Lesson: Range and Standard Deviation

Comment on Range and Standard Deviation

By "mean" you the mean "the average"?
Because I got confused when you solved the last example (at 8:30) of finding the standard deviation for the two sets, the mean for the first one was 10 according to the mean definition ( arrange from the lowest to the greatest and take the middle number)!
greenlight-admin's picture

Average and (arithmetic) mean are the same thing.
What you described (" arrange from the lowest to the greatest and take the middle number") is the MEDIAN

at 9.54 minutes, you say that, (1,1,2,4,7,13,30,34,47,80) here range 89, average age (mean) is 22 and SD is 24.6, but we know that SD is the average distance the data values are away from the mean, Now as its mean is 22, then the distance from the mean is (21,21,19,18,15,9,8,12,25,58) and the sum of this is 206, and the total number is 10 then average is 20.6, but you said that SD is 24.6, how can you get this? Can you explain it? Thanks
greenlight-admin's picture

In the video, we examine 2 different definitions of Standard Deviation: the formal definition and the informal definition.
At 9:45, I say "if we apply the FORMAL definition of Standard Deviation to both sets," we get a standard deviation of the top set is 24.6

When you applied the INFORMAL definition, you got 20.6. which is pretty close to what we found with the formal definition.

As I mention in the video, the informal definition will not yield the exact same standard deviation that you'd get using the formal definition. However, for the purposes of the GRE, it's close enough.

In fact, at 6:30 in the video, we used the informal definition to find the standard deviation, and when we were finished, the standard deviation we calculated was reasonably close to what we got when we used the formal definition.

As I said, for the purposes of the GRE, the informal definition is good enough to answer answer standard deviation question the GRE can throw your way.

I have been experiencing issues with the loading of the video :(
greenlight-admin's picture

Try clearing the cache on your browser.

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Let me know if that helps.

Concise and clear explanations. Thanks for the video! It is really helpful in my prep for GRE.

http://greprepclub.com/forum/qotd-10-the-table-above-shows-the-frequency-distribution-of-2499.html

Please answer this.

TO find range i assumed 160 is max hight (since they asked for min possible) and 140 as min hight, i got 20 as range.

There are 4 possible range.

1) if we think height is dispersed from 140 to 164 then range will be 24.
2) 144 and 164 then range will be 20
3) 140 and 160 then range will be 20
4) 144 and 160 then range will be 16

since they have asked for lease possible range then we should assume that 6 students are of 144 hight and 4 students are of 160 hight, then range will yield minimum. so B is answer.

in case, if they had ask for highest possible range then it would 24.

Thank you so much for your response Brent! :)
greenlight-admin's picture

That's correct.

http://greprepclub.com/forum/each-of-the-following-linear-equations-defines-y-2297.html

Please answer this question.

instead of finding 1st, 2nd, 3rd, 4th so on till 100th.

will find 1st and 100th numbers of each.
A) y= 1/3 when x =1 and y = 100/3 when x=100 range will be 33

B) y=1/2 when x=1 and y=100/2 when x=100 range will be 49.5

C) y=1 when x=1 and y=100 when x=100 range will be 99

D) y=52 when x =1 and y=250 when x=100 range will be 198

E) y=-17 when x=1 and y=280 when x=100 range will be 297

looking at the range more spread is there at E, therefore SD will also be greatest.
hence E.

Is this a valid approach?
greenlight-admin's picture

In many cases, the set with the greatest range will also have the greatest standard deviation. But this isn't always the case. Consider these two sets:

Set A: {-10, 0, 0, 0, 0, 0, 0, 10}
Set B: {-8, -8, -8, -8, 8, 8, 8, 8}

In both sets, the mean is 0.

Even though set A has the greater range, set B has the greater standard deviation. So, it also helps to examine some of the numbers in the set as well.

yes exactly, but since all given are linear function, hence our assumption should be true.

Anyhow it is always better to test for some data, in your solution for the above question i found its tedious to find mean and SD. what procedure would be best to find mean or SD in the above kind of question?
greenlight-admin's picture

For these kinds of questions, you need not find the mean or SD. You should be able to just examine each set and see which one has the greatest dispersion.

