Set K consists of 4 integers. What is the median of K?

Hello,

Since the set K doesn't consist of evenly placed integers, having mean =3 doesn't help. I is insufficient
Mode tell nothing about median II is insufficient.

Combining i and ii

average is 3 of 4 integers, Sum=12 .
From II Mode is 3 . ie 3 has highest frequency Set K can be {3,3,3,3 } or {2,3,3,4} or {1,3,3,5,} .
Any case Median is 3.

OA =C

Hi All,

A few questions on Test Day are going to involve statistics concepts: mean, median, mode, range, standard deviation. When you see these concepts in DS questions, you have to consider the 'terms' that are used and the various possibilities that exist within those 'restrictions.'

We're told that Set K consists of 4 INTEGERS. We're asked for the MEDIAN of Set K. To find the median of this group, we have to put the numbers in order (from least to greatest) then take the average of the 'middle 2' terms.

Fact 1: The average (arithmetic mean) of K is 3.

For the average of 4 integers to be 3, the SUM must be 12.

IF the set is.....
{3,3,3,3] then the median is 3

IF the set it.....
{1,2,3,6} then the median is 2.5
Fact 1 is INSUFFICIENT

Fact 2: The mode of K is 3.

This tells us that 3 is the MOST COMMON integer (so it must appear MORE than once and be most frequent).

IF the set is....
{3,3,3,3} then the median is 3

IF the set is....
{3,3,4,5} then the median is 3.5
Fact 2 is INSUFFICIENT

Combined, we know...
The average is 3
The mode is 3

This means that we need AT LEAST two 3s and the sum of everything still must be 12

IF the set is....
{3,3,3,3} then the median is 3

IF the set is....
{2,3,3,4} then the median is 3

IF the set is....
{0,3,3,6} then the median is 3

The median will ALWAYS be 3.
Combined, SUFFICIENT

Final Answer:

GMAT assassins aren't born, they're made,
Rich

Set K consists of 4 integers. What is the median of K?

(1) The average (arithmetic mean) of K is 3.

(2) The mode of K is 3.

In the video explanation, it is said that since the mode is 3 the set of integers is {3,3,3,3} or {3,3,x,y}. I have a doubt why is the mode a minimum of two values here? Why cant the mode be repeated thrice in the set as follows {3,3,3,x}

Can someone please help?

Clearly, statements 1st and 2nd itself are not sufficient. Combining them, we get three different combinations (1,5),(2,4) and (3,3) and, interestingly enough in each of these cases we would get an unique value '3' for the median.

yes, I got that. However, definition of mode means value which is repeated the most times in a set. Why does the mode have max value of 2 here? and not 3?

Hi AmitLobo,

The mode can be repeated thrice, no problem. But, notice that in this problem {3,3,3,x} => is same as {3,3,3,3}. x has to be equal to 3 (sum = 12 from st.1).

Hope it helps.

Nice question. To prove that statement (2) is insufficient, we need to specify that there are 2 cases that satisfy condition (2) but they lead to 2 different result.

Case 1. Set K = {3, 3, 3, 3}. It's clear that the median of K is 3.

Case 2. Set K = {3, 3, 3, x}. No matter what x is, the median of K is still 3. This case has the same result as Case 1, so we could left it out.

Case 3. Set K = {3, 3, x, y}. And now, the median of K could be different. For example, if K = {3, 3, 5, 7} then the median of K is 4.

Hence, we need just 2 cases: case 1 & case 3 to prove that (2) is insufficient. No need to mention case 2 here. That's why the OE lefts out that case.

Dear AmitLobo,

I'm happy to respond.

First of all, I'm sorry if you have been waiting for a response all this time. This Magoosh subforum on GMAT Club sometimes gets more attention and sometimes less. I assume that you know that, since you are a Magoosh student, you can email the student help team either at help@magoosh.com or by clicking the purple "Help" button in the lower right-hand corner of any Magoosh page. You usually get a response from student help in about 24 hours.

