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Math Expert V
Joined: 02 Sep 2009
Posts: 55804

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11 00:00

Difficulty:   85% (hard)

Question Stats: 42% (01:05) correct 58% (00:59) wrong based on 161 sessions

HideShow timer Statistics Is the range of set $$S$$ greater than its mean?

(1) All elements of set $$S$$ are negative

(2) The median of set $$S$$ is negative

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Math Expert V
Joined: 02 Sep 2009
Posts: 55804

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Official Solution:

Notice that the range of ANY set is more than or equal to zero.

(1) All elements of set $$S$$ are negative. The mean of a set with all negative elements is certainly negative so less than its range (which as discussed is always non-negative). Sufficient.

(2) The median of set $$S$$ is negative. So, there is at least one negative term is the set. Now, consider two cases:

A. If all elements in set $$S$$ are negative then we have the same scenario as above so $$Range \gt Mean$$;

B. If not all elements in set $$S$$ are negative then $$Range=Largest-Smallest$$, which will mean that $$\text{Range} \gt \text{Largest Element}$$ (that's because the smallest element in set $$S$$ is negative. For example consider the following set {-1, -1, 2}: the range of that set is $$Range=2-(-1)=3 \gt 2)$$. For the same reason the mean will be less than the largest element, so $$Range \gt Largest \gt Mean$$.

So, in any case $$Range \gt Mean$$. Sufficient.

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Manager  Joined: 18 Jan 2014
Posts: 164
Location: India
GMAT 1: 740 Q50 V40 Show Tags

Bunuel wrote:
Official Solution:

Notice that the range of ANY set is more than or equal to zero.

(1) All elements of set $$S$$ are negative. The mean of a set with all negative elements is certainly negative so less than its range (which as discussed is always non-negative). Sufficient.

(2) The median of set $$S$$ is negative. So, there is at least one negative term is the set. Now, consider two cases:

A. If all elements in set $$S$$ are negative then we have the same scenario as above so $$Range \gt Mean$$;

B. If not all elements in set $$S$$ are negative then $$Range=Largest-Smallest$$, which will mean that $$\text{Range} \gt \text{Largest Element}$$ (that's because the smallest element in set $$S$$ is negative. For example consider the following set {-1, -1, 2}: the range of that set is $$Range=2-(-1)=3 \gt 2)$$. For the same reason the mean will be less than the largest element, so $$Range \gt Largest \gt Mean$$.

So, in any case $$Range \gt Mean$$. Sufficient.

What if the set has only one number Set: {x}. Can we rule out this option saying that range is not defined for set consisting of one number? Or range is 0 for this set?
Math Expert V
Joined: 02 Sep 2009
Posts: 55804

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popov wrote:
Bunuel wrote:
Official Solution:

Notice that the range of ANY set is more than or equal to zero.

(1) All elements of set $$S$$ are negative. The mean of a set with all negative elements is certainly negative so less than its range (which as discussed is always non-negative). Sufficient.

(2) The median of set $$S$$ is negative. So, there is at least one negative term is the set. Now, consider two cases:

A. If all elements in set $$S$$ are negative then we have the same scenario as above so $$Range \gt Mean$$;

B. If not all elements in set $$S$$ are negative then $$Range=Largest-Smallest$$, which will mean that $$\text{Range} \gt \text{Largest Element}$$ (that's because the smallest element in set $$S$$ is negative. For example consider the following set {-1, -1, 2}: the range of that set is $$Range=2-(-1)=3 \gt 2)$$. For the same reason the mean will be less than the largest element, so $$Range \gt Largest \gt Mean$$.

So, in any case $$Range \gt Mean$$. Sufficient.

What if the set has only one number Set: {x}. Can we rule out this option saying that range is not defined for set consisting of one number? Or range is 0 for this set?

The range of a single element set is 0.
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I think this is a high-quality question and the explanation isn't clear enough, please elaborate. Hi,

what would be the range for a set (-1, -2,-3,-4)? would it be -3 or 3? Why range is always positive?
Math Expert V
Joined: 02 Sep 2009
Posts: 55804

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Saurav Arora wrote:
I think this is a high-quality question and the explanation isn't clear enough, please elaborate. Hi,

what would be the range for a set (-1, -2,-3,-4)? would it be -3 or 3? Why range is always positive?

The range is the difference between the largest and smallest elements of a set. The range of {-1, -2, -3, -4} is therefore -1 - (-4) = 3.
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Joined: 12 Jul 2013
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Bunuel wrote:
Saurav Arora wrote:
I think this is a high-quality question and the explanation isn't clear enough, please elaborate. Hi,

what would be the range for a set (-1, -2,-3,-4)? would it be -3 or 3? Why range is always positive?

The range is the difference between the largest and smallest elements of a set. The range of {-1, -2, -3, -4} is therefore -1 - (-4) = 3.

Hi

Elements : { -8,-9,-10 }
Range = -2
Mean = -9

Range > Mean

Elements : { -1,-2,-3,-4 ...... -10 }
Range= -9
Mean = -5

Mean > Range

Hence Statement 1 is insufficient .
Please let me know if otherwise
Math Expert V
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Posts: 55804

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rt1601 wrote:
Bunuel wrote:
Saurav Arora wrote:
I think this is a high-quality question and the explanation isn't clear enough, please elaborate. Hi,

what would be the range for a set (-1, -2,-3,-4)? would it be -3 or 3? Why range is always positive?

The range is the difference between the largest and smallest elements of a set. The range of {-1, -2, -3, -4} is therefore -1 - (-4) = 3.

Hi

Elements : { -8,-9,-10 }
Range = -2
Mean = -9

Range > Mean

Elements : { -1,-2,-3,-4 ...... -10 }
Range= -9
Mean = -5

Mean > Range

Hence Statement 1 is insufficient .
Please let me know if otherwise

The correct answer to the question is D, so there MUST be something wrong with your solution...

Let's see: the range is the difference between the largest and smallest elements of a set. The range CANNOT be negative. The range of {-8, -9, -10} is -8 - (-10) = 2 and the range of {-1, -2, -3, -4 ...... -10} is -1 - (-10) = 9.
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Concentration: General Management, Finance
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This question is quite tricky but a really good one! I fell for A only as I did not consider the negative numbers and itsimplications on the range to a sufficient degree.
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https://gmatclub.com/forum/verbal-strategies-268700.html#p2082192

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Intern  Joined: 17 Apr 2018
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I think this is a high-quality question and I agree with explanation. is range always greater than mean and median... if not, in which case?
Math Expert V
Joined: 02 Sep 2009
Posts: 55804

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zeeniagoyal wrote:
I think this is a high-quality question and I agree with explanation. is range always greater than mean and median... if not, in which case?

No. For example, take {1, 2}. Range = 1, while median = mean = 1.5.
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