Hi there! can you please provide some clarity to this question?

The numbers in data set S have a standard deviation of 5. If a new data set is formed by adding 3 to each number in S, what is the standard deviation of the numbers in the new data set?

A 2
B 3
C 5
D 8
E 15
greenlight-admin's picture

Great question.

IMPORTANT CONCEPT: Adding the same value to every value in a set has no effect on the standard deviation.

For example, notice that in the set {5, 7, 9, 11} each value is 2 greater than the one before it.

If we add 10 to each value in that set, we get the new set {15, 17, 19, 21}. In this NEW set, each value is 2 greater than the one before it. So, the standard deviation of the NEW set will be equal to the standard deviation of the original set.

Now onto the question that you posed....

Since we're adding the same value (3) to every value in set
S, the standard deviation of the new set will still be 5.

Answer: C

https://greprepclub.com/forum/heights-of-the-female-students-in-a-certain-class-is-13-2-in-1761.html

Hi Brent - can you elaborate on Answer Choice A? Specifically on how we'd find the range of the heights of all students.

http://www.urch.com/forums/gre-math/152259-problem.html

Hi Brent,

I have solved the above question in the following manner. Is it correct?

Since its given that the six number are in increasing order and let the range be x, we can infer that 2a - 3.7 = x.
Therefore, a = (x + 3.7)/2

Lets plug in each answer choice in place of x.

When x = 4,
a = (4 + 3.7)/2 which equals 3.85.
But since a has to be in between 4.1 and 8.5. Eliminate option A.

Now when x = 5.2,
a = (5.2 + 3.7)/2 which equals 4.45.
In this case a is between 4.1 and 8.5 but 2a = 8.9 which has to be greater than 9.2. Eliminate option B.

Now when x = 7.3,
a = (7.3 + 3.7)/2 which equals 5.5.
This value is in between 4.1 and 8.5 and also 2a = 11 is greater than 9.2. Keep option C.

Likewise if we follow the above methodology, we have C,D,E as answer.

Thanks,
Suraj
greenlight-admin's picture

Question link: http://www.urch.com/forums/gre-math/152259-problem.html

Great approach, Suraj!!!

Cheers,
Brent

https://greprepclub.com/forum/qotd-10-the-table-above-shows-the-frequency-distribution-of-2499.html
Why isn't the answer 20
greenlight-admin's picture

Question link: https://greprepclub.com/forum/qotd-10-the-table-above-shows-the-frequenc...

The range COULD be 20 if the shortest person were 142 cm tall and the tallest person were 162 cm tall.

However, the question asks for the LEAST possible range of the heights.

Notice, if the shortest person were 144 cm tall and the tallest person were 160 cm, then the range would be 16.

In fact, 16 is the LEAST possible range of the heights.

Does that help?

Cheers,
Brent

https://greprepclub.com/forum/set-n-is-a-set-of-x-distinct-positive-integers-where-x-10153.html
How does the statement when X>2 comes in? I thought it would required the lease distinct value of the set to be greater than 2
greenlight-admin's picture

Question link: https://greprepclub.com/forum/set-n-is-a-set-of-x-distinct-positive-inte...

I think you're reading the given information incorrectly.

GIVEN: Set N is a set of x distinct positive integers where x > 2.

S, x is the NUMBER of integers in set N.
So, the fact that x > 2 is telling us that there are more than two values in set N.
In other words, x does not tell us anything about the VALUES of the numbers in set N.

Does that help?

Cheers,
Brent

https://greprepclub.com/forum/the-frequency-distributions-shown-above-represent-two-groups-8090.html

I understood everything except how symmetrical graphs have same mean?
greenlight-admin's picture

Question link: https://greprepclub.com/forum/the-frequency-distributions-shown-above-re...

I didn't mean to imply that all graphs that are symmetrical must have the same mean. This just happens to be the case with this question.

