# Lesson: Range and Standard Deviation

## Comment on Range and Standard Deviation

### I really like your videos.﻿

I really like your videos.﻿

### I have a question - you say

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

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

### hi - maybe a silly error on

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

### You are using the informal

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)

### https://gmatclub.com/forum/if

https://gmatclub.com/forum/if-m-is-a-negative-integer-and-k-is-a-positive-integer-126819.html

Hi Brent, would you say using the informal definition here is what lead me to the wrong answer of picking choice 3 as sufficient? That is, I picked 0 as its eligible since we can take the negative number to be -1 and positive number to be 5, and them once we apply the informal definition, I got the "average distance away from the mean" to be 0. Hence I thought the choice 3 was valid .. but apparently not according to other posts.

### Question link: https:/

I think you're confusing the mean with average DISTANCE from the mean.
If we let -1 and 5 be M and K, then the set is: {-7, -5, -3, -1, 0, 1, 3, 5, 7}
Mean = 0

-7 is 7 units from the mean
-5 is 5 units from the mean
-3 is 3 units from the mean
-1 is 1 units from the mean
0 is 7 units from the mean
1 is 1 units from the mean
3 is 3 units from the mean
5 is 5 units from the mean
7 is 7 units from the mean

Average DISTANCE from the mean = (7 + 5 + 3 + 1 + 0 + 1 + 3 + 5 + 7)/9
= 32/9
≈ 3.5

Does that help?

Cheers,
Brent

### Yes I saw the video all over

Yes I saw the video all over again and found the issue. Thanks Brent!

### Hi Brent,

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

### Good question.

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...

STATEMENTS 1 AND 2 COMBINED
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

Cheers,
Brent

### Thanks Brent

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

### Hi,

Hi,

Can the Standard Deviation be equal to Mean?

Thanks

### Yes, that's possible. The

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.

Cheers,
Brent

### Hi Brent, could you please

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.

### If we list the number of

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.

Cheers,
Brent

### Is the variance not tested by

Is the variance not tested by the GMAT? What about the Interquartile range?
Thank you,
Ana

### The test-makers say that

The test-makers say that variance is tested on the GMAT, but I've never seen an official question that actually tests it (other than a question in the Diagnostic test in the Official Guide for GMAT Review).

Variance is covered in this video: https://www.gmatprepnow.com/module/gmat-statistics/video/809

Interquartile range is NOT tested.

Cheers,
Brent

### https://gmatclub.com/forum/is

https://gmatclub.com/forum/is-the-standard-deviation-of-a-less-than-the-standard-deviation-of-set-242477.html

Hi Brent, aren't the standard deviations for case B of your solution toward the end = 0? Rather than greater? Thank you. Please correct me if I am wrong. Thanks beforehand!

### Link to my solution: https:/

Link to my solution: https://gmatclub.com/forum/is-the-standard-deviation-of-a-less-than-the-...

I'm not sure what you mean.
The only time we get a standard deviation of ZERO is when all of the values in the set are EQUAL.
Which set are you referring to?

Cheers,
Brent

### https://gmatclub.com/forum/if

https://gmatclub.com/forum/if-s-is-a-set-of-four-numbers-x-y-z-and-w-is-the-range-of-the-numbe-249456.html

Hi Brent, does the question want us to assume the numbers to be in an ascending or descending manner? And I do not understand when you mentioned that if we add more numbers to the set, the range doesn't change ..

You have example of 3,10 and the range to be 7. However, if we add an element say 2, doesn't the range change from being 7 to an 8?

Sorry for the lengthy question!

### Link to my solution: https:/

Link to my solution: https://gmatclub.com/forum/if-s-is-a-set-of-four-numbers-x-y-z-and-w-is-...

I didn't say "if we add more numbers to the set, the range doesn't change"

I said:
"The set {3, 10} has a range of 7
If we ADD more values to the set, we cannot make the range LESS THAN 7"

Likewise, for statement 1, we learn that the range of the set {x, w} is GREAT THAN 4
So, if we add more values to that set, the range cannot become less than 4.

Does that help?

Cheers,
Brent

### https://gmatclub.com/forum/a

https://gmatclub.com/forum/a-set-of-data-consists-of-the-following-5-numbers-47858.html

Hi Brent, why can it not be option C? Cos then we are only 0 units away from the mean ... If you could explain with a different approach, it would be particularly helpful!
Thanks beforehand.

### Question link: https:/

We want the NEW standard deviation (with the addition of two more numbers) to be close to the ORIGINAL standard deviation.

After some informal calculations, we see that the ORIGINAL standard deviation is about 2.4
In other words, in the ORIGINAL set of numbers, the average distance from the mean is about 2.4

So, in order for the NEW standard deviation to be close to 2.4, we need each of the two ADDITIONAL values to be about 2.4 away from the mean.

The mean = 4
3 is 1 units from the mean
5 is 1 units from the mean

On the other hand,
2 is 2 units from the mean
6 is 2 units from the mean

So, the addition of 2 and 6 (which are both 2 units from the mean) will change the standard deviation LESS THAN the addition of 3 and 5 (which are both 1 unit from the mean)

Does that help?

