15, –5, 12, –7, 10, 8

Find the mean, median and mode(s) for each data set. Write a sentence that compares each measure of center between the pairs of data sets.

5. Mark’s test scores: 82, 83, 74, 94, 76, 71, 94 Irina’s test scores: 95, 83, 79, 95, 82, 79, 89

6. Number of bowling pins knocked over by each roll of the bowling ball in a single game. Manuel: 7, 3, 5, 3, 10, 10, 8, 0, 7, 1, 9, 0, 8, 0, 10, 7, 3, 8 Isabel: 5, 4, 10, 0, 7, 10, 8, 2, 6, 6, 3, 4, 3, 9, 0, 8, 0

EXAMPLE 3

solution


Find the mean, median and mode(s) of each data set. Which number best describes the values in the data set? Explain why.

7. Yearly Salaries: $200,000 $70,000 $65,000 $68,000 $60,000 $30,000 $62,000 $70,000

8. Number of points in 7 games for a basketball team: 98, 87, 85, 102, 85, 93, 94

9. Home Prices: $1,000,000 $300,500 $320,000 $290,000 $250,000

10. Shoe sizes sold for a tennis shoe: 6, 7, 7, 7.5, 8, 8.5, 8.5, 8.5, 8.5, 9

11. Carolyn found the median for a set of data. Her work is below. She made a mistake. What was her mistake? Correct her work and find the median of the data set.

10 12 8 15 20 24 → → ← ← median = 8 + 15 _____ 2 = 11.5

For Exercises 12–15, find the following measures of variability: a. five-number summary b. range of the data c. interquartile range of the data 12. 4, 6, 8, 12, 20, 40 13. 24, 20, 27, 30, 24

14. 12, 17, 12, 20, 16, 12, 8, 19, 18, 6 15. 74, 76, 79, 68, 51, 59, 76, 60, 78

Find the mean absolute deviation of each data set.

16. 2, 10, 15, 20, 28 17. 15, 20, 25, 12, 21, 7, 11, 9

18. The two data sets below have the same median and mean, but not the same values. Use measures of variability to explain the differences between the two data sets.

Data Set 1: 2, 5, 11, 15, 22 Data Set 2: 10, 11, 11, 11, 12

19. The five-number-summary and mean for two data sets is shown below. Each data set has 30 values. Which data set has the least mean absolute deviation? Explain your answer.

Data Set 1: 10 ~ 20 ~ 30 ~ 40 ~ 50 mean = 32 Data Set 2: 60 ~ 62 ~ 64 ~ 66 ~ 68 mean = 65


For Exercises 20–22, find the: a. five-number summary b. range of the data c. interquartile range of the data d. mean absolute deviation of the data e. Compare the results and write sentences to compare the two data sets.

20. Money (in dollars) earned babysitting

January February March April May June

Kelsey 40 30 15 20 20 55

Rachel 25 35 60 10 75 35

21. Number of football passes completed in regular season games

Week 1 Week 2 Week 3 Week 4 Week 5 Week 6 Week 7 Week 8 Week 9

Wildcats 33 29 24 21 18 16 26 26 23

Tigers 32 31 30 16 24 27 15 33 26

22. Height (inches) of girls and boys in Ms. Pon’s seventh grade class.

48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78

× × ×××

××

× × ×

Height of girls (inches)

48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78

× ××

××

××

×

Height of boys (inches)

× × × ×

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