DSQ: Calculating arithmetic mean - GMAT Quantitative

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.

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.

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.

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.

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 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

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,

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.

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.

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:

.

Multiply the second equation by 2 on each side, and add it to the first equation.

Divide this sum by 5:

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 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

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.

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

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.

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:

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 .

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.