Lesson: Range and Standard Deviation

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I really like your videos.

I have a question - you say that it is necessary to square the numbers to account for the negative values?

If you were to just take the absolute value of all the values minus the mean, and then divide them by n, would that be the same as the standard deviation?

Nevermind, just watched the rest of the video! Thanks Brent :)

hi - maybe a silly error on my part? how'd you get SD of ~24.6 in the first set of numbers in your last example at the end of the video?

Average of 22. Take difference of each number (so absolute value of 1-22, 1-22, 3-22, and so on). Sum all those differences and divide by 10, the number of values in the set. I get ~20.6
gmat-admin's picture

You are using the informal definition to calculate the Standard Deviation of 20.6

I used the formal definition/formula to find that the Standard Deviation is approximately 24.6

On the GMAT, the informal definition is all you need (even though it yields a different SD)

Hi Brent,
Could you please use your magic and explain the problem below. I'd really like your take on it. Thanks.

Set X and Y have 5 numbers, respectively.Is the S.D of X > S.D of Y?
1) Range of X > Range of Y
2) Average of X is greater than that of Y
gmat-admin's picture

Good question.

First note that the average (arithmetic mean) of each set has no bearing the S.D. of each set. For example, even though the average of the set {3,3,3,3,3} is greater than the average of the set {1,1,1,1,1}, their standard deviations are the same.

Now let's jump straight to...

There are many sets that satisfy BOTH statements.
Consider these two possible cases:

CASE A: set X = {-2, -2, 0, 2, 2.1} and set Y = {-2, 0, 0, 0, 2}. In this case, the S.D. of set X IS GREATER THAN the S.D. of set Y.

CASE B: set X = {-2, 0, 0, 0, 2.1} and set Y = {-2, -2, 0, 2, 2}. In this case, the S.D. of set X IS LESS THAN the S.D. of set Y.

Since the COMBINED statements are insufficient, the correct answer is E


Thanks Brent

Just needed to see a couple more examples than what was provided on the practice sites to get it in my head. This is a really good practice problem that helps you to truly understand.

Thanks again


Can the Standard Deviation be equal to Mean?

gmat-admin's picture

Yes, that's possible. The most immediate example that comes to mind is a set of zeros {0, 0, 0, 0, 0}

Here, the mean = 0 and the standard deviation = 0.


Hi Brent, could you please help me with the answer to the question below?

The median number of business trips for 6 employees in a certain quarter is 10. The range of number of business trips for those employees that quarter is 9.

Determine whether the following statement is true, is false, or does not contain enough information: The fewest business trips could be 0.
gmat-admin's picture

If we list the number of trips taken by each employee, we will have a list of SIX NUMBERS.

If we write the values in ASCENDING order, let's say they look like this: {a, b, c, d, e, f}

Since we have an EVEN number of values, the median is the average of the 2 middlemost values.

So, we can write: (c + d)/2 = 10

There are two possible cases to consider:
CASE A: c = 10 and d = 10
CASE B: c is less than 10 and d is greater than 10

CASE A: If the RANGE of values is 9, one of the values is 10, then the SMALLEST POSSIBLE number must be greater than or equal to 1.
So, it's IMPOSSIBLE for the smallest value to be 0

CASE B: If the RANGE of values is 9, one of the values is greater than 10, then the SMALLEST POSSIBLE number must be greater than 1.
So, it's IMPOSSIBLE for the smallest value to be 0

So, as we can see, it's impossible for the fewest business trips to be 0.


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