02 Measures of Central Tendency Problem Set
MEASURES OF CENTRAL TENDENCY | PRACTICE PROBLEMS Complete the following to reinforce your understanding of the concept covered in this module.
PROBLEM 1: The tuition costs of colleges in the Texas university system are represented by the data below. What is the median cost of tuition for the overall university system closest too: $7568, $8650, $9225, $6720, $8840, $7820, $8260, $5912, $8432, $8990
A. $8,346 B. $8,650 C. $9,225 D. $9,328
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SOLUTION 1: The topic of MEDIAN can be referenced under the topic of ENGINEERING PROBABILITY AND STATISTICS on page 37 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. To calculate the median, the data set has to be ordered from either smallest to largest or largest to smallest, and then use symmetry to find the value in the middle. If there are an odd number of values, the median is the middle value. If there are an even number of values, the median is the average of two values in the middle. The elements, ordered, that we are working with are: $5912, $6720, $7568, $7820, $8260, $8432, $8650, $8840, $8990, $9225 Since there is not an even number of elements, the median is halfway between the two middle tuition costs, $8,260 and $8,432. Therefore, the median is:
ππππππ =
$8,260 + $8,432 = $8,346 2
Therefore, the correct answer choice is A. $π, πππ
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PROBLEM 2: The stem and left plot below represents the scores of students on an engineering exam. What is the median of the scores shown in the data below most close to: Stem
Leaf 5 6, 8, 9 6 1, 6, 9 7 4, 5, 7, 7, 9 , 9 8 2, 4, 6, 7, 7, 8, 8, 9 9 1, 3, 3, 4, 4, 5, 5, 5, 7
10 0, 0
A. 76 B. 87 C. 95 D. 100
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SOLUTION 2: The topic of MEDIAN can be referenced under the topic of ENGINEERING PROBABILITY AND STATISTICS on page 37 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. To calculate the median, the data set has to be ordered from either smallest to largest or largest to smallest, and then use symmetry to find the value in the middle. If there are an odd number of values, the median is the middle value. If there are an even number of values, the median is the average of two values in the middle. Referencing the leaf plot, there are 31 scores shown. The median is the middle score, or the 16π‘β value. In this case, the 16π‘β value is 87.
Therefore, the correct answer choice is B. ππ
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PROBLEM 3: The stem and left plot below represents the scores of students on an engineering exam. What is the mode of the scores closest too: Stem
Leaf 5 6, 8, 9 6 1, 6, 9 7 4, 5, 7, 7, 9 , 9 8 2, 4, 6, 7, 7, 8, 8, 9 9 1, 3, 3, 4, 4, 5, 5, 5, 7
10 0, 0
A. 76 B. 87 C. 95 D. 100
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SOLUTION 3: The topic of MODE can be referenced under the topic of ENGINEERING PROBABILITY AND STATISTICS on page 37 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
The mode is the score that appears the most often. In stem 9, there are three leaves with a value of 5, therefore, the mode is 95.
Therefore, the correct answer choice is C. ππ
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PROBLEM 4: The stem and left plot below represents the scores of students on an engineering exam. What is the median of the scores closest too: Stem
Leaf 3 2, 4 4 0, 2 5 1, 1, 3, 7 6 2, 3, 4, 4, 6, 8 7 1, 3, 5, 6, 7, 9 8 2, 5, 6, 8, 8 9 3, 6
A. 51 B. 64 C. 68 D. π΄ππ ππ π‘βπ ππππ£π
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SOLUTION 4: The topic of MEDIAN can be referenced under the topic of ENGINEERING PROBABILITY AND STATISTICS on page 37 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. To calculate the median, the data set has to be ordered from either smallest to largest or largest to smallest, and then use symmetry to find the value in the middle. If there are an odd number of values, the median is the middle value. If there are an even number of values, the median is the average of two values in the middle. There are 27 scores shown in the leaf and stem plot for the problem. The median is the middle score, or the 14π‘β value. In this case, the 14π‘β value is 68.
Therefore, the correct answer choice is C. ππ
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PROBLEM 5: The stem and left plot below represents the scores of students on an engineering exam. What is the mode of the scores closest too: Stem
Leaf 3 2, 4 4 0, 2 5 1, 1, 1, 3, 7 6 2, 3, 4, 4, 4, 6, 8 7 1, 3, 5, 6, 7, 9 8 2, 5, 6, 8, 8, 8 9 3, 6
A. 51 B. 64 C. 88 D. π΄ππ ππ π‘βπ ππππ£π
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SOLUTION 5: The topic of MODE can be referenced under the topic of ENGINEERING PROBABILITY AND STATISTICS on page 37 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
The MODE is the value in a data set that has the most frequency seen observation in a list of data set values It serves as a useful measure because it tells us what value is the one most likely to be observed.
