Problem Solving

BTGmoderatorDC wrote:A data set with a mean of 60 has a standard deviation of 3.5. Which of the following is the smallest number that falls within one standard deviation of the mean?

A. 53
B. 56
C. 59
D. 63.5
E. 65

OA C

Source: Manhattan GMAT

The standard deviation (SD) of a data set is a measure of how far the data points are deviated from the mean. One standard deviation means that data points are from 1*SD away from the mean.

Thus, the range of data points falling within 1SD are (Mean - 1*SD) to (Mean + 1*SD)

Thus, the number that falls within one standard deviation of the mean: (60 - 3.5) to (60 + 3.5) => 56.5 - 63.5.

The smallest number qualifying within 56.5 - 63.5 is 59.

The correct answer: C

Hope this helps!

-Jay
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BTGmoderatorDC wrote:A data set with a mean of 60 has a standard deviation of 3.5. Which of the following is the smallest number that falls within one standard deviation of the mean?

A. 53
B. 56
C. 59
D. 63.5
E. 65

-----------ASIDE-----------------------
A little extra background on standard deviations above and below the mean

If, for example, a set has a standard deviation of 4, then:
1 standard deviation = 4
2 standard deviations = 8
3 standard deviations = 12
1.5 standard deviations = 6
0.25 standard deviations = 1
etc


So, if the mean of a set is 9, and the standard deviation is 4, then:
2 standard deviations ABOVE the mean = 17 [since 9 + 2(4) = 17]
1.5 standard deviations BELOW the mean = 3 [since 9 - 1.5(4) = 3]
3 standard deviations ABOVE the mean = 21 [since 9 + 3(4) = 21]
etc.
-----------ONTO THE QUESTION-----------------------

A data set with a mean of 60 has a standard deviation of 3.5.
So, 1 standard deviation ABOVE the mean

BTGmoderatorDC wrote:A data set with a mean of 60 has a standard deviation of 3.5. Which of the following is the smallest number that falls within one standard deviation of the mean?

A. 53
B. 56
C. 59
D. 63.5
E. 65

-----------ASIDE-----------------------
A little extra background on standard deviations above and below the mean

If, for example, a set has a standard deviation of 4, then:
1 standard deviation = 4
2 standard deviations = 8
3 standard deviations = 12
1.5 standard deviations = 6
0.25 standard deviations = 1
etc

So, if the mean of a set is 9, and the standard deviation is 4, then:
2 standard deviations ABOVE the mean = 17 [since 9 + 2(4) = 17]
1.5 standard deviations BELOW the mean = 3 [since 9 - 1.5(4) = 3]
3 standard deviations ABOVE the mean = 21 [since 9 + 3(4) = 21]
etc.
-----------ONTO THE QUESTION-----------------------

A data set with a mean of 60 has a standard deviation of 3.5.
So, 1 standard deviation ABOVE the mean = 60 + 3.5 = 63.5
And 1 standard deviation BELOW the mean = 60 - 3.5 = 56.5

Which of the following is the smallest number that falls within one standard deviation of the mean?
The values within one standard deviation of the mean are all values that are greater than or equal to 56.5 AND less than or equal t 63.5
So, when we check the answer choices, we see that 59 is the smallest value that fits with the above range of values.

Answer: C

Cheers,
Brent

BTGmoderatorDC wrote:A data set with a mean of 60 has a standard deviation of 3.5. Which of the following is the smallest number that falls within one standard deviation of the mean?

A. 53
B. 56
C. 59
D. 63.5
E. 65

Since one standard deviation below the mean is 60 - 3.5 = 56.5, 53 and 56 will fall outside one standard deviation of the mean. So 59 is the smallest number in the given answer choices that falls within one standard deviation of the mean.

Alternate Solution:

One standard deviation below the mean is 60 - 3.5 = 56.5, and one standard deviation above the mean is 60 + 3.5 = 63.5. Thus, we are looking for the smallest data value that falls in the interval from 56.5 to 63.5. We see from the answer choices that 59 meets the requirement.

Answer: C