DSQ: Calculating arithmetic mean - GMAT Quantitative
Card 0 of 168
Data sufficiency question- do not actually solve the question
Find the mean of a set of 5 numbers.
1. The sum of the numbers is 72.
2. The median of the set is 15.
Data sufficiency question- do not actually solve the question
Find the mean of a set of 5 numbers.
1. The sum of the numbers is 72.
2. The median of the set is 15.
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Statement 2 does not provide enough information about the mean as it can vary greatly from the median. Statement 1 is sufficient to calculate the mean, because even though it is impossible to calculate the set of numbers, the mean is calculated by dividing the sum by the total number of incidences in the set.
Statement 2 does not provide enough information about the mean as it can vary greatly from the median. Statement 1 is sufficient to calculate the mean, because even though it is impossible to calculate the set of numbers, the mean is calculated by dividing the sum by the total number of incidences in the set.
How much greater is the average of the integers from 500 to 700 than the average of the integers from 60 to 90?
How much greater is the average of the integers from 500 to 700 than the average of the integers from 60 to 90?
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In this case, average is also the middle value.
In this case, average is also the middle value.
Is 3 the average of a sequence？
(1)There are 3 numbers in the sequence
(2)The mode of the sequence is 3
Is 3 the average of a sequence？
(1)There are 3 numbers in the sequence
(2)The mode of the sequence is 3
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For statement (1), since we don’t know the value of each number, we cannot calculate the average of the sequence. For statement (2), the mode of the sequence is 3 means that “3” occurs most times in the sequence, but we cannot get to the average because we don’t know the value of other numbers. If we look at the two conditions together, we will know that there are 2 or 3 “3”'s in the sequence, but we don’t know exactly how many times “3” occurs. If the 3 numbers are all “3”s, then we can answer the question; If not, then we cannot answer the question. Thus both statements together are not sufficient.
For statement (1), since we don’t know the value of each number, we cannot calculate the average of the sequence. For statement (2), the mode of the sequence is 3 means that “3” occurs most times in the sequence, but we cannot get to the average because we don’t know the value of other numbers. If we look at the two conditions together, we will know that there are 2 or 3 “3”'s in the sequence, but we don’t know exactly how many times “3” occurs. If the 3 numbers are all “3”s, then we can answer the question; If not, then we cannot answer the question. Thus both statements together are not sufficient.
Calculate the average of the 5 integers.
Statement 1: They are consecutive even integers.
Statement 2: The smallest of the integers is 8 less than the largest of the five integers.
Calculate the average of the 5 integers.
Statement 1: They are consecutive even integers.
Statement 2: The smallest of the integers is 8 less than the largest of the five integers.
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We are looking for an average here. Statement 1 tells us we are looking for even consecutive integers such as 2, 4, 6, 8, and 10. Statement 2 tells us the difference between the smallest and largest integer; however, the difference between the largest and smallest of five consecutive even (or odd) integers will ALWAYS be 8, regardless of what 5 consecutive integers we choose; therefore the two statements don't give us enough information to solve for the average.
We are looking for an average here. Statement 1 tells us we are looking for even consecutive integers such as 2, 4, 6, 8, and 10. Statement 2 tells us the difference between the smallest and largest integer; however, the difference between the largest and smallest of five consecutive even (or odd) integers will ALWAYS be 8, regardless of what 5 consecutive integers we choose; therefore the two statements don't give us enough information to solve for the average.
1. If the arithmetic mean of five different numbers is 50, how many of the numbers are greater than 50?
(1) None of the five numbers is greater than 100.
(2) Three of the five numbers are 24, 25 and 26, respectively.
1. If the arithmetic mean of five different numbers is 50, how many of the numbers are greater than 50?
(1) None of the five numbers is greater than 100.
(2) Three of the five numbers are 24, 25 and 26, respectively.
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For statement (1), there are different combinations that satisfy the condition. For example, the five numbers can be 
or the five numbers can be 
. Therefore, we cannot determine how many of the numbers are greater than 
by knowing the first statement.
For statement (2), even though we know three of them, the two unknown numbers can both be greater than 
, or one smaller and one greater. Thus statement (2) is not sufficient.
Putting the two statements together, we know that the sum of the two unknown numbers is
Since none of them is greater than 100, both of them have to be greater than 50. Therefore when we combine the two statements, we know that there are two numbers that are greater than 50.