The mode is the score that appears the most often. In this problem there are 3 modes, 51, 64, and 88, all occurring a total of 3 times each.
Therefore, the correct answer choice is D. π¨ππ ππ πππ πππππ
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PROBLEM 6: The population growth rate of a city is represented by the value below. What is the geometric mean of the population growth rate closest too: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 A. 7.5 B. 8.2 C. 9.1 D. 10.6
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SOLUTION 6: As we are told the values represent a growth rate, we know that we will be using the formula relating geometric relationships as indicated by the growth rate. The formula for the SAMPLE GEOMETRIC MEAN can be referenced under the topic of ENGINEERING PROBABILITY AND STATISTICS on page 37 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The SAMPLE GEOMETRIC MEAN is a measure of central tendency that uses multiplication rather than addition to summarize data values. The sample geometric mean is calculated using the formula: ππππππ πΊπππππ‘πππ ππππ = πβπ1 π2 π3 β¦ ππ Where: β’ π is the number of roots β’ ππ is the value of the root being evaluated As we know there are 12 roots, we can plug in the 12 values to calculate the sample geometric mean as: 10
β2 β 4 β 6 β 8 β 10 β 12 β 14 β 16 β 18 β 20 = 9.057
Therefore, the correct answer choice is C. π. π
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PROBLEM 7: Which of the following is NOT a measure of precision? A. ππ‘ππππππ πππ£πππ‘πππ B. ππππππ C. ππππππππ D. π΄ππ ππ π‘βπ ππππ£π
SOLUTION 7: The topic of MEASURE OF CENTRAL TENDENCY can be referenced under the subject of ENGINEERING PROBABILITY AND STATISTICS on page 37 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The median is the value defined within a lot of data points where there are an equal number of measurements above and below that particular value. Precision, which involves the spread of the data, does not have any relation to the median.
Therefore, the correct answer choice is B. π΄ππ πππ
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PROBLEM 1: The tuition costs of colleges in the Texas university system are represented by the data below. What is the median cost of tuition for the overall university system closest too: $7568, $8650, $9225, $6720, $8840, $7820, $8260, $5912, $8432, $8990
A. $8,346 B. $8,650 C. $9,225 D. $9,328
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SOLUTION 1: The topic of MEDIAN can be referenced under the topic of ENGINEERING PROBABILITY AND STATISTICS on page 37 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. To calculate the median, the data set has to be ordered from either smallest to largest or largest to smallest, and then use symmetry to find the value in the middle. If there are an odd number of values, the median is the middle value. If there are an even number of values, the median is the average of two values in the middle. The elements, ordered, that we are working with are: $5912, $6720, $7568, $7820, $8260, $8432, $8650, $8840, $8990, $9225 Since there is not an even number of elements, the median is halfway between the two middle tuition costs, $8,260 and $8,432. Therefore, the median is:
ππππππ =
$8,260 + $8,432 = $8,346 2
Therefore, the correct answer choice is A. $π, πππ
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PROBLEM 2: The stem and left plot below represents the scores of students on an engineering exam. What is the median of the scores shown in the data below most close to: Stem
Leaf 5 6, 8, 9 6 1, 6, 9 7 4, 5, 7, 7, 9 , 9 8 2, 4, 6, 7, 7, 8, 8, 9 9 1, 3, 3, 4, 4, 5, 5, 5, 7
10 0, 0
A. 76 B. 87 C. 95 D. 100
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SOLUTION 2: The topic of MEDIAN can be referenced under the topic of ENGINEERING PROBABILITY AND STATISTICS on page 37 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. To calculate the median, the data set has to be ordered from either smallest to largest or largest to smallest, and then use symmetry to find the value in the middle. If there are an odd number of values, the median is the middle value. If there are an even number of values, the median is the average of two values in the middle. Referencing the leaf plot, there are 31 scores shown. The median is the middle score, or the 16π‘β value. In this case, the 16π‘β value is 87.
