Arithmetic mean

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sachin_yadav: Master | Next Rank: 500 Posts; Posts: 212; Joined: Mon Dec 06, 2010 12:52 amFollowed by: 1 member; Location: India; Thanked: 5 times

by sachin_yadav » Fri Nov 07, 2014 9:04 am

Please help me in understanding the following points with the help of an example. These points are related to Arithmetic mean.

1.If in a set of numbers, the average = the highest or the lowest number, all the numbers will have to be equal.
2.If the average of a few consecutive integers is 0, then there will be an odd number of integers.

Regards
Sachin

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Fri Nov 07, 2014 9:15 am

sachin_yadav wrote:Please help me in understanding the following points with the help of an example. These points are related to Arithmetic mean.

1.If in a set of numbers, the average = the highest or the lowest number, all the numbers will have to be equal.
2.If the average of a few consecutive integers is 0, then there will be an odd number of integers.

Regards
Sachin

1. Without getting into a mathematical proof, the average value, is a value that best represents all of the values. How can the average turn out to be either the largest or smallest value AND be the value that best represents ALL values. They must be all equal

2. When all values are equally spaced, the mean = median. So, the median here is 0.
If the average = 0, then the SUM of all values = 0
So, each negative value must be canceled out by a positive value of equal size.
So, we have 0 as the middlemost values and PAIRS of values that cancel each other out (eg, -1 and 1)
0 plus a bunch of pairs means there is an ODD number of values altogether.

Cheers,
Brent

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sachin_yadav: Master | Next Rank: 500 Posts; Posts: 212; Joined: Mon Dec 06, 2010 12:52 amFollowed by: 1 member; Location: India; Thanked: 5 times

by sachin_yadav » Fri Nov 07, 2014 10:18 am

Thanks Brent,

Point 2 is fine.
Point 1: Ok, i think i am getting your point. Correct if i am wrong
average with the highest number = 4,4,4,4,5,5,5,5,5,5 = 4.6
average with the lowest number = 4,4,4,4,4,4,5,5,5,5 = 4.4

So, average cannot represent all values.

Sachin

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Fri Nov 07, 2014 11:15 am

sachin_yadav wrote:Thanks Brent,

Point 2 is fine.
Point 1: Ok, i think i am getting your point. Correct if i am wrong
average with the highest number = 4,4,4,4,5,5,5,5,5,5 = 4.6
average with the lowest number = 4,4,4,4,4,4,5,5,5,5 = 4.4

So, average cannot represent all values.

Sachin

I'm not sure what you mean by "average with the lowest number," but consider a few things.
4,4,4,4,5,5,5,5,5,5: average = 4.6 (in other words, 4.6 best represents all of the values)
4,4,4,4,4,4,5,5,5,5 = 4.4 (in other words, 4.4 best represents all of the values)
Notice that, in both cases, the average is BETWEEN the highest and lowest values (of 4 and 5)

A few other examples;
1,2,3,4: average = 2.5 (in other words, 2.5 best represents all of the values)
Notice that 2.5 is BETWEEN the greatest value (4) and the lowest value (1)

6,6,6,6,7: average = 6.2 (in other words, 6.2 best represents all of the values)
Notice that 6.2 is BETWEEN the greatest value (7) and the lowest value (6)

Now consider the following
7,7,7,7,7,7,7,7: average = 7 (in other words, 7 best represents all of the values)
Notice that 7 = the greatest value = the lowest value

Does that help?

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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sachin_yadav: Master | Next Rank: 500 Posts; Posts: 212; Joined: Mon Dec 06, 2010 12:52 amFollowed by: 1 member; Location: India; Thanked: 5 times

by sachin_yadav » Fri Nov 07, 2014 12:16 pm

I got it. The examples are pretty understandable.

Thanks

Sachin

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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 » Mon Nov 10, 2014 11:13 am

sachin_yadav wrote:Please help me in understanding the following points with the help of an example. These points are related to Arithmetic mean.

1.If in a set of numbers, the average = the highest or the lowest number, all the numbers will have to be equal.
2.If the average of a few consecutive integers is 0, then there will be an odd number of integers.

Let's take the first statement conceptually. If all the numbers the same except for one, the distinct number will do something to the average: if it's GREATER than the rest of the terms, it will RAISE the average; if it's LESS than the rest of the terms, it will LOWER the average.

If we wanted to prove this, suppose we have a set of (n + 1) numbers, n of which are equal to x, and one of which is equal to y. Further suppose that the average is x. This gives us the equation

n*x + y
------- = x
n+1

or nx + y = x(n+1)
or nx + y = nx + x
or y = x

So the terms MUST all be the same.

The second assertion is similar. If the average is 0, then the sum of the positive integers and the sum of negative integers must cancel out. Since we have consecutive integers, any positive integer x in the set must be accompanied by the negative integer -x. (For instance, if we have 1, we must also have -1.) Hence we must HAVE two (one pos, one neg) of every absolute value, plus 0, giving us an odd number of terms.

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Mathsbuddy: Master | Next Rank: 500 Posts; Posts: 447; Joined: Fri Nov 08, 2013 7:25 amFollowed by: 1 member; Thanked: 25 times

by Mathsbuddy » Tue Nov 11, 2014 5:15 am

sachin_yadav wrote:Thanks Brent,

Point 2 is fine.
Point 1: Ok, i think i am getting your point. Correct if i am wrong
average with the highest number = 4,4,4,4,5,5,5,5,5,5 = 4.6
average with the lowest number = 4,4,4,4,4,4,5,5,5,5 = 4.4

So, average cannot represent all values.

Sachin

Hi Sachin,

With your list of numbers for Point 1, your average of 4.6 does not equal the highest value of 5.
Therefore all values must be the same, e,g, 5, 5, 5, 5, 5. Top value = 5, average = 5.

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GMATinsight: Legendary Member; Posts: 1100; Joined: Sat May 10, 2014 11:34 pmFollowed by: 24 members; Location: New Delhi, India; Thanked: 205 times

by GMATinsight » Fri Nov 28, 2014 6:55 am

sachin_yadav wrote:Please help me in understanding the following points with the help of an example. These points are related to Arithmetic mean.

1.If in a set of numbers, the average = the highest or the lowest number, all the numbers will have to be equal.
2.If the average of a few consecutive integers is 0, then there will be an odd number of integers.

Regards
Sachin

In addition you may also note that Standard Deviation = 0 if the Average = Highest number or lowest number

Another view to verify the fact mentioned in point 1 is "Mean represent the middle value of the set and when the middle value is equal to Highest then certainly the lowest also must be equal to mean only.

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