Cheers,
Brent

### Yes that is very clear! Thank

Yes that is very clear! Thank you. I understand my fault :)

### https://gmatclub.com/forum/if

https://gmatclub.com/forum/if-m-is-a-negative-integer-and-k-is-a-positive-integer-126819.html
What if K amd M equal -1 and 5. then SD could be 0 right?

### Question link: https:/

Not quite. If K = -1 and M = 5, then the MEAN (not the standard deviation) is zero.
In order for the standard deviation of a set to be zero, all numbers in that set must be the SAME.

For example, the set {3,3,3,3,3,3} has standard deviation zero.

Does that help?

Cheers,
Brent

### Oh yes. I keep thinking that

Oh yes. I keep thinking that SD will be 0 when mean is 0 as we have the Arithmetic mean in the formula of SD.

Thanks!

### Hi Brent,

Hi Brent,

What is the reason behind not considering options c and e?

Thanks!
Kashaf

### Question link: https:/

I first tested answer choice A to show how the process works.
From there, I saved some time by scanning the remaining answer choices to see which one had values that were closest to 2.4 units away from the mean.
That said, students could also test answer choices B, C and E the same way I tested answer choices A and D.
The end result will still be the same.

Cheers,
Brent

### https://gmatclub.com/forum/if

https://gmatclub.com/forum/if-s-is-a-set-of-four-numbers-x-y-z-and-w-is-the-range-of-the-numbe-249456.html

Hey Brent can you help me understand this?
Why is this Key Concept true?
KEY CONCEPT: If the range of a set containing 2 values = k, then adding additional values to the set cannot decrease the range.
For example, the set {3, 10} has a range of 7
If we add more values to the set, we cannot make the range less than 7.

For the example above, are you saying we can under no circumstance add let's say 1 to the set? If so why is that?

### Question link: https:/

The range = (biggest # in the set) - (smallest # in the set)
So, for example, the range of the set {1, 2, 10} = 10 - 1 = 9

Notice that, if we add a number like 5 to the set, we get {1, 2, 5, 10}
Here, the range = 10 - 1 = 9
So, by adding a 5 to the set, the range REMAINS at 9.

If we add 8 to the set, we get {1, 2, 8, 10}
Here, the range = 10 - 1 = 9
So, by adding a 8 to the set, the range REMAINS at 9.

If we add 13 to the set, we get {1, 2, 10, 13}
Here, the range = 13 - 1 = 12
In this case the set, the range INCREASES from 9 to 12

If we add -7 to the set, we get {-7, 1, 2, 10}
Here, the range = 10 - (-7) = 17
In this case the set, the range INCREASES from 9 to 17

Notice that, by adding a number to the set, the range either REMAINS the same, or INCREASES.

Also recognize that, there is no number that we can add to the set that will DECREASE the range.

Does that help?

### Wow. Yes! Definitely helps.

Wow. Yes! Definitely helps. Thank you!

### Hi Brent,

Hi Brent,

https://gmatclub.com/forum/a-certain-list-consists-of-3-different-numbers-does-the-median-of-the-11156.html

Can you pls help with understanding how Statement 2 is sufficient? As in why "the sum cannot be 3 times smallest numbers or 3 times largest number"?

Thanks!

### Question link: https:/

Great question!
Let's say we have three different numbers.
Let x = the smallest number.
So, x + a = the middle number (for some positive value of a)
And x + b = the biggest number (for some positive value of b)

Note: In order for x+b to be the biggest number, it must be the case that b > a.

The sum of the three numbers = x + (x+a) + (x+b) = 3x + a + b

Let's first examine why the sum cannot be 3 times the smallest number.
We'll do this by assuming it's POSSIBLE for this to happen.
In other words: (sum of the three numbers) = 3(smallest number)
Substitute to get: 3x + a + b = 3(x)
Subtract 3x from both sides to get: a + b = 0
This equation is IMPOSSIBLE, since a and b are positive numbers.
This means it CAN'T be the case that the sum equals 3 times the smallest number.

Now let's examine why the sum cannot be 3 times the biggest number.
Once again we'll assume this is POSSIBLE.
In other words: (sum of the three numbers) = 3(biggest number)
Substitute to get: 3x + a + b = 3(x + b)
Expand: 3x + a + b = 3x + 3b
Subtract 3x from both sides to get: a + b = 3b
Subtract b from both sides to get: a = 2b
Since a and b are positive, and since b > a, it CAN'T be the case that a = 2b.
This means it CAN'T be the case that the sum equals 3 times the biggest number.

Finally, let's see what happens if the sum is equal to 3 times the middle number.
We get: 3x + a + b = 3(x + a)
Expand: 3x + a + b = 3x + 3a
Subtract 3x from both sides: a + b = 3a
Subtract a from both sides: b = 2a
Since a and b are positive, and since b > a, it's to have a situation in which 2b = a.

Does that help?

### Hi Brent,

Hi Brent,

I feel fortunate that I got to know about you. Your style of teaching is unique. I simply loved it.
I want to ask one thing, where can I get the notes/slides of all these chapters because then it would be easier to revise.