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nusmavrik
Joined: 26 Nov 2009
Last visit: 03 Apr 2022
Posts: 467
Given Kudos: 36
Status:Impossible is not a fact. It's an opinion. It's a dare. Impossible is nothing.
Affiliations: University of Chicago Booth School of Business
Location: Singapore
Concentration: General Management, Finance
Schools: Chicago Booth - Class of 2015
25 Aug 2010, 07:41
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E is a collection of four ODD integers and the greatest difference between any two integers in E is 4. The standard deviation of E must be one of how many numbers?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7
25 Aug 2010, 08:43
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I know this question, I've posted it in my topic: ps-questions-about-standard-deviation-85897.html
But there is a typo, it should be:
E is a collection of four ODD integers and the greatest difference between any two integers in E is 4. The standard deviation of E must be one of how many numbers?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7
Let the smallest odd integer be 1, thus the largest one will be 5. We can have following 6 types of sets:
1. {1, 1, 1, 5} --> mean=2 --> |mean-x|=(1, 1, 1, 3);
2. {1, 1, 3, 5} --> mean=2.5 --> |mean-x|=(1.5, 1.5, 0.5, 2.5);
3. {1, 1, 5, 5} --> mean=3 --> |mean-x|=(2, 2, 2, 2);
4. {1, 3, 3, 5} --> mean=3 --> |mean-x|=(2, 0, 0, 2);
5. {1, 3, 5, 5} --> mean=3.5 --> |mean-x|=(2.5, 0.5, 1.5, 1.5);
6. {1, 5, 5, 5} --> mean=4 --> |mean-x|=(3, 1, 1, 1).
CALCULATING STANDARD DEVIATION OF A SET {x1, x2, ... xn}:
1. Find the mean, \(m\), of the values.
2. For each value \(x_i\) calculate its deviation (\(m-x_i\)) from the mean.
3. Calculate the squares of these deviations.
4. Find the mean of the squared deviations. This quantity is the variance.
5. Take the square root of the variance. The quantity is th SD.
Expressed by formula: \(standard \ deviation= \sqrt{variance} = \sqrt{\frac{\sum(m-x_i)^2}{N}}\).
You can see that deviation from the mean for 2 pairs of the set is the same, which means that SD of 1 and 6 will be the same and SD of 2 and 5 also will be the same. So SD of such set can take only 4 values.
Answer: B.
Hope it's clear.
But there is a typo, it should be:
E is a collection of four ODD integers and the greatest difference between any two integers in E is 4. The standard deviation of E must be one of how many numbers?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7
Let the smallest odd integer be 1, thus the largest one will be 5. We can have following 6 types of sets:
1. {1, 1, 1, 5} --> mean=2 --> |mean-x|=(1, 1, 1, 3);
2. {1, 1, 3, 5} --> mean=2.5 --> |mean-x|=(1.5, 1.5, 0.5, 2.5);
3. {1, 1, 5, 5} --> mean=3 --> |mean-x|=(2, 2, 2, 2);
4. {1, 3, 3, 5} --> mean=3 --> |mean-x|=(2, 0, 0, 2);
5. {1, 3, 5, 5} --> mean=3.5 --> |mean-x|=(2.5, 0.5, 1.5, 1.5);
6. {1, 5, 5, 5} --> mean=4 --> |mean-x|=(3, 1, 1, 1).
CALCULATING STANDARD DEVIATION OF A SET {x1, x2, ... xn}:
1. Find the mean, \(m\), of the values.
2. For each value \(x_i\) calculate its deviation (\(m-x_i\)) from the mean.
3. Calculate the squares of these deviations.
4. Find the mean of the squared deviations. This quantity is the variance.
5. Take the square root of the variance. The quantity is th SD.
Expressed by formula: \(standard \ deviation= \sqrt{variance} = \sqrt{\frac{\sum(m-x_i)^2}{N}}\).
You can see that deviation from the mean for 2 pairs of the set is the same, which means that SD of 1 and 6 will be the same and SD of 2 and 5 also will be the same. So SD of such set can take only 4 values.
Answer: B.
Hope it's clear.
09 Nov 2010, 19:06
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gettinit
J is a collection of four odd integers whose range is 4. The standard deviation of J must be one of how many numbers?
a 3
b 4
c 5
d 6
e 7
Please explain your thought process on this one. Thanks
a 3
b 4
c 5
d 6
e 7
Please explain your thought process on this one. Thanks
This is a good question though I did not like the wording very much. Instead of 'SD of J must be one one how many numbers', 'How many distinct values can SD of J take' is better. Anyway,
First I thought J is a set of four odd integers with range 4 so I said J = {1, x, y, 5}
Now x and y can take 3 different values: 1, 3 or 5
Either both x and y are same. This can be done in 3 ways.
Or x and y are different. This can be done in 3C2 ways = 3 ways
Total x and y can take values in 3 + 3 = 6 ways
Let me enumerate them for clarification:
{1, 1, 1, 5}, {1, 3, 3, 5}, {1, 5, 5, 5}, {1, 1, 3, 5}, {1, 1, 5, 5}, {1, 3, 5, 5}
These are the 6 ways in which you can choose the numbers.
Important thing: SD of {1, 1, 1, 5} and {1, 5, 5, 5} is same. Why?
SD measures distance from mean. It has nothing to do with the actual value of mean and actual value of numbers.
In {1, 1, 1, 5}, mean is 2. Three of the numbers are distance 1 away from mean and one number is distance 3 away from mean.
In {1, 5, 5, 5}, mean is 4. Three of the numbers are distance 1 away from mean and one number is distance 3 away from mean.
Similarly, {1, 1, 3, 5} and {1, 3, 5, 5} will have the same SD.
Then, {1, 3, 3, 5} will have a distinct SD and {1, 1, 5, 5} will have a distinct SD.
In all, there are 4 different values that SD can take in such a case.
Note: It doesn't matter what the actual numbers are. SD of 1, 3, 5, 7 is the same as SD of 12, 14, 16, 18. For detailed explanation of SD and how to calculate it, check the theory or Stats.
