A set S had 20 different positive integers. A 21st integer is added

Question

A set S had 20 different positive integers. A 21st integer, equivalent to the mean of the set, was added to the set. Which of the following statistics for the group  may  have changed?

I. Mean
II. Median
III. Standard Deviation

A. II only
B. III only
C. I and II only
D. II and III only
E. I, II, and III

Krunaal · Answer

Mean

Let  T  be the sum of 20 different +ve integers, therefore mean =  \frac{T}{20}

Now, for 21 integers, new sum =  T + \frac{T}{20}  =  \frac{21T}{20} , and mean =  \frac{21T}{20*21}  =  \frac{T}{20}

Thus, the mean  does not  change.

Median

Let S = {1, 2, 3, 4,.......10, 11,........17, 18, 19, 100}

Mean = approx. 13.5

Median = 10.5

After adding the mean (13.5) to the list the median becomes 11

Thus, median  may change .

SD

Since we have 20 different +ve integers, we have an SD > 0

Now, whenever an integer y is added to a set where mean = m, if |y - m| < SD, then SD is reduced.

Here, |y - m| = 0 < SD. Therefore, SD is reduced.

Thus, SD  may change .

Answer D.

GMATinsight · Answer

CONCEPT 
- Inclusion of another value equal to Mean of a set in the set of number does NOT change mean of the set
- Values included equal to Mean in the set ALWAYS Decrease the standard deviation of the set

I. Mean -  Will remain same 
II. Median -  May or may not change 
III. Standard Deviation -  Will reduce

Answer: Option D

Related Video:

https://www.youtube.com/watch?v=32H3ZgxOpSU

Bunuel · Answer

Expert's Global Video Solution:

https://www.youtube.com/watch?v=TXMgJREiDVY

PSKhore · Answer

They've asked to add Mean, shouldn't you be adding T/20 and not T?

BinodBhai · Answer

Krunaal  the ques says, "a 21st integer term equivalent to the mean of the set is added to the set" ... my question is even consecutive terms can't have integer mean. In your answer, the mean you're adding is 13.5 which isn't an integer. I don't think question is phrased correctly.

When I tried doing this with a set of 20 integers - consecutive multiples of 2 from 2 to 40, I got B as my answer.

Bunuel · Answer

The mean of an even number of consecutive integers is always an integer and a half. The mean of an odd number of consecutive integers is the middle term, so an integer. For example, the mean of 1, 2, 3 is 2. So, the mean of a list of consecutive integers can be an integer if the list consists of an odd number of terms.

The integer part is irrelevant for this question. Yes, technically the example given in the solution violates the condition, but that's not the main point.

Adding a number equal to the mean to any list of numbers will not change the mean.

The median, however, could change. For example, in the list 1, 2, 6, 11, the mean is 5 and the median is 4. Adding 5 to the list changes the median to 5.

As for the standard deviation, adding a number equal to the mean will shrink the spread, thus reducing the standard deviation.

A set S had 20 different positive integers. A 21st integer is added

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A set S had 20 different positive integers. A 21st integer is added

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