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Standard Deviation Query

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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by surajgarg » Thu Jul 22, 2010 9:57 am
How does one approach SD related problems which are presented in the examples below.

1. If the mean of a data set is 75 and the standard deviation is 10, what is the range of scores that fall within one standard deviation of the mean?

2. The mean score of a class on a test was 60 and the standard deviation was 15. If Elena's score was within 2 standard deviations of the mean, what is the lowest score she could have received?

What is meant by within 'x' SD of the mean?

Also pls help me with some notes on SD that would help in the GMAT. I have the ones put up by papgust :)
Looking for something different from that.[/url]
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by Stuart@KaplanGMAT » Thu Jul 22, 2010 10:14 am
surajgarg wrote:How does one approach SD related problems which are presented in the examples below.

1. If the mean of a data set is 75 and the standard deviation is 10, what is the range of scores that fall within one standard deviation of the mean?

2. The mean score of a class on a test was 60 and the standard deviation was 15. If Elena's score was within 2 standard deviations of the mean, what is the lowest score she could have received?

What is meant by within 'x' SD of the mean?

Also pls help me with some notes on SD that would help in the GMAT. I have the ones put up by papgust :)
Looking for something different from that.[/url]
The easiest way to attack these sorts of questions is to plot the mean and the SDs on a number line. Let's work with your second question, because the first will be a snap once we do the second.

The mean is 60, so we put that in the middle of our diagram:

-----------------------------60-------------------------------

The standard deviation is 15, so we count off in blocks of 15 both below and above the mean:

30-----------45------------60------------75------------90

Each of those intervals is one standard deviation. So, we'd say that 75 is one SD above the mean and 45 is one SD below the mean.

The question tells us that her score is within 2 SDs of the mean, so we count off 2 intervals in either direction - that takes us to 30 (below) and 90 (above). Accordingly, she scored somewhere between 30 and 90 (inclusive). The question asks for her lowest possible score: choose 30.

Now try the first question you posted!
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by surajgarg » Thu Jul 22, 2010 10:20 am
Stuart Kovinsky wrote:
surajgarg wrote:How does one approach SD related problems which are presented in the examples below.

1. If the mean of a data set is 75 and the standard deviation is 10, what is the range of scores that fall within one standard deviation of the mean?

2. The mean score of a class on a test was 60 and the standard deviation was 15. If Elena's score was within 2 standard deviations of the mean, what is the lowest score she could have received?

What is meant by within 'x' SD of the mean?

Also pls help me with some notes on SD that would help in the GMAT. I have the ones put up by papgust :)
Looking for something different from that.[/url]
The easiest way to attack these sorts of questions is to plot the mean and the SDs on a number line. Let's work with your second question, because the first will be a snap once we do the second.

The mean is 60, so we put that in the middle of our diagram:

-----------------------------60-------------------------------

The standard deviation is 15, so we count off in blocks of 15 both below and above the mean:

30-----------45------------60------------75------------90

Each of those intervals is one standard deviation. So, we'd say that 75 is one SD above the mean and 45 is one SD below the mean.

The question tells us that her score is within 2 SDs of the mean, so we count off 2 intervals in either direction - that takes us to 30 (below) and 90 (above). Accordingly, she scored somewhere between 30 and 90 (inclusive). The question asks for her lowest possible score: choose 30.

Now try the first question you posted!
Thanks Stuart. I understood the solution approach to the problems. But I am still not very clear on the theory.

Does it mean that if we know the SD and the mean we just need to add/subtract the SD to the mean to get the numbers on either side of the mean? And how is that?
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by Stuart@KaplanGMAT » Thu Jul 22, 2010 12:25 pm
surajgarg wrote:Thanks Stuart. I understood the solution approach to the problems. But I am still not very clear on the theory.

Does it mean that if we know the SD and the mean we just need to add/subtract the SD to the mean to get the numbers on either side of the mean? And how is that?
If by "numbers on either side of the mean" you mean "the actual values that exist in that particular set", then no. From merely the SD and the mean you cannot reconstruct a set, since there are an infinite number of sets that will generate the same results (this is true even in most cases in which you know the number of terms; the one exception is when SD=0, since that means that every term equals the mean).
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by divineacclivity » Sun Sep 29, 2013 6:30 pm
Stuart Kovinsky wrote:
surajgarg wrote:How does one approach SD related problems which are presented in the examples below.