If we take Distribution A and draw a vertical line through the middle of 30, we see that each side is a mirror image of the other. This tells us that Distribution A has a mean of 30.

Likewise, if we take Distribution B and draw a vertical line through the middle of 30, we see that each side is a mirror image of the other. This tells us that Distribution B has a mean of 30.

Does that help?

Cheers,
Brent

http://www.urch.com/forums/gre-math/152259-problem.html

hey Brent I did not understand your explanation for this? Could you please elaborate?
greenlight-admin's picture

Question link: http://www.urch.com/forums/gre-math/152259-problem.html

Glad to help!
Here's a better/different solution of mine: https://greprepclub.com/forum/3-7-4-1-a-8-5-9-2-2a-the-six-numbers-shown...

Please let me know if that helps.

Cheers,
Brent

https://greprepclub.com/forum/qotd-10-the-table-above-shows-the-frequency-distribution-of-2499.html

Hi!

Here, if we were asked to find simply the range, what would be the formula?
greenlight-admin's picture

Question link: https://greprepclub.com/forum/qotd-10-the-table-above-shows-the-frequenc...

If the question had asked us for the range of the heights (rather than the least possible range of the heights), then we wouldn't be able to answer the question.

The reason for that is because, rather than having EXACT heights, we're given INTERVALS in which the heights lie (e.g., 140 - 144 cm, 145 - 149 cm).

For example, let's say we have Al and Bob. Al is EXACTLY 180 cm tall, and Bob is EXACTLY 186 cm tall.
So, range = (greatest value) - (smallest value)
= 186 - 180
= 6
So, the range here is 6cm
----------------------------------------
Conversely, let's say Al's and Bob's heights are given as INTERVALS. Al's height is BETWEEN 160 and 170 cm, , and Bob's height is BETWEEN 170 and 180 cm.
Here, we can't determine the range of heights.

For example, it COULD be the case that Al is 165 cm and Bob is 175 cm
In this case, range = 175 - 165 = 10 cm

However, it COULD also be the case that Al is 168 cm and Bob is 172 cm
In this case, range = 172 - 168 = 4 cm

As you can see, when we're dealing with INTERVALS, we cannot definitively determine the range.

Does that help?

Cheers,
Brent

Got it, thanks!

Hello Brent,

For the following question https://greprepclub.com/forum/the-frequency-distributions-shown-above-represent-two-groups-8090.html

Why did you consider 10,20,30,40 & 50 to calculate mean. one should consider frequency for calculating mean right and then judge the SD for both sets.
greenlight-admin's picture

Question link: https://greprepclub.com/forum/the-frequency-distributions-shown-above-re...

We COULD use the strategy you've described, but that will take us much longer to answer the question. So, it still works, but we can speed things up if we use some number sense.

Hello Brent,

I have question related to https://greprepclub.com/forum/set-n-is-a-set-of-x-distinct-positive-integers-where-x-10153.html

I considered my set to be (3,4,5) now we see mean is 4 so we judge SD . Now i multiply set by -1 and find new set to be (-3,-4,-5) mean is -4. SD is still same as positive number would then the answer be C.

Also i have a set (2,3,4) mean is 3 .I multiply it with 5. my new set is (10,15,20) mean is 15. My mean has changed my median has changed by my SD for set (2,3,4) is 1 and SD for set (10,15,20) SD is 5. So SD changed too right.

I think i am confused to what happens to SD when a set is multiplied by a negative number and a positive number.
greenlight-admin's picture

Question link: https://greprepclub.com/forum/set-n-is-a-set-of-x-distinct-positive-inte...

It all comes down to how dispersed the values are.
For example, the sets {3, 4, 5} and {-3, -4, -5} are equally dispersed, so they have the same SD.
Likewise the sets {3, 4, 5} and {-55, -56, -57} are equally dispersed, so they have the same SD.

Conversely, the values in the set {30, 40, 50} are more dispersed than the values in the set {-55, -56, -57}, So, the SD of the 1st set is greater than the SD of the 2nd set.

Does that help?

Cheers,
Brent

Yes it does , thank you. :)

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