Mean
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ISEE Upper Level Quantitative Reasoning › Mean
Mark's numeric grade in his Spanish class is determined by five equally weighted hourly tests and a final, weighted twice as much as an hourly test. The highest score possible on each is 100.
Going into finals week, Mark's hourly test scores are 92, 66, 84, 77, and 87. What must Mark score on his final, at minimum, in order to achieve a grade of 80 or better for the term?
Explanation
Mark's grade is a weighted mean in which his hourly tests have weight 1 and his final has weight 2. If we call 
which simplifies to
Since Mark wants his score to be 80 or better, we solve this inequality:
Mark must score 77 or better on his final.
Mark's numeric grade in his Spanish class is determined by five equally weighted hourly tests and a final, weighted twice as much as an hourly test. The highest score possible on each is 100.
Going into finals week, Mark's hourly test scores are 92, 66, 84, 77, and 87. What must Mark score on his final, at minimum, in order to achieve a grade of 80 or better for the term?
Explanation
Mark's grade is a weighted mean in which his hourly tests have weight 1 and his final has weight 2. If we call
which simplifies to
Since Mark wants his score to be 80 or better, we solve this inequality:
Mark must score 77 or better on his final.
A gymnastics contest has seven judges, each of whom rates each contestant's performance on a scale from 0 to 10. A contestant's score is calculated by disregarding the highest and lowest scores, and taking the mean of the remaining five scores.
The seven judges rated Sally's performance with the following seven scores: 
Which of these quantities is the greater?
(a) Sally's score
(b) Sue's score
(b) is greater.
(a) is greater.
(a) and (b) are equal.
It cannot be determined from the information given.
Explanation
To calculate whether Sally or Sue has the higher average, it is only necessary to add, for each contestant, all of their scores except for their highest and lowest. Since both sums are divided by 5, the higher sum will result in the higher mean score.
(a) For Sally, the highest and lowest scores are 9.7 and 9.1. The sum of the other five scores is:
(b) For Sue, the highest and lowest scores are 10.0 and 9.1. The sum of the other five scores is:
Sue's total - and, subsequently, her score - is higher than Sally's, so (b) is the greater quantity.
A gymnastics contest has seven judges, each of whom rates each contestant's performance on a scale from 0 to 10. A contestant's score is calculated by disregarding the highest and lowest scores, and taking the mean of the remaining five scores.
The seven judges rated Sally's performance with the following seven scores:
Which of these quantities is the greater?
(a) Sally's score
(b) Sue's score
(b) is greater.
(a) is greater.
(a) and (b) are equal.
It cannot be determined from the information given.
Explanation
To calculate whether Sally or Sue has the higher average, it is only necessary to add, for each contestant, all of their scores except for their highest and lowest. Since both sums are divided by 5, the higher sum will result in the higher mean score.
(a) For Sally, the highest and lowest scores are 9.7 and 9.1. The sum of the other five scores is:
(b) For Sue, the highest and lowest scores are 10.0 and 9.1. The sum of the other five scores is:
Sue's total - and, subsequently, her score - is higher than Sally's, so (b) is the greater quantity.
This semester, Mary had five quizzes that were each worth 10% of her grade. She scored 89, 74, 84, 92, and 90 on those five quizzes. Mary also scored a 92 on her midterm that was worth 25% of her grade, and a 91 on her final that was also worth 25% of her class grade. What was Mary's final grade in the class?
89
87
85
91
Explanation
To find her average grade for the class, we need to multiply Mary's test scores by their corresponding weights and then add them up.
The five quizzes were each worth 10%, or 0.1, of her grade, and the midterm and final were both worth 25%, or 0.25.
average = (0.1 * 89) + (0.1 * 74) + (0.1 * 84) + (0.1 * 92) + (0.1 * 90) + (0.25 * 92) + (0.25 * 91) = 88.95 = 89.
Looking at the answer choices, they are all spaced 2 percentage points apart, so clearly the closest answer choice to 88.95 is 89.
This semester, Mary had five quizzes that were each worth 10% of her grade. She scored 89, 74, 84, 92, and 90 on those five quizzes. Mary also scored a 92 on her midterm that was worth 25% of her grade, and a 91 on her final that was also worth 25% of her class grade. What was Mary's final grade in the class?
89
87
85
91
Explanation
To find her average grade for the class, we need to multiply Mary's test scores by their corresponding weights and then add them up.
The five quizzes were each worth 10%, or 0.1, of her grade, and the midterm and final were both worth 25%, or 0.25.
average = (0.1 * 89) + (0.1 * 74) + (0.1 * 84) + (0.1 * 92) + (0.1 * 90) + (0.25 * 92) + (0.25 * 91) = 88.95 = 89.
Looking at the answer choices, they are all spaced 2 percentage points apart, so clearly the closest answer choice to 88.95 is 89.
Consider the following data set:
Which of these numbers is greater than the others?
The midrange of the set
The median of the set
The mode of the set
The mean of the set
It cannot be determined from the information given
Explanation
The median of the set is the fifth-highest value, which is 70; this is also the mode, being the most commonly occurring element.
The mean is the sum of the elements divided by the number of them. This is
The midrange is the mean of the least and greatest elements, This is
The midrange is the greatest of the four.
Consider the following data set:
Which of these numbers is greater than the others?
The midrange of the set
The median of the set
The mode of the set
The mean of the set
It cannot be determined from the information given
Explanation
The median of the set is the fifth-highest value, which is 70; this is also the mode, being the most commonly occurring element.
The mean is the sum of the elements divided by the number of them. This is
The midrange is the mean of the least and greatest elements, This is
The midrange is the greatest of the four.
Sally's final score in economics is calculated by taking the mean of the best four of her five test scores. Sally received a final score of 78. Her first four test scores were 90, 80, 65, and 70. Which is the greater quantity?
(A) Her fifth test score
(B) 65
(A) is greater
(B) is greater
(A) and (B) are equal
It is impossible to determine which is greater from the information given
Explanation
Had Sally scored 65 or less on her fifth test, that would have been the dropped score, and her final score would have been the mean of 90, 80, 65, and 70. This is the sum of the scores divided by four:
Since Sally's mean was greater than this (78), it can be deduced that her fifth score was better than 65, and that the 65 was dropped. Therefore, (A) is greater.
Sally's final score in economics is calculated by taking the mean of the best four of her five test scores. Sally received a final score of 78. Her first four test scores were 90, 80, 65, and 70. Which is the greater quantity?
(A) Her fifth test score
(B) 65
(A) is greater
(B) is greater
(A) and (B) are equal
It is impossible to determine which is greater from the information given
Explanation
Had Sally scored 65 or less on her fifth test, that would have been the dropped score, and her final score would have been the mean of 90, 80, 65, and 70. This is the sum of the scores divided by four:
Since Sally's mean was greater than this (78), it can be deduced that her fifth score was better than 65, and that the 65 was dropped. Therefore, (A) is greater.