Therefore, the correct answer choice is B. ππ
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PROBLEM 3: The stem and left plot below represents the scores of students on an engineering exam. What is the mode of the scores closest too: Stem
Leaf 5 6, 8, 9 6 1, 6, 9 7 4, 5, 7, 7, 9 , 9 8 2, 4, 6, 7, 7, 8, 8, 9 9 1, 3, 3, 4, 4, 5, 5, 5, 7
10 0, 0
A. 76 B. 87 C. 95 D. 100
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SOLUTION 3: The topic of MODE can be referenced under the topic of ENGINEERING PROBABILITY AND STATISTICS on page 37 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
The mode is the score that appears the most often. In stem 9, there are three leaves with a value of 5, therefore, the mode is 95.
Therefore, the correct answer choice is C. ππ
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PROBLEM 4: The stem and left plot below represents the scores of students on an engineering exam. What is the median of the scores closest too: Stem
Leaf 3 2, 4 4 0, 2 5 1, 1, 3, 7 6 2, 3, 4, 4, 6, 8 7 1, 3, 5, 6, 7, 9 8 2, 5, 6, 8, 8 9 3, 6
A. 51 B. 64 C. 68 D. π΄ππ ππ π‘βπ ππππ£π
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SOLUTION 4: The topic of MEDIAN can be referenced under the topic of ENGINEERING PROBABILITY AND STATISTICS on page 37 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. To calculate the median, the data set has to be ordered from either smallest to largest or largest to smallest, and then use symmetry to find the value in the middle. If there are an odd number of values, the median is the middle value. If there are an even number of values, the median is the average of two values in the middle. There are 27 scores shown in the leaf and stem plot for the problem. The median is the middle score, or the 14π‘β value. In this case, the 14π‘β value is 68.
Therefore, the correct answer choice is C. ππ
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PROBLEM 5: The stem and left plot below represents the scores of students on an engineering exam. What is the mode of the scores closest too: Stem
Leaf 3 2, 4 4 0, 2 5 1, 1, 1, 3, 7 6 2, 3, 4, 4, 4, 6, 8 7 1, 3, 5, 6, 7, 9 8 2, 5, 6, 8, 8, 8 9 3, 6
A. 51 B. 64 C. 88 D. π΄ππ ππ π‘βπ ππππ£π
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SOLUTION 5: The topic of MODE can be referenced under the topic of ENGINEERING PROBABILITY AND STATISTICS on page 37 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
The MODE is the value in a data set that has the most frequency seen observation in a list of data set values It serves as a useful measure because it tells us what value is the one most likely to be observed.
The mode is the score that appears the most often. In this problem there are 3 modes, 51, 64, and 88, all occurring a total of 3 times each.
Therefore, the correct answer choice is D. π¨ππ ππ πππ πππππ
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PROBLEM 6: The population growth rate of a city is represented by the value below. What is the geometric mean of the population growth rate closest too: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 A. 7.5 B. 8.2 C. 9.1 D. 10.6
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SOLUTION 6: As we are told the values represent a growth rate, we know that we will be using the formula relating geometric relationships as indicated by the growth rate. The formula for the SAMPLE GEOMETRIC MEAN can be referenced under the topic of ENGINEERING PROBABILITY AND STATISTICS on page 37 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The SAMPLE GEOMETRIC MEAN is a measure of central tendency that uses multiplication rather than addition to summarize data values. The sample geometric mean is calculated using the formula: ππππππ πΊπππππ‘πππ ππππ = πβπ1 π2 π3 β¦ ππ Where: β’ π is the number of roots β’ ππ is the value of the root being evaluated As we know there are 12 roots, we can plug in the 12 values to calculate the sample geometric mean as: 10
β2 β 4 β 6 β 8 β 10 β 12 β 14 β 16 β 18 β 20 = 9.057
Therefore, the correct answer choice is C. π. π
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PROBLEM 7: Which of the following is NOT a measure of precision? A. ππ‘ππππππ πππ£πππ‘πππ B. ππππππ C. ππππππππ D. π΄ππ ππ π‘βπ ππππ£π
SOLUTION 7: The topic of MEASURE OF CENTRAL TENDENCY can be referenced under the subject of ENGINEERING PROBABILITY AND STATISTICS on page 37 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The median is the value defined within a lot of data points where there are an equal number of measurements above and below that particular value. Precision, which involves the spread of the data, does not have any relation to the median.
Therefore, the correct answer choice is B. π΄ππ πππ
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