For statement (1), there are different combinations that satisfy the condition. For example, the five numbers can be
For statement (2), even though we know three of them, the two unknown numbers can both be greater than
Putting the two statements together, we know that the sum of the two unknown numbers is
Since none of them is greater than 100, both of them have to be greater than 50. Therefore when we combine the two statements, we know that there are two numbers that are greater than 50.
What is the arithmetic mean of a data set with twenty data values, each of which is a positive integer?
-
The sum of the odd values is 1,144 and the sum of the even values is 856.
-
The sum of the lowest ten values is 400 and the sum of the greatest ten values is 1,600.
What is the arithmetic mean of a data set with twenty data values, each of which is a positive integer?
-
The sum of the odd values is 1,144 and the sum of the even values is 856.
-
The sum of the lowest ten values is 400 and the sum of the greatest ten values is 1,600.
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The arithmetic mean of a data set is the sum of the values divided by the number of values in the set. Since we know that there are 20 values, all we need is the sum of the values. The sum can be easily deduced to be 2,000 from either one of the statements, so the arithmetic mean can be determined to be 
.
The answer is that either statement alone is sufficient.
The arithmetic mean of a data set is the sum of the values divided by the number of values in the set. Since we know that there are 20 values, all we need is the sum of the values. The sum can be easily deduced to be 2,000 from either one of the statements, so the arithmetic mean can be determined to be
The answer is that either statement alone is sufficient.
In western North Carolina, three towns lie along scenic rte 129 (the old Tallulah road). From north to south they are - Bear Creek, Sweet Gum and Robbinsville. Each town boasts of 6 establishments that sell moonshine (18 establishments total). Henry decides to test the alcohol content in all 18 establishemnts to see if there is a significant difference between the proof level in each town. He gathers the folloiwng evidence -
Robbinsville - 171, 170, 166, 180, 170, 177; avg=172.3; variance=26.67
Sweet Gum - 181, 177, 164, 190, 181, 180; avg = 178.8; variance=71.77
Bear Creek - 170, 180, 171, 191, 188, 188; avg=181.3; variance=83.87
The null hypothesis is: the 3 averages are the same.
Use the ANOVA F-test to see if we can reject the null hypothesis at the 95% level of confidence. Give both the F value and the percentile value (p) for the test.
Hint - ANOVA compares the variation between samples (MSB) to the variation within samples (MSW). If MSB is much greater than MSW, then we reject the null hypothesis and conclude that the 3 samples are significantly different. The F statistic is calculated MSB/MSW.
In western North Carolina, three towns lie along scenic rte 129 (the old Tallulah road). From north to south they are - Bear Creek, Sweet Gum and Robbinsville. Each town boasts of 6 establishments that sell moonshine (18 establishments total). Henry decides to test the alcohol content in all 18 establishemnts to see if there is a significant difference between the proof level in each town. He gathers the folloiwng evidence -
Robbinsville - 171, 170, 166, 180, 170, 177; avg=172.3; variance=26.67
Sweet Gum - 181, 177, 164, 190, 181, 180; avg = 178.8; variance=71.77
Bear Creek - 170, 180, 171, 191, 188, 188; avg=181.3; variance=83.87
The null hypothesis is: the 3 averages are the same.
Use the ANOVA F-test to see if we can reject the null hypothesis at the 95% level of confidence. Give both the F value and the percentile value (p) for the test.
Hint - ANOVA compares the variation between samples (MSB) to the variation within samples (MSW). If MSB is much greater than MSW, then we reject the null hypothesis and conclude that the 3 samples are significantly different. The F statistic is calculated MSB/MSW.
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The number of sample for each town = 6; N=6; E = 6-1 = 5
Overall average (of all 18 establishments) = 177.5 = O
Number of towns (columns) = 3; V1=3-1=2
total samles = 18; V2=18-3 = 15
For an F value of 2.13 with V1=2 and V2 = 15, p = .153
So, we can not reject the null hypothesis because our data would occur 15% of the time assuming the 3 averages are equal.
Note - the cutoff F value for 95% is 3.68
The number of sample for each town = 6; N=6; E = 6-1 = 5
Overall average (of all 18 establishments) = 177.5 = O
Number of towns (columns) = 3; V1=3-1=2
total samles = 18; V2=18-3 = 15
For an F value of 2.13 with V1=2 and V2 = 15, p = .153
So, we can not reject the null hypothesis because our data would occur 15% of the time assuming the 3 averages are equal.
