For a certain set of n numbers, where n > 1, is the average (arithmeti

SOLUTION

For a certain set of n numbers, where n > 1, is the average (arithmetic mean) equal to the median?

(1) If the n numbers in the set are listed in increasing order, then the difference between any pair of successive numbers in the set is 2. This implies that the set is even;y spaced. In any evenly spaced set the mean (average) is equal to the median. Sufficient.

(2) The range of the n numbers in the set is 2(n- 1). This is completely useless. Not sufficient.

Answer: A.

P.S. The range is the difference between the smallest and largest elements of the set.
_________________

For a certain set of n numbers, where n > 1, is the average (arithmetic mean) equal to the median?

(1) If the n numbers in the set are listed in increasing order, then the difference between any pair of successive numbers in the set is 2.
(2) The range of the n numbers in the set is 2(n- 1).

Sol:

St 1: The above statement basically means that the terms in the set are in Arithmetic Progression and thus Mean=Median.

Note that if it is given that Mean=Median then it does not mean that the set is in order

St 2: The range of n numbers is 2(n-1)

Consider n=3

Then trange =4

Now if Set consists of elements 1,3 and 5 then Mean=Median
but set consists of 1,4,5 then Mean is not equal to Median

Ans is A

650 levl is okay
_________________

A) x-4,x-2,x,x+2,x+4 (condition is successive difference is 2)
average = (x-4+x+4)/2 = 2x/2 = x
median = x

x-4,x-2,x,x+2,x+4,x+6 (condition is successive difference is 2)
average = (x-4+x+6)/2 = (2x+2)/2 = 2(x+1)/2 = x+1
median = x+1

SUFFICENT

B) What is a range? largest number - smallest number. It doesn't tell us much about the mean or median.
You can try some numbers here.
Constraint here is n > 1 which means you can take only 2 or 3 nos and find their average and that should be sufficient to answer the question.

INSUFFICIENT
Answer A

Difficulty level - 650

I thought that 2(n-1) was the range for both n consecutive even/odd integers
Therefore, we still have an evenly spaced set
Thus, 2 was sufficient

Could someone please advice where am I going wrong here?

Thanks!
Cheers
J

But the reverse is not necessarily true: not all n-element sets which have the range equal to 2(n - 1) are consecutive odd or consecutive even integers.

For example, {2, 3, 6} has the range equal to 2(n - 1) but this set is not of that type.
_________________

If numbers are listed in increasing order (consecutive even numbers, consecutive odd numbers and consecutive multiples), median=mean. Hence, since the condition 1) is always yes, the condition is sufficient. The correct answer choice is A.


l For cases where we need 3 more equations, such as original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 80% chance that E is the answer (especially about 90% of 2 by 2 questions where there are more than 3 variables), while C has 15% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since E is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, C or D.

A neat little trick to remember is that for any series that is an Arithmetic progression, namely difference between each successive term is constant, the median is always = mean. I'll try to prove it below. Let us say there are n terms. There are two pssibillities n is odd or n is even. Let us say the constant difference is d (2 in the case of this problem).

1st term: a, 2nd term = a+d, 3rd term = a+2*d .... nth term = a+(n-1)*d
Adding all you get Sum = a*n +[ (1+2+3...+(n-1))]*d = a*n +[ n*(n-1)/2]*d ( another interesting result sum of the first n-1 integers is n*(n-1)/2]. Hence the Average = Sum/n = a+(n-1)/2*d

n is odd: Median = (n+1)/2 th term = a+(n-1)/2*d = Average so if n is odd we have proved avg always equal to mean.
n is even: Median = average of n/2 and (n/2+1)th term =[ a+(n/2-1)*d+a+(n/2)*d ]/2 = a+(n-1)*d/2 = Average so if n is even too we have proved avg always equal to mean. Thus (1) is always sufficient to answer such questions of if avg = median.

(2) Since only range is given we cant determine anything about the numbers in between so this informatino is insufficient. Hence answer is A.
_________________

I am confused about median. Median is required to be part of set or not?

Bunuel

We are required to state whether the set mean and median are equal or not. It can be there in set or not depending upon the number of elements in n. Since the no. In set is at consecutive distance.
If no. Of elements are odd the middle value is mean as well as median.
If no. Of elements are even. Then also average of two middle values is median and average of all will be same as median as well.

Posted from my mobile device
_________________

for option B, shouldn't we consider only those sets where the formula 2n-2 works? i.e only consecutive even numbers?

difference between any pair....is confusing i thought its talking about ....(a,b).....(c,d)....then total 4 numbers...and difference is between them (something like that)

Question Stem Analysis:

We are given a set containing more than one numbers. We need to determine whether the mean is equal to the median for this set. Recall that in an evenly spaced set, the mean is equal to the median.

Statement One Alone:

\(\Rightarrow\) If the n numbers in the set are listed in increasing order, then the difference between any pair of successive numbers in the set is 2.

This means that the set is an evenly spaced set. Thus, as we noted above, the mean is equal to the median for this set. Statement one alone is sufficient.

Eliminate answer choices B, C, and E.

Statement Two Alone:

\(\Rightarrow\) The range of the n numbers in the set is 2(n- 1).

If our set is {1, 3}, then n = 2. Note that the range of this set is 3 - 1 = 2, which is equal to 2(n - 1) = 2(2 - 1) = 2. In this case, the mean is equal to the median.

If our set is {1, 2, 5}, then n = 3. Note that the range of this set is 5 - 1 = 4, which is equal to 2(n - 1) = 2(3 - 1) = 4. However, the mean of this set (which is 8/3) is not equal to the median (which is 2).

Since there is more than one possible answer to the question, statement two alone is not sufficient.

Answer: A
_________________