Standardizing Normally Distributed Random Variables
I discuss standardizing normally distributed random variables (turning variables with a normal distribution into something that has a standard normal distribution). I work through an example of a probability calculation, and an example of finding a percentile of the distribution. It is assumed that you can find values from the standard normal distribution, using either a table or a computer.
The mean and variance of adult female heights in the US is estimated from data found in a National Health Statistics Report:
McDowell MA, Fryar CD, Ogden CL, Flegal KM. Anthropometric reference data for children and adults: United States, 2003-2006. National health statistics re- ports; no 10. Hyattsville, MD: National Center for Health Statistics. 2008.
For those using R, here is the R code for the examples used in this video:
American female heights example (approximately normally distributed with a mean of 162.2 and a standard deviation of 6.8).
Finding the probability that a randomly selected female is taller than 170.5 cm.
Easiest way:
1-pnorm(170.5,162.2,6.8)
[1] 0.111121
Standardizing route:
1-pnorm((170.5-162.2)/6.8,0,1)
[1] 0.111121
The default in R's pnorm is the standard normal distribution (mean=0, SD=1), so the mean and SD can be left out when dealing with the standard normal.
1-pnorm((170.5-162.2)/6.8)
[1] 0.111121
Finding the probability that a randomly selected female has a height between 150.5 and 170.5.
Easiest way:
pnorm(170.5,162.2,6.8)-pnorm(150.5,162.2,6.8)
[1] 0.8462162
Standardizing route:
pnorm((170.5-162.2)/6.8)-pnorm((150.5-162.2)/6.8)
[1] 0.8462162
10th percentile of heights of adult American females.
Easiest:
qnorm(.1,162.2,6.8)
[1] 153.4854
Alternatively, via the standard normal distribution:
qnorm(.1)
[1] -1.281552
That's the 10th percentile of the standard normal distribution. Converting to the distribution of heights,
-1.281552*6.8+162.2
[1] 153.4854
Пікірлер: 226
Videos like these.. 4 years old and still saving the lives of students around the world. Good shit.
@jbstatistics
6 жыл бұрын
Thanks for the compliment! I tried to make them stand the test of time (e.g. examples that play well through time, no Justin Bieber or Rebecca Black references :).
@prernarawat4322
5 жыл бұрын
@@jbstatistics thanks to uuu
@fernandofrio5863
4 жыл бұрын
And up until 6 yeaaars
@ejaazzz
3 жыл бұрын
Up until 7 years now
@loganmauricio8832
2 жыл бұрын
Sorry to be offtopic but does anyone know a trick to get back into an Instagram account? I was stupid forgot the password. I would appreciate any assistance you can offer me!
Just want to thank you for this even after 7 years
In fewer than 3 days of taking notes from your videos, I finally understood what I couldn't figure out in more than 10 days of classes. I got a 96% on my first exam thanks to your videos, and the more I watch the more I understand. Thank you, sir.
@mahelapriyadarshana4196
Жыл бұрын
For me, it was not a 10 days but a whole semester 🙃 Thank you for providing such helpful videos. Keep it ip @jbstatistics 👍
Hey I just stopped by to tell you how good professor you are! Thanks for making those videos. Just like other comments told you,you are able to explain things that others try and fail and even create more confusion. Thanks again! Cheers!
@jbstatistics
8 жыл бұрын
+Marlom Oliveira Thanks Marlom! I'm very glad I can be of help. All the best.
Man you are awesome! You just made my life so much easier. I tried so many channels and two different teachers and so far you can explain in 10 minutes what others take hours! Thank you!
@jbstatistics
8 жыл бұрын
+Jorge Hurtado You're very welcome Jorge!
of all the videos ive watched today, yours are the best and makes things easier. You're so far the best in explaining these statistics topics😊🤓
you actually saved me. Thank you for explaining this in a digestable manner. I wish I had found you earlier!
So glad I found this channel. Thanks so much for the step by step explanation. Your video was very helpful.
One of the best videos out there. Thanks man !
I am having a really hard time understanding my statistics lectures and these videos are amazing. Better than Khan Academy and everything else I have found. I have an online exam today so it is nice that these videos are very short. I have to come back when I am doing the homework.