Note - the cutoff F value for 95% is 3.68
A meteorologist is attempting to calculate the average (arithmetic mean) temperature highs for the past week. What is the arithmetic mean of the high temperatures for the last week?
1. The mean high temperature for the past 3 days is 75 degrees. The mean high temperature for the 4 days before that was 80 degrees.
2. The high temperatures for the last 7 days are as follows (in degrees): 80, 81, 79, 80, 77, 75, 73
A meteorologist is attempting to calculate the average (arithmetic mean) temperature highs for the past week. What is the arithmetic mean of the high temperatures for the last week?
1. The mean high temperature for the past 3 days is 75 degrees. The mean high temperature for the 4 days before that was 80 degrees.
2. The high temperatures for the last 7 days are as follows (in degrees): 80, 81, 79, 80, 77, 75, 73
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Statement 1 can be used to find the arithmetic mean by combining the information. For statement 1, the average for the last seven days is found by reversing the arithmetic mean equation. Let 
be the sum of the degrees for the past 3 days. Let 
be the sum of the degrees for the 4 days before that. Then we get
and 
so we can solve and find 
and 
.
Also, we know 
= the arithmetic mean for the last week.
So 
Statement 2 can be used to find the arithmetic mean using the arithmetic mean formula. This is the total sum, divided by the number of days. Thus,
Statement 1 can be used to find the arithmetic mean by combining the information. For statement 1, the average for the last seven days is found by reversing the arithmetic mean equation. Let
Also, we know
So
Statement 2 can be used to find the arithmetic mean using the arithmetic mean formula. This is the total sum, divided by the number of days. Thus,
When assigning a score for the term, a professor takes the mean of all of a student's test scores.
Joe is trying for a score of 90 for the term. He has one test left to take. What is the minimum that Joe can score and achieve his goal?
Statement 1: He has a median score of 85 so far.
Statement 2: He has a mean score of 87 so far.
When assigning a score for the term, a professor takes the mean of all of a student's test scores.
Joe is trying for a score of 90 for the term. He has one test left to take. What is the minimum that Joe can score and achieve his goal?
Statement 1: He has a median score of 85 so far.
Statement 2: He has a mean score of 87 so far.
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Knowing the median score is neither necessary nor helpful. What will be needed is the sum of the scores so far and the number of tests Joe has taken. But the number of tests taken is not given, and without this, there is no way to find the sum either.
Knowing the median score is neither necessary nor helpful. What will be needed is the sum of the scores so far and the number of tests Joe has taken. But the number of tests taken is not given, and without this, there is no way to find the sum either.
Choose the answer that best describes sufficient data to solve the problem.
3 numbers are given in increasing order. The arithemetic mean of the first two is 5 less than the arithmetic mean of all three. The sum of the first two numbers is equal to the arithmetic mean of the last two. What is the first number?
I. The second number is given.
II. The arithmetic mean of the first and third numbers is given.
Choose the answer that best describes sufficient data to solve the problem.
3 numbers are given in increasing order. The arithemetic mean of the first two is 5 less than the arithmetic mean of all three. The sum of the first two numbers is equal to the arithmetic mean of the last two. What is the first number?
I. The second number is given.
II. The arithmetic mean of the first and third numbers is given.
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3 numbers are given in increasing order.
I. The second number is given.
is determined
II. The arithmetic mean of the first and third numbers is given.
is given, and therefore 
is given.
We are given a system of equations by the prompt:
The arithemetic mean of the first two is 5 less than the arithmetic mean of all three.
The sum of the first two numbers is equal to the arithmetic mean of the last two.
Combining these two, we get: 
Thus, the second statement doesn't actually provide us with any information (we are still left with 2 equations and 3 variables, which cannot be solved for any particular number).
On the other hand, if y is determined, then 
relates 
and 
. Similarly, 
relates 
and 
. Since these give different equations, we could use them to solve for both 
and 
.
So the first gives sufficient data while the second does not.
3 numbers are given in increasing order.
I. The second number is given.
II. The arithmetic mean of the first and third numbers is given.
We are given a system of equations by the prompt:
The arithemetic mean of the first two is 5 less than the arithmetic mean of all three.
The sum of the first two numbers is equal to the arithmetic mean of the last two.
Combining these two, we get:
Thus, the second statement doesn't actually provide us with any information (we are still left with 2 equations and 3 variables, which cannot be solved for any particular number).
On the other hand, if y is determined, then
So the first gives sufficient data while the second does not.
