Lesson: Normal Distributions

Comment on Normal Distributions

http://greprepclub.com/forum/the-random-variable-x-is-normally-distributed-1726.html

http://greprepclub.com/forum/qotd-13-the-figure-above-shows-the-standard-normal-2502.html

could you explain these two sums
greenlight-admin's picture

Sorry for taking so long!
To properly explain those two questions, I wanted to create some graphics.

My solutions can be found here:

- http://greprepclub.com/forum/the-random-variable-x-is-normally-distribut...

- http://greprepclub.com/forum/qotd-13-the-figure-above-shows-the-standard...

http://greprepclub.com/forum/a-random-variable-y-is-normally-distributed-with-a-mean-of-1811.html

Theoretically i have understood. since mean is 200 and SD is 10, event greater than 220 will fall into 3rd region that is after 95 percent, which is very rare to happen, so probability will be less.

But how do i know how much less it is, what would be the value? How probability 1/6 make sense in normal distribution table?
greenlight-admin's picture

Question link: http://greprepclub.com/forum/a-random-variable-y-is-normally-distributed...

As you've stated, only 5 percent of data points are more than 2 standard deviations away from the mean.

This means 2.5% of the data points are more than 2 standard deviations ABOVE the mean.
And 2.5% of the data points are more than 2 standard deviations BELOW the mean.

In other words, the probability is 2.5/100 that a data point is more than 2 standard deviations ABOVE the mean

So, P(Y is greater than 220) = 2.5/100

Since 1/6 > 2.5/100, the correct answer is B

http://greprepclub.com/forum/the-figure-above-shows-the-probability-distribution-2350.html

In this, while calculating probability. I have added all the probabilities in the interval 1-2, 2-3 and 3-4. i got 0.76.

then i thought, probability is (possible number of events) / (total event), so i put answer as 1/0.76.

please correct my understanding.

is it like, in probability distribution we will plot only probability of the event ( X can take 0.30 (Second highest probability) between value 1 and 2), hence we just need to add the probability?

In normal distribution, how will be the plot of random variable X, in the sense will plot value of X, is it or else?
greenlight-admin's picture

Question link: http://greprepclub.com/forum/the-figure-above-shows-the-probability-dist...

First of all, all probabilities range from 0 to 1, inclusive. Since 1/0.76 is greater than 1, this is an invalid probability.

Here's how we need to look at it.

P(1 < x < 4) = P(1 < x < 2 OR 2 < x < 3 OR 3 < x < 4)
= P(1 < x < 2) + P(2 < x < 3) + P(3 < x < 4)
= 0.30 + 0.32 + 0.16
= 0.76

Normal distribution i understand and also percentile but when it comes to the questions, it is really difficult for me. DO I need to divide any percentile give into half and project it on the bell curve for eg 75th percentile into 37.5 each and put beyond 34% on each side of the bell and see where the value of 75th percentile lie? Especially when mean is not given I get confused.
greenlight-admin's picture

No, you would be required to perform that kind of calculation.

Hi,
As per your explanation https://greprepclub.com/forum/the-random-variable-x-is-normally-distributed-1726.html

Help on the following question please:

The 75th percentile on a test corresponded to a score of 700, while the 25th percentile corresponded to a score of 450.

QUANTITY A: 800
QUANTITY B: A 95th percentile score

Source:MGRE 5lb, OC : D
greenlight-admin's picture

I should note that your question is considerably different from the linked question you posted.

In the linked question, we're told that the random variable x is NORMALLY distributed.
In the question above, the test scores are not necessarily normally distributed.

Here's my solution.
If a score of 700 is in the 75th percentile, then 75% of all scores are LESS THAN 700
If a score of 450 is in the 25th percentile, then 25% of all scores are LESS THAN 450

There are infinitely many scenarios that meet these two conditions. Here are two such scenarios:

Scenario 1:There are 100 test scores, and the scores are:
25 scores of 200
1 score of 450 [so, 450 is in the 25th percentile]
49 scores of 600
1 score of 700 [so, 700 is in the 75th percentile]
19 scores of 750
1 score of 800 [so, 800 is in the 95th percentile]
14 scores of 900
In this case, the 95th percentile score is 800
So, we get:
QUANTITY A: 800
QUANTITY B: 800
Here, the two quantities are EQUAL