Thank you so much! I now can pass my class because of you!
video of 7 years old and it is saving my life for the statistics exam. Definitely awesome
Your video rescued us from the hell of our quiz in stats! Thank you so much!!
@jbstatistics
7 жыл бұрын
You are very welcome Amy!
I usually never comment, but god damn after watching 1 minute of your video I actually understood more than 1 hour of trying to understand my textbook.... you are amazing!
Your videos are helping me to finally understand econometrics. Thank you so much.
@jbstatistics
7 жыл бұрын
You are very welcome Adam. All the best!
even 10 years later, you're still saving people from failing their exams last-minute :D
I wish I could have found this channel earlier in the semester. THIS helped so much !!!!!!!!!! Thanks!!!
@jbstatistics
8 жыл бұрын
+Valeria Santoyo I'm glad I could help!
I just watched the fast version in 2X and now watching this makes me feel I am watching it in -2X :D
Best Statistics videos on KZread.
Thanks! nicely explained and really helpful .
Really amazing video.Thanks a lot!
Thanks for these videos. They're some of the more sensible ones out there ...
@jbstatistics
10 жыл бұрын
You are welcome Brolnox.
amazing, thank you so much. so easy with this explanation. you deserve an award.
@jbstatistics
5 жыл бұрын
Thanks for the very kind words!
This is a great explaination!
Still relevant. Love this!
VERY WELL EXPLAINED, THANK YOU
Thank you so much for helping me shine light on the previously dark world that is statistics :D If I would get an A its honestly 99% because of you!!
@jbstatistics
6 жыл бұрын
You are very welcome. I'm glad to be of help!
thank you for all of your videos
@jbstatistics
6 жыл бұрын
You are very welcome!
Thank you for this!
You are Incredible.Thanks For the Videos.
Yes, videos from 2006 and up are still helping.
You're a lifesaver. Thanks!
Perfect pace, good stuff!
@jbstatistics
7 жыл бұрын
Thanks!
Very useful ❤thanks for this😂 helping us in 2018 too! Phew
you can't imagine how helpful your videos have been ! thank you
@jbstatistics
9 жыл бұрын
You are very welcome!
Superb video...!!! Very useful!! Thank you.
@jbstatistics
6 жыл бұрын
You are very welcome!
Very well explained and clear, well done.
@jbstatistics
8 жыл бұрын
Thanks!
You make it so simple....thank you so much 🤩👍
@jbstatistics
2 жыл бұрын
You are very welcome!
Thank you so much !
Could you explain why in statistical inference one should never standardize a variable according to a sample distribution of the same data and why that would be circular reasoning?
What i was looking for. Thanks
@jbstatistics
6 жыл бұрын
You are very welcome!
hi can u please explain the difference between central and non central chi square distribution..I want to know the mean and variance for each of them..
thanks this helps me a lot.
Great explanation
Thanks for useful video and if possible please make video for Random Process. Thanks again.
Best stats videos on KZread
@jbstatistics
5 жыл бұрын
Thanks!
When I look at 1.22 in the table, I find 0.8888 and NOT 0.111. What is wrong?
@jbstatistics
7 жыл бұрын
Your table is giving you the area to the left of the value of z. In my example, I needed to find the area to the right. Since the area under the entire curve is 1, the area to the right of 1.22 is 1-0.8888 = 0.1112.
@RedRiverChannel
7 жыл бұрын
ok, thank you!
@wiiliskaako5875
6 жыл бұрын
you then subtract 1
Could you or someone explain me with more details how to find the probability when there are two values like (a
Standardizing is something I'm running into a lot lately, and I'm curious about something. If I have a variable x and I want to identify outliers, I could standardize and remove anything above +/- 1.96, or whichever cutoff I choose. I wonder, if x is not normally distributed (e.g. are quite skewed), are z-scores no longer valid? I think that because z-scores are calculated using the mean, but since x is not normal, the mean is no longer a reasonable value for this. Is this correct?
I'm glad you find my videos helpful. The random variable X has a mean of mu (E(X) = mu). (X-mu) is itself a random variable, with a mean of 0. E(X-mu) = E(X) - E(mu) = mu - mu = 0. As a simple example, suppose we have a distribution with a mean of 3. If we create a new distribution by subtracting 3 from every possible value, then the mean of the new distribution would be 0.