1. If the mean of a data set is 75 and the standard deviation is 10, what is the range of scores that fall within one standard deviation of the mean?

2. The mean score of a class on a test was 60 and the standard deviation was 15. If Elena's score was within 2 standard deviations of the mean, what is the lowest score she could have received?

What is meant by within 'x' SD of the mean?

Also pls help me with some notes on SD that would help in the GMAT. I have the ones put up by papgust :)
Looking for something different from that.[/url]
The easiest way to attack these sorts of questions is to plot the mean and the SDs on a number line. Let's work with your second question, because the first will be a snap once we do the second.

The mean is 60, so we put that in the middle of our diagram:

-----------------------------60-------------------------------

The standard deviation is 15, so we count off in blocks of 15 both below and above the mean:

30-----------45------------60------------75------------90

Each of those intervals is one standard deviation. So, we'd say that 75 is one SD above the mean and 45 is one SD below the mean.

The question tells us that her score is within 2 SDs of the mean, so we count off 2 intervals in either direction - that takes us to 30 (below) and 90 (above). Accordingly, she scored somewhere between 30 and 90 (inclusive). The question asks for her lowest possible score: choose 30.

Now try the first question you posted!
2. The mean score of a class on a test was 60 and the standard deviation was 15. If Elena's score was within 2 standard deviations of the mean, what is the lowest score she could have received?

So, would that also mean that Elena's score is NOT a part of the set whose standard deviation is 15?
I say so because standard deviation itself means that the lowest and highest elements of a set are 'standard deviation' value away from the mean i.e. the lowest elements of the set could be 60 - 15 = 45 and the highest could be 60+15=75. Am I wrong in that sense?
Please correct me. Thanks.
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by vinay1983 » Tue Oct 01, 2013 4:06 am
divineacclivity wrote:
Stuart Kovinsky wrote:
surajgarg wrote:How does one approach SD related problems which are presented in the examples below.

1. If the mean of a data set is 75 and the standard deviation is 10, what is the range of scores that fall within one standard deviation of the mean?

2. The mean score of a class on a test was 60 and the standard deviation was 15. If Elena's score was within 2 standard deviations of the mean, what is the lowest score she could have received?

What is meant by within 'x' SD of the mean?

Also pls help me with some notes on SD that would help in the GMAT. I have the ones put up by papgust :)
Looking for something different from that.[/url]
The easiest way to attack these sorts of questions is to plot the mean and the SDs on a number line. Let's work with your second question, because the first will be a snap once we do the second.

The mean is 60, so we put that in the middle of our diagram:

-----------------------------60-------------------------------

The standard deviation is 15, so we count off in blocks of 15 both below and above the mean:

30-----------45------------60------------75------------90

Each of those intervals is one standard deviation. So, we'd say that 75 is one SD above the mean and 45 is one SD below the mean.

The question tells us that her score is within 2 SDs of the mean, so we count off 2 intervals in either direction - that takes us to 30 (below) and 90 (above). Accordingly, she scored somewhere between 30 and 90 (inclusive). The question asks for her lowest possible score: choose 30.

Now try the first question you posted!
2. The mean score of a class on a test was 60 and the standard deviation was 15. If Elena's score was within 2 standard deviations of the mean, what is the lowest score she could have received?

So, would that also mean that Elena's score is NOT a part of the set whose standard deviation is 15?
I say so because standard deviation itself means that the lowest and highest elements of a set are 'standard deviation' value away from the mean i.e. the lowest elements of the set could be 60 - 15 = 45 and the highest could be 60+15=75. Am I wrong in that sense?
Please correct me. Thanks.
See SD means variation from the mean of any set. Here we are told 1 SD is 15, so 2 SD means + 30 to 60.

Something like this 30,45,6075,90.

Technically her score might lie between 30 and 90, but in the question you are referring to, we are asked about her lowest score possible.

So by the above information we can safely conclude that she might have scored 30, which is withing 2 SD of the mean score 60. 29 would be outside 2 SD from the mean.

I hope you are clear now.
You can, for example never foretell what any one man will do, but you can say with precision what an average number will be up to!
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