Scenario 1:There are 100 test scores, and the scores are:
25 scores of 200
1 score of 450 [so, 450 is in the 25th percentile]
49 scores of 600
1 score of 700 [so, 700 is in the 75th percentile]
19 scores of 750
1 score of 850 [so, 850 is in the 95th percentile]
14 scores of 900
In this case, the 95th percentile score is 850
So, we get:
QUANTITY A: 800
QUANTITY B: 850
Here, Quantity B is GREATER

Answer: D

Cheers,
Brent

Thanks, If a ques. donot mention it is normally distributed, it should be considered
greenlight-admin's picture

If a question does not specifically state that a population is normally distributed, then we cannot assume that it is normally distributed.

https://greprepclub.com/forum/the-random-variable-x-is-normally-distributed-1726.html

I don't really understand the solutions mentioned there but I solved this question like this:

200 values - 30 percentile
20 values - 3 percentile
x - 75

20 * 75 = 3x => x = 500

The answer is B. Is this method okay?
greenlight-admin's picture

Question link: https://greprepclub.com/forum/the-random-variable-x-is-normally-distribu...
Unfortunately, the fact that the strategy worked was purely coincidental.
The strategy won't work, because we need to use the fact that the values are normally distributed.

Here's an example to show why:

Let's say set X = {1, 2, 3, 4, 5, . . . . . 100} (i.e., all positive integers from 1 to 100)
In this case, 31 is in the 30th percentile, since 30 of the 100 values are less than 31

Now, let's say set Y = {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, . . . 1, 31} (i.e., there are 99 ones and 1 thirty-one)
In this case, 31 is in the 99th percentile, since 99 of the 100 values are less than 31

As you can see, if a set is NOT normally distributed, it's hard to make strong conclusions about the values in the set.

Does that help?

Cheers,
Brent

Practice Questions
Question: 5
Page: 156
Difficulty: hard

Hi Brent,

This was a tricky problem and I am not 100% on the explanation.

''So, where should that line go?''

I understand that 750 is the average of 650 and 850. But how do you determine where to draw the line corresponding to 75th percentile and thus divide the area between 650 and 850 into 2 equal proportions? Please explain.
greenlight-admin's picture

It's a VERY tricky question.
Here's my full solution: https://greprepclub.com/forum/the-random-variable-x-is-normally-distribu...

Please let me know if that helps.

Cheers,
Brent

https://greprepclub.com/forum/a-random-variable-y-is-normally-distributed-with-a-mean-of-1811.html
Why can't the probability be 2.5%.
greenlight-admin's picture

Question link: https://greprepclub.com/forum/a-random-variable-y-is-normally-distribute...

It depends on how precise we want to be.

On page 149 of The GRE Official Guide (3rd edition), we're told that 2% of of the values in a normal distribution are more than 2 units of standard deviation from the mean.

Other resources like to be more precise and say that 2.5% of of the values in a normal distribution are more than 2 units of standard deviation from the mean.

Regardless of whether we use 2% or 2.5%, the correct answer is still B (since both of those values are still less than 1/16)

Cheers,
Brent

hey Brent can you tell me what reasonably symmetrical means?
greenlight-admin's picture

Here's a page with pictures to go along with the definition of "symmetrical": https://www.learner.org/courses/learningmath/geometry/session7/part_a/in...

In the context of the above video, if we examine a normal distribution, it will appear to be symmetrical on either side of the mean. When I say "reasonably symmetrical," I mean that the distribution may not be PERFECTLY symmetrical

Does that help?

Cheers,
Brent

https://greprepclub.com/forum/a-random-variable-y-is-normally-distributed-with-a-mean-of-1811.html

Hey brent In the lesson above you have said that m-2d and m+2d= 13.5 and in the solution you have given 14 and 2 respectively. which one is correct? or is both okay?
greenlight-admin's picture

I had a similar discussion above in the comment section.

Both are fine. It really depends on how precise we want to be.

On page 149 of The GRE Official Guide (3rd edition), we're told that 2% of of the values in a normal distribution are more than 2 units of standard deviation from the mean. Other resources like to be more precise and say that 2.5% of of the values in a normal distribution are more than 2 units of standard deviation from the mean.

Rest assured, the GRE test-makers won't create a question that exploits these minor differences.