@cogitateandabet
4 жыл бұрын
that's what i needed...thanks man
@adityashikhar6143
3 жыл бұрын
@@cogitateandabet Man I didn't get it. Can you please explain
Great stuff! I never thought I would ever be interested in probibility and statistics (having had a subpar education in the field), but apparently I was wrong!
@jbstatistics
6 жыл бұрын
That's good to hear! And trust me, there are more interesting things in probability and statistics than standardizing normally distributed random variables!
@eeltauy
6 жыл бұрын
I'm hooked!
How do you find the area between -1.72 and 1.22 without using any software computations? or is there a formula for it?:-o
@jbstatistics
7 жыл бұрын
There isn't a formula -- the area is obtained by numerical integration. If you don't have access to software that will find the area (e.g. R, SAS, Excel), then you would need to look up the relevant values in the standard normal table and make the appropriate calculation.
@chilledstudentvloggr4732
7 жыл бұрын
Is this by using the (+ and -) zscore table? And thank you for responding! Your videos helped me a lot!
@jbstatistics
7 жыл бұрын
Yes, you'd use areas from the Z table to find the appropriate area.
sir plz help me to solve this... ** Given a normal distribution with a mean of 60 and sd of 14, n=100 what percentage of cases will lie between the score of50 and 65??
Thank you.
Thanks, this really helps!
@jbstatistics
7 жыл бұрын
You are very welcome.
Great videos, thank you for taking the time upload these lessons! How did you find the Z value for the 10th percentile on this distribution curve? I am familiar finding the value of the 10th percentile (using norminv fx in excel) but not sure how you derived Z score of 10th percentile I look forward to your response,
@jbstatistics
8 жыл бұрын
+promar In the video I needed to find the 10th percentile of the standard normal distribution. This can be found using software, or a standard normal table if software is not available. I used the R command qnorm(.1,0,1) to find it. I know little of Excel, but the NORMINV command would perform the same function. NORMINV(0.1,0,1) would result in the correct value of -1.281552.
god bless u for making these
1. I watched your videos on how to read standard tables but still can not understand how the area got to be 0.111. I think we should look at the table with positive numbers and that gives an area of 0.8888. The table with negative numbers gives an area of 0.111, as you said in the video, but I do not see why we should look at that table in the first place. 2. In the percentiles section, you have marked 0.1 on the left side of our graph. Left side is for negative values, right? Why should 0.1 not come to the right side? I would be glad if you could help me. Great video. Thank you.
@abdulbasitraza6204
5 жыл бұрын
we have to subtract the .8888 from 1 in order to have the area greater than z=1.22 and that becomes equal to .111
Hey!!! Can u suggest some websites to practice questions based on these concepts? Help appreciated
Can someone explain why the 10th percentile is a value that has an area 10% to the left. Why not the right?
@BonyS3k
8 жыл бұрын
+Jono Hermes in this example it means, that you are taller, than 10% of people.
@paulwary
7 жыл бұрын
The equivalent situation on the right side would be the 90th percentile, ie a height that is taller than 90% of the population. 100% is the area under the entire curve.
simply superb
@jbstatistics
5 жыл бұрын
Thanks!
Thank you
Simply Great
@jbstatistics
5 жыл бұрын
Thanks!
thank you so much
Good Video
in what video did you compute for the area?
EACH LECTURE IS SHORT AND VERY VERY EXCELLENT -LANGUAGE IS VERY CLEAR - EXCELLENT THANK U AMARJIT ADVOCATE DELHI HIGH COURT INDIA
@jbstatistics
8 жыл бұрын
+Amarjeet Singh You are very welcome!
Hello JB statistics. I really like your videos they are so simple to follow and understand with appropriate examples. Please, could you do a video for a transformation of random variables and also Convergence of Random variables.. i tried to find something to learn on this but i couldn't. Or else if its already available please give the name or link so that i can check it out. Stay blessed, your videos are life saver trust me.
