Mean - ISEE Upper Level (grades 9-12) Quantitative Reasoning
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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?
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?
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.
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.
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Consider the following data set:
Which of these numbers is greater than the others?
Consider the following data set:
Which of these numbers is greater than the others?
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.
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.
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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?
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?
Mark's grade is a weighted mean in which his hourly tests have weight 1 and his final has weight 2. If we call 
his final, then his term average will be
,
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 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.
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Which is true about the mean and the median of the following data set:
Which is true about the mean and the median of the following data set:
The mean of the data set is the sum of the elements divided by 5:
The median of the data set is the middle value when the values are arranged in ascending order, which here is the third-highest value. This is 79.
The mean exceeds the median by 
The mean of the data set is the sum of the elements divided by 5:
The median of the data set is the middle value when the values are arranged in ascending order, which here is the third-highest value. This is 79.
The mean exceeds the median by
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The course average for a chemistry class is the mean of five test scores. Anne has scores 
so far; Barb has scores 
so far. Which is the greater quantity?
(a) The score Anne must make to average 
(b) The score Barb must make to average 
The course average for a chemistry class is the mean of five test scores. Anne has scores
(a) The score Anne must make to average
(b) The score Barb must make to average
The only real comparison that needs to be made is between the two students' totals; the one with the lesser total needs a greater score to average 
.
(a) Anne's total: 
(b) Barb's total: 
Barb has fewer points so she needs more points to average 
. This makes (b) greater.
The only real comparison that needs to be made is between the two students' totals; the one with the lesser total needs a greater score to average .
(a) Anne's total:
(b) Barb's total:
Barb has fewer points so she needs more points to average
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A student's course average is determined by calculating the mean of five tests. Chuck is trying for an average of 
in the course; his first four test scores are 
Which is the greater quantity?
(a) The score Chuck needs on the fifth test to achieve his goal
(b) 
A student's course average is determined by calculating the mean of five tests. Chuck is trying for an average of
Which is the greater quantity?
(a) The score Chuck needs on the fifth test to achieve his goal
(b)
For Chuck to achieve an average of 
, his scores on the five tests must total 
. At current, his scores total 
, so he needs 
points to achieve his average. This makes (a) greater.
For Chuck to achieve an average of
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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: 
They rated Sue's performance with the following seven scores: 
Which of these quantities is the greater?
(a) Sally's score
(b) Sue's score
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
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.
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.
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A gymnastics meet has seven judges. After each routine, each judge assigns a merit-based score from 0 to 10; a contestant's score for the routine is the mean of all the judges' scores except for the highest and the lowest.
The seven judges individually assigned the following scores to one of Kathy's routines: 
Which is the greater quantity?
(a) Kathy's score for the routine
(b) 9.5
A gymnastics meet has seven judges. After each routine, each judge assigns a merit-based score from 0 to 10; a contestant's score for the routine is the mean of all the judges' scores except for the highest and the lowest.
The seven judges individually assigned the following scores to one of Kathy's routines:
Which is the greater quantity?
(a) Kathy's score for the routine
(b) 9.5
The highest and lowest scores of the seven are 9.9 and 9.3, so Kathy's score is the mean of the other five:
This makes (a) greater.
The highest and lowest scores of the seven are 9.9 and 9.3, so Kathy's score is the mean of the other five:
This makes (a) greater.
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Which is the greater quantity?
(a) The mean of the data set 
(b) 
Which is the greater quantity?
(a) The mean of the data set
(b)
The sum of the elements in the data set 
is
Divide by 5 to get the mean:
The sum of the elements in the data set
Divide by 5 to get the mean:
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Which is the greater quantity?
(a) The mean of the data set: 
(b) 
Which is the greater quantity?
(a) The mean of the data set:
(b)
The sum of the elements in the data set 
is:
Divide by 5 to get the mean:
(b) 
(b) is greater.
The sum of the elements in the data set
Divide by 5 to get the mean:
(b)
(b) is greater.
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Which is the greater quantity?
(a) The mean of the data set: 
(b) 
Which is the greater quantity?
(a) The mean of the data set:
(b)
(a) The mean of the data set is the sum of its elements divided by 
:
(b) 
(b) is greater.
(a) The mean of the data set is the sum of its elements divided by
(b)
(b) is greater.
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A student's grade in Professor Kalton's abstract algebra class is the mean of his or her five test scores.
Philip outscored Kellie on the first test by 8 points and on the second test by 5 points. They scored the same on the third test. Kellie outscored Philip by 7 points on the fourth test and by 6 points on the fifth.
Which is the greater quantity?
(a) Philip's grade
(b) Kellie's grade
A student's grade in Professor Kalton's abstract algebra class is the mean of his or her five test scores.
Philip outscored Kellie on the first test by 8 points and on the second test by 5 points. They scored the same on the third test. Kellie outscored Philip by 7 points on the fourth test and by 6 points on the fifth.
Which is the greater quantity?