Cheers,
Brent

https://greprepclub.com/forum/the-graph-represents-the-normally-distributed-scores-on-a-te-10166.html

Don't we assume that normal distributions are symmetrical about the mean?
greenlight-admin's picture

Question link: https://greprepclub.com/forum/the-graph-represents-the-normally-distribu...

Yes, normal distributions are definitely symmetrical about the mean. However, as you can from my solution, if we are shading a portion of the distribution, the shaded portion need not be symmetrical about the mean.

Does that help?

Cheers,
Brent

https://greprepclub.com/forum/the-random-variable-x-is-normally-distributed-1726.html

I think this is a great question to understand the dynamics of bell curve. At first, I got really anxious when you shifted the line(which divides 60th and 90th percentile) closer to A. Later I realized that this was necessary to divide the two areas under A and B equally. From there, we could see that 75th percentile isn't exactly 750 which is between 650 and 850.

Do I make any sense?

Thanks! Brent
greenlight-admin's picture

Question link: https://greprepclub.com/forum/the-random-variable-x-is-normally-distribu...

That's a perfect analysis of my solution!

Cheers,
Brent

https://greprepclub.com/forum/the-graph-represents-the-normally-distributed-scores-on-a-te-10166.html

Hi Brent, why can't we use this approach: Mean+SD= 600, this is the first case and Mean-SD=500 for the second case. We can then solve for SD or mean..since mean value will remain the same?

Thanks!
greenlight-admin's picture

Question link: https://greprepclub.com/forum/the-graph-represents-the-normally-distribu...

If we say Mean + SD = 600, then we're suggesting that a score of 600 is 1 standard deviation above the mean.

Likewise, if we say Mean - SD = 500, then we're suggesting that a score of 500 is 1 standard deviation below the mean.

Unfortunately, there's nothing in the question that tells us that this is so.
That said, the misleading diagram certainly LOOKS like this is so, but we can't trust the diagram.

Does that help?

Cheers,
Brent

Hi Brent,

Your lectures are awesome, and finally math makes sense to me.Thank you!!
greenlight-admin's picture

Thanks for that!

https://greprepclub.com/forum/the-scores-for-the-500-students-who-took-ms-johnson-s-fina-3631.html

I got the right answer... however I don't understand why we don't consider the 16% as below the mean (i.e., 1 SD below the mean)? Is it because the context of the question suggests that it should be ABOVE the mean (i.e., an exam score of 92) or is it for some other reason? Any clarification would really be appreciated!
greenlight-admin's picture

Link: https://greprepclub.com/forum/the-scores-for-the-500-students-who-took-m...

You're correct about context.

The given information, tells us about the students who scored at least 92 points and students who scored at or below 56. So, these are the only scores we need to concern ourselves with.

A score of 92 is 1 standard deviation ABOVE the mean
A score of 56 is 2 standard deviations BELOW the mean

So, these are the only scores we need to consider.

Cheers,
Brent

https://greprepclub.com/forum/the-scores-for-the-500-students-who-took-ms-johnson-s-fina-3631.html

Hi Brent,
I am not able to understand how 16% of all values are 1 SD above the mean? Could you please shed some light on this.

Thanks,
Ketan
greenlight-admin's picture

Question link: https://greprepclub.com/forum/the-scores-for-the-500-students-who-took-m...

This is the nature of all normal distributions.

For example, in any normal distribution:
- about 50% of the population is ABOVE the mean and 50% is BELOW the mean.
- about 2% of the population is more than 2 standard deviations BELOW the mean
- about 68% of the population is less than 1 standard deviation from the mean
etc.

One of the characteristics of all normal distributions is that 16% of the population is more than 1 standard deviation ABOVE the mean.
See the graph at https://greprepclub.com/forum/the-scores-for-the-500-students-who-took-m...

Cheers,
Brent

https://greprepclub.com/forum/the-graph-represents-the-normally-distributed-scores-on-a-te-10166.html

I looked at the solution to this question. It seems to me that even for statistics we cannot assume that figures are drawn to scale. This IS the case for geometry also. But for data interpretation questions we can approximate by looking at the graphs. This all seems a little confusing. Could you kindly give a guideline about all the cases when we can make approximation from the figure and when not?
greenlight-admin's picture

Question link: https://greprepclub.com/forum/the-graph-represents-the-normally-distribu...

There are two times in which we can assume a diagram is drawn to scale:
1) The question is a data interpretation question
2) We are explicitly told the diagram is drawn to scale.