@jbstatistics
8 жыл бұрын
+Alan Lwanga Hi Alan. Thanks for the compliment, and I'm glad to be of help. I doubt that I'll have time in the near future to make videos on those topics. I'm sure there are some good resources out there, but I don't have any suggestions. All the best.
you're the best bro
the last question really confused me i do not understand where we got the z= -1,28 from and dat was the reason i watched this video can someone please explain @jbstatistics
@jbstatistics
6 жыл бұрын
-1.28 is the 10th percentile of the standard normal distribution, and it can be found using software or a standard normal table. If you don't know how to use the standard normal table, then it would be helpful to watch one of my videos on using the standard normal table (e.g. kzread.info/dash/bejne/X4mgzKucocLYlsY.html).
@charlottemkhize
6 жыл бұрын
jbstatistics thank you so much Sir, I have a statistic exam tomorrow and your quick response has helped me a lot thank you thank you and I do get it now ☺
hi, you are great at explaining sir. I have question . I look at the value 1.22 on the table it have the probability 0.8888 and -1.22 had probability of 0.1112 . if the standard deviation is 1.22 how come the prob is 0.1111 ? thanks
@bhagirathtallapragada4699
8 жыл бұрын
+Seerat Raseen Yes apparently the negative value has been mistaken for the positive one! had the same doubt.
@jbstatistics
8 жыл бұрын
+Bhagirath Tallapragada The question is asking for the probability that a randomly selected female is *taller* than 170.5 cm. So we need the area to the *right* of 170.5 under the appropriate normal curve. This is equal to the area to the *right* of the calculated z-score (1.22) under the standard normal curve. When you run off to the z table and look up 1.22, the table gives the area to the *left*, which is not what we need. The standard normal distribution is symmetric about 0, so the area to the *right* of 1.22 is equal to the area to the *left* of -1.22. This is why when you look up -1.22 in the table, you find the final answer. You can also find the final answer by subtracting the area to the left of 1.22 from 1.
i have a problem can you help me about this exersise N(32 , 2 ) P(27
@mohitpuri3045
7 жыл бұрын
solve copying procedure as he did
@mobilbear
7 жыл бұрын
for your first case, μ==E=32, σ=Var=2, then your Z1=(27-32)/2=-2.5, your Z2=(29-32)/2=-1.5. So problem now is P(-2.5
Man you are amazing. you should teach in my University !!!
@jbstatistics
7 жыл бұрын
Thanks! You never know what the future might bring :)
THANKK YOUU ♥️
i cant thank you enough.
As I opened my Chinese Version book of probability in Gausion's distribution chapter. I thought...w*f but after your explaination... Thx
how could we know that 0. 1 lies on left but not on the right part please explain i couldn't understand
Hello Thank you for the videos Can anyone please tell How we can calculate this expression " P(-1.72
@huntersikari
8 жыл бұрын
Got it ,,we have some table with values!
@pikapikacheww_
4 жыл бұрын
@@huntersikari care to explain how to get area for -1.72 ?
@matt5931
Жыл бұрын
@@pikapikacheww_ If using R you could write: pnorm(1.22) - pnorm(-1.72) to get the %. -1.72 and 1.22 are Z scores (x-mean/sd) for each side. I am terrible at explaining and 2 years late but I hope this helps someone.
How would you calculate the probability of the exercise with the adult American female and her height if the equation said P(150.5 < X -10 < 170.5)?
@jbstatistics
3 жыл бұрын
Isolate X by adding 10 everywhere, then apply the same techniques from this video.
Can I ask what if the question is "Ten texpayers are selected at random. What is the probability that 3 of the taxpayers will get their refunds at least 16 weeks?" but before this question it alreasy state that the average is 12 and stdev 3..the variable is the amount of time.. Can I know how to calculate it.. Thank you
How did you get get .846?
@InstansVisio
8 жыл бұрын
+MadMax and how do u get 0.111 ?
@jbstatistics
8 жыл бұрын
+Killan TRAPSA I have videos outlining how to find areas under the standard normal curve. You can use software or a standard normal table to find that area.
@mobilbear
7 жыл бұрын
see how it is calculated: See, P(Z1-1.72)=0.9573, what you actually want is the Z1 and Z2. Because the left side of Z2 = 1-Z2 = 0.0427, and we would know that the "AND" part will be Z1 - (1-Z2) = 0.8888-0.0427=0.8461
@mobilbear
7 жыл бұрын
+Killan TRAPSA1, because in the previous example, it is to ask P(Z>1.22), while, actually from the Normal Distribution value table, you can get 1.22 as 0.8888, which means value smaller than 1.22, as you already knew that the total shadowing is 1, then the P(Z>1.22)= 1-0.8888=0.1112, hope this helps
@wiiliskaako5875
6 жыл бұрын
you subtract 0.047 from 0.8888 and if u r asking how i got this numbers its the area of -1.72 and 1.22
Going through almost all your videos, haha.
thank you !!
