Median of an even set

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uniyal01: Junior | Next Rank: 30 Posts; Posts: 12; Joined: Fri Jan 15, 2016 9:15 am; Thanked: 2 times

by uniyal01 » Mon Jul 04, 2016 2:27 pm

Is it true that when the number of terms in a set is even, then it can't be concluded that 50% of terms are greater/less than or equal to the median?
But then, the I found the following statement in an official answer: The average of a set of consecutive integers is also the middle of the set, with an equal number of terms greater and smaller than it.

Is that statement only ALWAYS true for when the number of consecutive terms are NOT even?

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OptimusPrep: Master | Next Rank: 500 Posts; Posts: 410; Joined: Fri Mar 13, 2015 3:36 amFollowed by: 8 members; Location: Worldwide; Thanked: 120 times; GMAT Score:770

by OptimusPrep » Mon Jul 04, 2016 8:07 pm

uniyal01 wrote:Is it true that when the number of terms in a set is even, then it can't be concluded that 50% of terms are greater/less than or equal to the median?
But then, the I found the following statement in an official answer: The average of a set of consecutive integers is also the middle of the set, with an equal number of terms greater and smaller than it.

Is that statement only ALWAYS true for when the number of consecutive terms are NOT even?

Median of a set with even number of terms = average of middle two terms when the set is in ascending/descending order
Median of a set with odd number of terms = middle term when the set is in ascending/descending order

In an equidistant set such as a set of consecutive integers, the median will always be the average of all the terms.

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by [email protected] » Mon Jul 04, 2016 9:04 pm

Hi uniyal01,

When dealing with GMAT questions, you really have to pay careful attention to the information that you're given. The MEDIAN of a group and the AVERAGE of that group are not necessarily the same thing.

However, when dealing with CONSECUTIVE INTEGERS, the median and the average will be the SAME (and the same number of terms will be above the median as below it).

For example, a group with an odd number of terms....

1, 2, 3, 4, 5

Average = 15/5 = 3
Median = 3

And a group with an even number of terms...

2, 3, 4, 5

Average = 14/4 = 3.5
Median = (3+4)/2 = 3.5

If you're NOT dealing with CONSECUTIVE INTEGERS though, these rules don't necessarily hold true.

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

Contact Rich at [email protected]

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800_or_bust: Master | Next Rank: 500 Posts; Posts: 199; Joined: Sat Apr 26, 2014 10:53 amFollowed by: 4 members; Thanked: 16 times; GMAT Score:780

by 800_or_bust » Tue Jul 05, 2016 6:25 am

uniyal01 wrote:Is it true that when the number of terms in a set is even, then it can't be concluded that 50% of terms are greater/less than or equal to the median?
But then, the I found the following statement in an official answer: The average of a set of consecutive integers is also the middle of the set, with an equal number of terms greater and smaller than it.

Is that statement only ALWAYS true for when the number of consecutive terms are NOT even?

I'm confused. You can NEVER conclude that 50% of the terms are greater than or equal to the median, and 50% are less than or equal to the median. In an extreme case (i.e. a set with SD=0) all 100% of the terms could be equal to the median, in which case 100% of the terms are greater than or equal to the median and 100% of the terms are less than or equal to the median. This would be true regardless of whether the set had an even or odd number of terms.

800 or bust!

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Matt@VeritasPrep: GMAT Instructor; Posts: 2630; Joined: Wed Sep 12, 2012 3:32 pmFollowed by: 119 members; Location: East Bay all the way; Thanked: 625 times; GMAT Score:780

by Matt@VeritasPrep » Thu Jul 07, 2016 3:46 pm

800_or_bust wrote:You can NEVER conclude that 50% of the terms are greater than or equal to the median, and 50% are less than or equal to the median.

We certainly can.

{1, 2, 3, 4}

Only 1 and 2 satisfy x â‰¤Â median, and only 3 and 4 satisfy x â‰¥Â median, so again we'd have 50% â‰¤Â and 50% â‰¥.

To satisfy x â‰¤Â y, you don't need to have some x = y.

Save $100 on any VP online course!

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ceilidh.erickson: GMAT Instructor; Posts: 2095; Joined: Tue Dec 04, 2012 3:22 pmFollowed by: 247 members; Thanked: 1443 times

by ceilidh.erickson » Sat Jul 09, 2016 9:49 am

uniyal01 wrote:Is it true that when the number of terms in a set is even, then it can't be concluded that 50% of terms are greater/less than or equal to the median?
But then, the I found the following statement in an official answer: The average of a set of consecutive integers is also the middle of the set, with an equal number of terms greater and smaller than it.

Is that statement only ALWAYS true for when the number of consecutive terms are NOT even?

I think the fundamental thing that you're missing here is that when you have an EVEN number of terms in a consecutive set, the median/average is NOT one of your terms. Look at Rich's example:
2, 3, 4, 5
Average = 14/4 = 3.5
Median = (3+4)/2 = 3.5
3.5 is NOT a term in the set. Thus, half the terms are less than 3.5, and half are greater.

When you have an ODD number of terms in a consecutive set, the median/average IS one of the terms in the set. Again look at Rich's example:
1, 2, 3, 4, 5
Average = 15/5 = 3
Median = 3
3 IS a term in the set. Two terms are less, two terms are greater. (That's not the same as 50% less, 50% greater, because the 3 accounts for 20% of the terms in this particular set).

Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education

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ceilidh.erickson: GMAT Instructor; Posts: 2095; Joined: Tue Dec 04, 2012 3:22 pmFollowed by: 247 members; Thanked: 1443 times

by ceilidh.erickson » Sat Jul 09, 2016 9:50 am

800_or_bust wrote: I'm confused. You can NEVER conclude that 50% of the terms are greater than or equal to the median, and 50% are less than or equal to the median. In an extreme case (i.e. a set with SD=0) all 100% of the terms could be equal to the median, in which case 100% of the terms are greater than or equal to the median and 100% of the terms are less than or equal to the median. This would be true regardless of whether the set had an even or odd number of terms.

We're specifically talking about consecutive / evenly spaced sets here, so the example of all terms being equal does not apply.

Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education

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