Problem Solving

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

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 got it. The examples are pretty understandable.

Thanks

Sachin

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.

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.

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.