Doubts on measures of Central Tendency

Question

Hi Everyone,

Just wanted to understand a concept:

Say we have a set of numbers with us and we are asked a few questions as below:

1. What numbers should be added to the list so that 
a. Mean remains the same.
b. Mean decreases
c. Mean increases
d. Standard Deviation increases
e. Standard Deviation decreasese
f. Standard Deviation remains the same

saurabh9gupta · Answer

Bunuel   mikemcgarry   chetan2u  - please help on this

saurabh9gupta · Answer

So I have found out the partial answer to this:

If we have a set of numbers and mean has some surprising properties

If say we have a set of numbers say - 1,2,3,4,5,6,7,8,9,10
Mean = 5.5
SD - 3.02765

And the property is - 
For mean to remain the same - the sum of numbers to be added should be n * original mean

Example - if I have to add 2 numbers, their sum should be 11 ( 2 * 5.5)
and my mean remains the same

For mean to increase - add numbers which are more than the orginal mean
For mean to decrease- add numbers which are less than the orginal mean

For Std. Dev, I am facing issues..
now can you guys help out?

Bunuel   chetan2u

chetan2u · Answer

hi...

Few immediate thoughts on this..
1) Increase each number by the same positive number
Mean will increase, but the spread of the number will remain the same, so SD will remain same as it is dependent on the spread around mean.
2) Decrease each number by the same positive number
Mean will decrease, but the spread of the number will remain the same, so SD will remain same as it is dependent on the spread around mean.
3) Multiply each number by the same positive integer >1.
Mean will increase, and the spread of the number will also increase, so SD increases.
Example 1, 3, 5... multiply y 2.. 2, 6, 10... spread has increased now
4) Division will have the opposite effect

gvij2017 · Answer

I am happy to respond here and hope that my explanation will appease your query.

1. Mean remain same. 
Now look, mean can be same in only in two case - first,when all the elements of set are same and you are adding the same no in set and second, when you add 0 in a set which is symmetric about 0. 
other wise even addition of 0, positive or negative number will change the mean- slightly or significantly depending on the value to be added.

2. mean decrease:
Add any value below mean of original set.

3. Mean increase 
Add any value above mean of original set.

4. Change in SD after addition of a number. 
It is quite tedious to answer. 
I did simple exercise here.

Let's take set A= (1, 2,3 4)
 Case 1:
mean= 2.5 
SD = 1.118

as we know SD=  
 
Case 2
Add 2.5 in set A 
mean= 2.5
SD= 1

SD reduced after addition of mean in original set.

Case 3 
add 2 in set A- a value below original mean
 Mean= 2.4
SD= 1.019

Case- 4 
Add 3 in set A- a value above original mean (By same delta from mean- 0.5 as in case 3) 
mean= 2.6 
SD= 1.019

Case 5 = Lets take value x for which SD remain same. 
Mean = 2.4 + x/5
SD= 1.118 
I calculated x= 4.74 and -3.74, therefore for x>4.74 and x<-3.74, SD will increase from original SD.

Friend! this is my explanation. Please give feedback.

gvij2017 · Answer

I did a small calculation in above post. But still I am little confused on the behavior of SD when we add value in a set. 
Case-1 When I add value equal to original mean, the revised SD reduced greatly. 
Case- 2 As I am moving far from mean revised SD is increasing from the SD in case-1.
Case- 3 At certain values addition, SD remains as SD in original case. beyond these revised SD will be greater than original.

What I have faced in many questions related to SD is that question simply ask when SD will be more affected by addition of new element or deletion of existing element in set or whether SD will increase or decrease when you change all elements.

I need your help to understand it clearly.

ScottTargetTestPrep · Answer

a) To keep the mean the same, the average of the numbers we add to the list must equal the mean. For instance, for the data set {1, 5, 9}, the mean is 5. If we add 5 to this list, the mean of the new list {1, 5, 5, 9} will also be 5. If we add 4 and 6 to the list (note the average of 4 and 6 is 5), we will obtain {1, 4, 5, 6, 9}, and the mean remains unchanged.

b) To decrease the mean, we simply add numbers whose average is less than the mean of our data set. For instance, for the same data set {1, 5, 9}, if we add 4 to the list (which is less than the mean), we will obtain {1, 4, 5, 9}. The mean of this set is 4.75. If we add 3 and 6 to this list (note the average of 3 and 6 is 4.5, which is less than the mean), we will obtain {1, 3, 5, 6, 9}. The mean of this new list is 4.8. 
 
c) To increase the mean, we add numbers whose average is greater than the mean of the data set. For instance, if we add 6 to the list {1, 5, 9}, the mean of the new list {1, 5, 6, 9} is 5.25. If we add 4 and 7 to this list (the average of which is 5.5), we will obtain {1, 4, 5, 7, 9}, and the mean of this new list is 5.2.

d) If we have a set of n elements with mean m and standard deviation s, adding elements that are s√(1+1/n) more or less than the mean will increase the standard deviation. For example, for the set {10 , 12}, we have n = 2, m = 11 and s = 1; therefore for this set, s√(1+1/n) corresponds to √(1+1/2) = s√(3/2), which is roughly 1.22. Adding 12.1 to this set will not increase the standard deviation since 12.1 is less than 11 + 1.22 = 12.22; but anything greater than 12.22 will.

e) To decrease the standard deviation, we should add number(s) close to the mean. For instance, for the same list {1, 5, 9}, if we add 6 to the list, the standard deviation of {1, 5, 6, 9} is about 2.86. The number that will decrease the standard deviation the most is, of course, the mean. If we add the mean (which is 5) to the list {1, 5, 9}, the standard deviation of the new list {1, 5, 5, 9} is about 2.83. There is no way we can decrease the standard deviation more than that by adding a single value. That being said, as you will see below, there is no hard and fast rule about the exact numbers to select that will definitely reduce the standard deviation, and finding those numbers would require some pretty intense math.

f) Keeping the standard deviation the same is complicated business. If we are adding only one value to the list, we would have to solve a quadratic equation, which is far beyond the scope of the GMAT, and we will obtain two solutions. Any value other than those two solutions will change the standard deviation. For the set {1, 5, 9}, the two values that will keep the standard deviation the same are 5 - (8√2)/3 and 5 + (8√2)/3. If we add any one of those values, the standard deviation will remain the same. If we add a value between those two values, the standard deviation will decrease, and if we add a value that is not between those two values, the standard deviation will increase.

gvij2017 · Answer

I am grateful to you. Your elaborated explanation cleared all hazy doubts. Thanks.

blitzkriegxX · Answer

I'd like to add some more important info and reiterate a few points to what Scott has explained well above -

A)  When standard deviation is 0,  then we can have 3 things possible - 
1. The numbers are all same. 
2. The set has only 1 number. 
3. The set has only 0.

B)  Adding a constant to each value  in the set  doesn't effect  SD.

C)  Multiplying a constant to each value  in the set  changes  the SD.

D)  SD is always  greater than or equal to zero.

E)  If you have to choose any one value to decrease the SD , then you better let the chosen value be the mean.

F)  You never have to calculate SD in Gmat.  Just calculate the mean and see how far spread the numbers are.

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