In a group of 10 people, the median height is 70 inches, the average

Explanation:

Median for 10 people = 70
Means 2 person in the middle is having average = 70
So it can be same height also, i.e. 70, 70 or more values
as another person with Height 74 is added in a group of 10 people, so Median will be 6th person in the group when the height is in ascending.
So it can be or cannot be changed.

The average must be changed.

Range, Highest - lowest in Height, will not be affected.

IMO-A

This is my reasoning, please correct me if I am wrong!

In a set of a odd number of integers the median will always be an integers as the middle number will be an integer, hence the median can not stay the same if it started out as a rational (70,5). The average must change unless the added number happens to correspond to the current average, in this question that is NOT the case, as such the average needs to change. As no alternative includes all three alternatives we need not consider the range. The range only has to change if the extreme values changes, in this case we do not have any information that regarding the extreme values thus we can not with certainty say that the range has changes.

Since mean and median are different, so we know that this is NOT an evenly spaced set.

There are 10 people

_ _ _ _ _ _ _ _ _ _

Now, we know that the range is 12 so the smallest term is X and the largest term is X+12.

Median = \(\frac{a_5 + a_6}{2} = 70\)

=\(a_5 + a_6 = 140\)

We take \(a_5\) and \(a_6\) as close as possible so we get \(a_5 = a_6 = 70\)

Our set as of now looks like

_ _ _ _ 70 70 _ _ _ _

When 74 is added the median changes from \(\frac{a_5 + a_6}{2}\) to \(a_6 = 70\)

The median remains the same, but since a new term is added , which is far away from 70.5 , the average will change. We need to check if the range changes or not because if (smallest integer < 74 < greatest integer), then the range remains same else the range changes as well.


We are given that the average = 70.5 so Sum = 705 \((70.5 \times 10)\)

Now we need to take 2 integers such that there is a difference of 12 between the two. Let's take the two integers as 64 and 76.

Till here, our set looks like

64 _ _ _ 70 70 _ _ _ 76

Let's maximize the value of 76 so that our set looks like

64 _ _ _ 70 70 76 76 76 76

We need to fill the 3 gaps and the remaining sum is \(705 - 76 \times 4 - 140 - 64 = 197\)
We will distribute 197 in the remaining three gaps such that each number is less than 69.

64 65 65 67 70 70 76 76 76 76

Now, if we add 74 to this set, the updated set becomes

64 65 65 67 70 70 74 76 76 76 76

The median, range remains same but the average changes.

Now, is it possible to take a number outside the range.

What if our set is

61 _ _ _ 70 70 _ _ _ 73

In this case is it possible to reach the sum 705? Let's check, we maximize the values so that our set looks like

61 70 70 70 70 70 73 73 73 73

The sum of this set becomes 703 so we need 2 extra and the way to do that is changing our smallest integer to 62 and largest integer as 74.
Now, when we add another 74 to the list the range will not change.

Hence, only the average changes.

OA, A

In a group of 10 people, the median height is 70 inches, the average (arithmetic mean) height is 70.5 inches, and the range of the heights is 12 inches. If an additional person who is 74 inches tall joins the group, which of the three statistics must change?

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