@jbstatistics
5 жыл бұрын
You are very welcome!
where on earth did you get the -1.28? I cannot find any table that says this. Please, anyone?
@jbstatistics
4 ай бұрын
It's the 10th percentile of the standard normal distribution, found with software or a standard normal table.
How do you find the -1.28 using software though? (ie. using R). I understand how to get the value from the standard normal table but I don't know the formula in R. Do you know the formula?
@jbstatistics
5 жыл бұрын
The R command qnorm(.1) returns -1.281552 (the 0.1th quantile of the standard normal distribution). qnorm(.1,0,1) would also work (returning the 0.1th quantile of a normal distribution with mean = 0 and SD = 1, which is of course the SND).
@simonpaisley101
5 жыл бұрын
@@jbstatistics Thank you so much! That helps a lot!
I dont get where did you get 0.846 in q2, I know its from the table but where.. and how..
Good presentation
@jbstatistics
5 жыл бұрын
Thanks!
ur simply awsome thnx
@jbstatistics
8 жыл бұрын
+Fikri Saoudi You are very welcome!
I still did not get how you ended up with the 0.11 probability. can you please explain how can I calculate the probability of 0.11
What software can I use
Hi can you elaborate more on how you found the area for the second example (around 7:42) with the P(-1.72
@jbstatistics
9 жыл бұрын
P(-1.72
@jordanpeterson2800
9 жыл бұрын
jbstatistics Don't quite understand how you came up with 0.846 out of -1.72 and 1.22 at the 8:00 mark. Could you elaborate?
@alix14359
8 жыл бұрын
+jbstatistics please would you be able to explain this
@mobilbear
7 жыл бұрын
+1, i don't understand either, please help to answer, thanks
@mobilbear
7 жыл бұрын
ah! i got it. +537yaya, +Jordan, +Alix, See, P(Z1-1.72)=0.9573, what you actually want is the Z1 and Z2. Because the left side of Z2 = 1-Z2 = 0.0427, and we would know that the "AND" part will be Z1 - (1-Z2) = 0.8888-0.0427=0.8461
thanks
Hi can you help he solve this question. ..if a mean is 280 and sd is 12..in a normal distributed random variable. .what's the probability between 275 than 290
@jbstatistics
10 жыл бұрын
I suggest you watch the example in this video that starts at 5:50. Cheers.
Hi, just wanted to confirm, the std. deviation if 6.8 is a value given in the question on 10th percentile calculation. Or is it something that is calculated from somewhere?
@jbstatistics
8 жыл бұрын
+Shirin Tejani That was a given -- it represents the true standard deviation of heights of adult American women. (In reality, the true standard deviation is unknown. The value given here is based on sample data, but is likely very close to the true value. In this question we are pretending that the true standard deviation is a known quantity.)
@ShirinTejani
8 жыл бұрын
+jbstatistics Thanks! (PS - the videos are really great!)
@jbstatistics
8 жыл бұрын
+Shirin Tejani Thanks! I'm glad you find them helpful!
I do not know how did you get the 0.111
How did you get 0.111?
what did you do to find 0.846 ???
@minjeonkim9655
3 жыл бұрын
ye im also wondering
@nostro1940
2 жыл бұрын
Think about it, the whole area to the left of 1,22 is 76% You need to subtract the area to the left of -1,72 (which is 4,2%) to get the total area of 72,15% He is wrong
Sorry, why do you say that by subtracting the mean from X you necessarily get a new mean of 0? I lost you in that reasoning. Thanks! (Very helpful videos)
We can find the appropriate areas using software or a standard normal table. I have a playlist "Using a Standard Normal Table" which contains videos for the two main table types. Cheers.
Your videos have been fucking helpful for my short summer class. THANK YOU!
@jbstatistics
7 жыл бұрын
You are very welcome!