(a) Philip's grade
(b) Kellie's grade
You do not need to take the two means; just compare the sums, since each will be divided by 5.
Let 
be Kellie's total points. Then since Philip outscored Kellie by 8 points and 5 points on two tests and scored fewer than Kellie by 7 points and 6 points on two tests, Philip's score is
.
Philip and Kellie scored the same number of points, making their mean test scores the same.
You do not need to take the two means; just compare the sums, since each will be divided by 5.
Let
Philip and Kellie scored the same number of points, making their mean test scores the same.
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A student's grade in Professor Jackson's Shakespeare class is the mean of his or her four best test scores out of five.
Craig and his brother Jerry have been in a friendly competition to see who can get the best grade in the class.
Craig outscored Jerry on the first test by 9 points and on the fifth test by 5 points. Jerry outscored Craig by 6 points on the second test and by 8 points on the fourth. Their scores were identical on the third.
Which is the greater quantity?
(a) Craig's grade
(b) Jerry's grade
A student's grade in Professor Jackson's Shakespeare class is the mean of his or her four best test scores out of five.
Craig and his brother Jerry have been in a friendly competition to see who can get the best grade in the class.
Craig outscored Jerry on the first test by 9 points and on the fifth test by 5 points. Jerry outscored Craig by 6 points on the second test and by 8 points on the fourth. Their scores were identical on the third.
Which is the greater quantity?
(a) Craig's grade
(b) Jerry's grade
This question cannot be answered.
Let 
stand for Jerry's total score after his lowest test is thrown out.
We need to compare the sums after the lowest test for each student is disregarded, since each will be divided by the four tests. But it is not known which test will be thrown out for each student.
If, for example, the first test is thrown out for both Craig and Jerry, Craig's total will be
,
and Jerry's score will be higher.
If the second test is thrown out for both Craig and Jerry, Craig's total will be
,
and Craig's score will be higher.
This question cannot be answered.
Let
We need to compare the sums after the lowest test for each student is disregarded, since each will be divided by the four tests. But it is not known which test will be thrown out for each student.
If, for example, the first test is thrown out for both Craig and Jerry, Craig's total will be
and Jerry's score will be higher.
If the second test is thrown out for both Craig and Jerry, Craig's total will be
and Craig's score will be higher.
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Katie's grade in her Shakespeare class is the mean of her best five test scores out of six tests taken.
Her test scores are 
.
Which is the greater quantity?
(a) Katie's grade
(b) 
Katie's grade in her Shakespeare class is the mean of her best five test scores out of six tests taken.
Her test scores are
Which is the greater quantity?
(a) Katie's grade
(b)
Take the sum of all of her test scores except for her lowest and divide that by the number of test scores included.
Take the sum of all of her test scores except for her lowest and divide that by the number of test scores included.
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John's grade in his economics class is the mean of his best five test scores out of the six tests he takes.
John's first five test scores are: 
Which is the greater quantity?
(a) The lowest score John must take to achieve a score of at least a score of 
for the term
(b)
John's grade in his economics class is the mean of his best five test scores out of the six tests he takes.
John's first five test scores are:
Which is the greater quantity?
(a) The lowest score John must take to achieve a score of at least a score of
(b)
John's current point sum is 
.
Even if John achieves a score lower than 
on his sixth test, he will have at least 
points and a minimum mean of at least 
. He can even score zero points on his last test and keep his average above what he wants, making (b) greater.
John's current point sum is
Even if John achieves a score lower than
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Consider the following set of numbers:
Quantity A: Median of the set
Quantity B: Mean of the set
Consider the following set of numbers:
Quantity A: Median of the set
Quantity B: Mean of the set
The median of the set of numbers is determined by arranging the numbers in numerical order and finding the middle number. In this case there are two middle numbers, 
and 
, so we find the average of those numbers, which gives us 
.
The mean is found by dividing the sum of elements by the number of elements in the set:
Quantity B is larger.
The median of the set of numbers is determined by arranging the numbers in numerical order and finding the middle number. In this case there are two middle numbers,
The mean is found by dividing the sum of elements by the number of elements in the set:
Quantity B is larger.
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Compare 
and 
:
The average of 
The average of 
Compare
The average of a list of terms can be found as follows:
So we can write:
So 
is greater than 
The average of a list of terms can be found as follows:
So we can write:
So
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Which is greater:
The mean of the data set 
Which is greater:
Mean of a data set 
is the sum of the data set values divided by the number of data:
So we have:
So the mean of the data set is greater than 
Mean of a data set
So we have:
So the mean of the data set is greater than
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Which one is greater:
The mean of the data set 
Which one is greater:
Mean of a data set 
is the sum of the data set values divided by the number of data:
So we have:
Mean of a data set
So we have:
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Which one is greater:
The mean of the data set 
Which one is greater:
Mean of a data set 
is the sum of the data set values divided by the number of data:
So we have:
So the mean of the data set is smaller than 
.
Mean of a data set
So we have:
So the mean of the data set is smaller than
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