Neither of these conditions are met in this question, we we can't make any assumptions.

Cheers,
Brent

https://greprepclub.com/forum/the-scores-for-the-500-students-who-took-ms-johnson-s-fina-3631.html

For this question:
I calculated 92 as being 1 SD above the mean and 56 being around 2 SD below the mean.

I then subtracted 92 from 56 and considered this to be 3 SD
36 = 3 SD
12 = SD

So the mean = 92 - 12 = 80
So A < B

Is my reasoning correct?
greenlight-admin's picture

Question link: https://greprepclub.com/forum/the-scores-for-the-500-students-who-took-m...

The math you used in your solution suggests that you're using the same strategy I used in my video solution. The only difference is that you didn't formally write the equations m + d = 92 and m - 2d = 56

So, I think your approach is perfectly valid.

Cheers,
Brent

could you help me answer this question it was on Kaplan test but I feel like the wording is very confusing
https://brainly.com/question/13031225
greenlight-admin's picture

Question link: https://brainly.com/question/13031225

This question is beyond the scope of the GRE.
For GRE questions involving normal distributions, you need only know the percentiles for 1, 2 and 3 standard deviations from the mean.
The linked question above requires additional information that requires a SCIENTIFIC calculator (which is not available on the GRE)

Cheers,
Brent

they provided the following information.
percentile SD below the mean
10 1.28
20 0.84
30 0.52
40 0.25
the table shows selected percentiles and corresponding SD below the mean in a normal distribuiton
greenlight-admin's picture

Weird, I don't see any tables. That said, I don't have an account with Brainly, so it seems I'm now being prevented from seeing everything on the page (unless I buy a subscription).

The table you've provided are the "z-scores" of a normal distribution.
This is beyond the scope of the GRE (even with the table).

I suggest you stick with official GRE questions.

Cheers,
Brent

Hi Brent,

How would you answer this question:

The scores on a given history test are normally distributed about a mean of 72 points. A score of 78 is in the nth percentile, while a score of 84 is in the mth percentile.

Quantity A
n – 50

Quantity B
m – n
greenlight-admin's picture

Tricky!!
I needed to make a few graphics, so I posted the question to GRE Prep Club.
You'll find my solution here: https://greprepclub.com/forum/the-scores-on-a-given-history-test-are-nor...

Cheers,
Brent

Thank you Brent for the illustrative answer.

Do you mind explaining more how did you come up with these 2 equations, still don't get it.

We're told that 78 is in the nth percentile
So n – 50 = the PERCENTAGE of scores between 78 and 72

We're told that 84 is in the mth percentile
Likewise, m – n = the PERCENTAGE of scores between 84 and 78

Thanks again!
greenlight-admin's picture

Let's say there are 100 scores in total.

Now consider this possible situation:
72 is in the 50th percentile, which means 50 scores are less than 72.
78 is in the 70th percentile, which means 70 scores are less than 78.
70 - 50 = 20
So, about 20 scores lie between 72 and 78.
In other words, 20% of the scores lie between 72 and 78.

Likewise, if 72 is in the 50th percentile, and 78 is in the nth percentile, then n - 50 = the percentage of scores between 72 and 78.

it make sense now, thank you!

hi brent! How would you solve this if it were normally distributed? https://greprepclub.com/forum/the-75th-percentile-on-a-test-corresponded-to-a-score-of-10154.html

I did the following:
700-450 = 250 points are in between 25th and 75th percentile. Which is 5 points per percentile

So 95th percentile would be 800 and the answer would be C.

Is this correct if we assumed normal distribution?
greenlight-admin's picture

Question link: https://greprepclub.com/forum/the-75th-percentile-on-a-test-corresponded...

With that change, the question would never be an official question, since we'd need a scientific calculator to answer it.

That said, our solution would begin by finding the mean.
From the given info, we know that 25% of scores are LESS than 450, and 25% of scores are GREATER than 700
Given the symmetry of the normal distribution, we know that the mean lies in the middle of 450 and 700 at 575
If the mean is 575, we need to know the number of standard deviations (away from the mean of 575) that correspond to the 75th and 25th percentiles. This part would require a scientific calculator.
Once we know those values, we can determine the standard deviation of the set.
Once we know the standard deviation, we can answer the (new) question.

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