What is the mean of 
, 
, 
, 
, 
, and 
?
Statement 1: 
Statement 2: 
What is the mean of
Statement 1:
Statement 2:
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The mean of a data set requires you to know the sum of the elements and the number of elements; you know the latter, but neither statement alone provides any clues to the former.
However, if you know both, you can add both sides of the equations as follows:
Rewrite as:
and divide by 9:
Now you know the sum, so divide it by 6 to get the mean:
.
The mean of a data set requires you to know the sum of the elements and the number of elements; you know the latter, but neither statement alone provides any clues to the former.
However, if you know both, you can add both sides of the equations as follows:
Rewrite as:
and divide by 9:
Now you know the sum, so divide it by 6 to get the mean:
If 
and 
, then what is the mean of 
, 
, 
, 
, and 
?
If
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Multiply the second equation by 2 on each side, and add it to the first equation.
Divide this sum by 5:
Multiply the second equation by 2 on each side, and add it to the first equation.
Divide this sum by 5:
Data sufficiency question
Determine the mean of a number set.
1. The mode is 7.
2. The median is 7.
Data sufficiency question
Determine the mean of a number set.
1. The mode is 7.
2. The median is 7.
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The median gives information about the center of a set of numbers, but is insufficient for calculating the mean. Additionally, the mode merely indicates which number is the most repeated value. Therefore, more information is required to calculate the mean.
The median gives information about the center of a set of numbers, but is insufficient for calculating the mean. Additionally, the mode merely indicates which number is the most repeated value. Therefore, more information is required to calculate the mean.
A professor records the average class grade for each exam. The average class grades for the semester are respectively:
What is the average class grade for the semester?
(1) 
(2) 
A professor records the average class grade for each exam. The average class grades for the semester are respectively:
What is the average class grade for the semester?
(1)
(2)
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The mean class grade is the sum of all average class grades divided by the number of grades:
Using Statement (1):
Therefore Statement (1) is sufficient to calculate the arithmetic mean for these grades.
Using statement (2):
Therefore 
Therefore Statement (2) is sufficient to calculate the arithmetic mean for these grades.
Each Statement ALONE IS SUFFICIENT to answer the question
The mean class grade is the sum of all average class grades divided by the number of grades:
Using Statement (1):
Therefore Statement (1) is sufficient to calculate the arithmetic mean for these grades.
Using statement (2):
Therefore
Therefore Statement (2) is sufficient to calculate the arithmetic mean for these grades.
Each Statement ALONE IS SUFFICIENT to answer the question
What is the value of 
?
(1) 
(2) The arithmetic mean of the numbers in the list is 
.
What is the value of
(1)
(2) The arithmetic mean of the numbers in the list is
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Statement (1) provides the value of y in terms of x, which is not enough to determine the value of x in the list we are given.
Statement (2) gives the arithmetic mean of the list. We can the write the following equation:
However, we cannot find the value of x using the information in Statement (2) only.
Using the information in Statement (1), we can replace y by x-4 in the previous equation:
We need both statements to calculate the value of x.
Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Statement (1) provides the value of y in terms of x, which is not enough to determine the value of x in the list we are given.
Statement (2) gives the arithmetic mean of the list. We can the write the following equation:
However, we cannot find the value of x using the information in Statement (2) only.
Using the information in Statement (1), we can replace y by x-4 in the previous equation:
We need both statements to calculate the value of x.
Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
During a particularly hectic week at Ballard High, Robert drank 5, 8, 3, 6, 2, 9, and 14 cans of Slurp Soda, respectively, on each of the 7 days. What is the product of Robert's mean and median soda consumption for that week?
During a particularly hectic week at Ballard High, Robert drank 5, 8, 3, 6, 2, 9, and 14 cans of Slurp Soda, respectively, on each of the 7 days. What is the product of Robert's mean and median soda consumption for that week?
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To find the mean, we find the sum of all of the values, and divide by how many there are:
To find the mean we rearrange the values in ascending numerical order and select the middle value:
The product, then, is 
To find the mean, we find the sum of all of the values, and divide by how many there are:
To find the mean we rearrange the values in ascending numerical order and select the middle value:
The product, then, is
Give the arithmetic mean of the first and third terms of an arithmetic sequence.
Statement 1: The fifth term of the sequence is 130.
Statement 2: The second term of the sequence is 100.
Give the arithmetic mean of the first and third terms of an arithmetic sequence.
Statement 1: The fifth term of the sequence is 130.