In case you haven't already gotten an answer, I will provide one here. In a GMAT DS question, it's very easy to use picking numbers to prove that individual statements are insufficient--as soon as two different choices of numbers produce two different answers to the prompt question, we know that the statement is insufficient. At that stage of the solution, there is no reason to investigate every possibility.

At that stage, the instructor was evaluating the second statement, "The mode of K is 3," and trying to decide whether it is sufficient.
The choice of {3,3,3,3} produces a definitive answer to the prompt question--the median would be 3
The choice {3, 3, 11, 50} produces another answer to the prompt question--the median would be 7
Right there--BAM! Two different choices consistent with this statement produce two different answers to the prompt question. We are already done. We have determined without a doubt that the second statement, alone and by itself, is not sufficient.

Yes, the set could be {3, 3, 3, 51}, but there's no reason to investigate anything else--we have already determined what we wanted to determine about statement #2 at that stage of the problem, namely, that it is insufficient.

On Data Sufficiency, it's very important to be strategic: know exactly the information you need, and don't spend time exploring mathematical options that don't get you closer to the information you need.

Does all this make sense?
Mike

Official Explanation:

For this problem, we have few constraints, so we are going to have to start by picking numbers. Remember that picking numbers is an excellent way to demonstrate that a single statement or the pair of statements is not sufficient. If two possible choices consistent with a statement produce different answers to the prompt question, then BAM! we know right away that the statement is not sufficient. Proving that an individual statement or two statements are sufficient takes some logical reasoning: we can't establish sufficiency with picking numbers, only the absence of sufficiency.

Remember: all we know are the four elements of the set are integers: they could be positive, zero, or negative.

Statement #1: The average of K is 3.

We will approach this with picking numbers.

Choice #1 = {3, 3, 3, 3}

This has an average of 3 and the median is 3

Choice #2 = {0, 0, 0, 12}

This also has an average of 3 but the median is 0

Right there, BAM! Two different choices consistent with statement #1 lead to two different answers to the prompt question. Statement #1, alone and by itself, has to be not sufficient.


Statement #2: The mode of K is 3.

We will approach this with picking numbers.

Choice #1 = {3, 3, 3, 3}

This has a mode of 3 and the median is 3

Choice #2 = {3, 3, 5, 23}

This has a mode of 3 but a median of 4.

Again, BAM! Two different choices consistent with statement #2 lead to two different answers to the prompt question. Statement #2 alone and by itself, has to be not sufficient.


Combined statements:

Choice #1 = {3, 3, 3, 3}

Choice #2 = {2, 3, 3, 4}

Choice #3 = {– 5, 3, 3, 11}

Those three example of sets that have both a mean of 3 and a mode of 3. All of these have medians of 3. Picking numbers here doesn't prove anything, but it does suggest a pattern of reasoning.

Certainly in the set {3, 3, 3, 3}, both statements are true and the median is 3.

We absolutely can't have three 3's and one different number, because there's no way that set could possibly have an average of 3.

We could have two 3's, and two other numbers, but think about it.

We can't have {3, 3, bigger, biggest}, because that would have an average higher than 3.

Similarly, we can have {smallest, smaller, 3, 3} because that would have an average less than 3.

If we have two 3's and two other numbers, it absolute must be the case that one is larger than three and one is smaller than three, by equal amounts, so that they average out to 3. This would have to be:

{smaller, 3, 3, bigger}

This kind of set always has a median of 3.

Thus, for all possible cases consistent with the combined statements, the only possible answer to the prompt question is 3. Combined, the statements produce a definitive answer, so they are sufficient together.

Answer = (C)

Set K consists of 4 integers. What is the median of K?

Let's denote the 4 integers : x, y, z, w

(1) The average (arithmetic mean) of K is 3.

(x + y + z + w)/4 = 3 --> (x + y + z + w) = 12

1 + 2 + 3 + 6
1 + 3 + 4 + 4

Insufficient (different medians)

(2) The mode of K is 3.

1 3 3 5 ---> median = 3
1 1 3 3 ---> median = 2

Insufficient

C:

1 3 3 5
1 3 3 3
3 3 3 3

The median must be 3

Sufficient

C