Statement 2: The second term of the sequence is 100.
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An arithmetic sequence is one in which each term is formed by adding the same number to its preceding term - the common difference.
Let 
be the first term, and 
be the common difference. The first five terms are
The arithmetic mean of two numbers is half the sum of the numbers. The arithmetic mean of the first and third terms is
,
which is the second term. Statement 2 alone gives this number as 100.
Now assume Statement 1 alone. Consider these two sequences, both of which can be seen to be arithmetic with fifth term 130:
The arithmetic mean of the first and third terms differ, as can be seen by looking at the second terms; in the first sequence, it is 127, and in the second, it is 100. That makes Statement 2 inconclusive.
An arithmetic sequence is one in which each term is formed by adding the same number to its preceding term - the common difference.
Let
The arithmetic mean of two numbers is half the sum of the numbers. The arithmetic mean of the first and third terms is
which is the second term. Statement 2 alone gives this number as 100.
Now assume Statement 1 alone. Consider these two sequences, both of which can be seen to be arithmetic with fifth term 130:
The arithmetic mean of the first and third terms differ, as can be seen by looking at the second terms; in the first sequence, it is 127, and in the second, it is 100. That makes Statement 2 inconclusive.
Give the arithmetic mean of 
and 
.
Statement 1: 
Statement 2: 
Give the arithmetic mean of
Statement 1:
Statement 2:
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The arithmetic mean of 
and 
is equal to 
.
Statement 1 alone gives us this value directly.
From Statement 2 alone, the value can be determined by dividing both sides by 4:
The arithmetic mean of
Statement 1 alone gives us this value directly.
From Statement 2 alone, the value can be determined by dividing both sides by 4:
Give the arithmetic mean of the second and third terms of an arithmetic sequence.
Statement 1: The fourth term of the sequence is 120.
Statement 2: The first term of the sequence is 0.
Give the arithmetic mean of the second and third terms of an arithmetic sequence.
Statement 1: The fourth term of the sequence is 120.
Statement 2: The first term of the sequence is 0.
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Assume Statement 1 alone. The sequences
and
are both arithmetic, each term being the previous term plus the same number (in the first case, this common difference is 40; in the second, it is 20). The fourth term is 120 in both cases, The second and third terms of the first sequence have arithmetic mean 
; the second and third terms of the second sequence have arithmetic mean 
. Therefore, the mean of those two terms cannot be determined for certain. A similar argument holds for Statement 2 alone being insufficient.
Now assume both statements. Let 
be the common difference of the sequences mentioned in Statement 2. By Statement 2, 0 is the first term, so the sequence will be
By the first statement, the fourth term is 120, so
The second terms is 
and the third term is 
, and their arithmetic mean is .
Assume Statement 1 alone. The sequences
and
are both arithmetic, each term being the previous term plus the same number (in the first case, this common difference is 40; in the second, it is 20). The fourth term is 120 in both cases, The second and third terms of the first sequence have arithmetic mean
Now assume both statements. Let
By the first statement, the fourth term is 120, so
The second terms is
Give the arithmetic mean of and .
Statement 1: A rectangle with length 
and width 
has area 500.
Statement 2: A triangle with base of length 
and height 
has area 250.
Give the arithmetic mean of
Statement 1: A rectangle with length
Statement 2: A triangle with base of length
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The area of a rectangle is the product of its length and width; the area of a triangle is half the product of its height and the length of its base. Therefore, from Statement 1, we get that
.
From Statement 2, we get that
, or, equivalently,
, or
In other words, the two statements are equivalent, so one of two things happens - either statement alone is sufficient, or both together are insufficient. We show that the latter is the case:
Case 1: 
The mean of the two is 
.
Case 2: 
The mean of the two is 
.
Therefore, knowing the area of the rectangle with these dimensions is not helpful to determining their arithmetic mean. This makes Statement 1, and, equivalently, both statements together, unhelpful.
The area of a rectangle is the product of its length and width; the area of a triangle is half the product of its height and the length of its base. Therefore, from Statement 1, we get that
From Statement 2, we get that
In other words, the two statements are equivalent, so one of two things happens - either statement alone is sufficient, or both together are insufficient. We show that the latter is the case:
Case 1:
The mean of the two is
Case 2:
The mean of the two is
Therefore, knowing the area of the rectangle with these dimensions is not helpful to determining their arithmetic mean. This makes Statement 1, and, equivalently, both statements together, unhelpful.