Statistics Lecture 6.5: The Central Limit Theorem for Statistics. Using z-score, Standard Score

[@tradingchallengeto]

11 years passed,still some students who're struggling using your lessons,thank you very much Professor Leonard,God Bless You

[@ProfessorLeonard]

Point of Clarification: The formulas for the “Mean of Sample Means = Population Mean” and “Standard Deviation of Sample Means = Population Std. Dev / sqrt (n)” are ALWAYS TRUE!! (No matter the size of “n”). I actually state this in Section 6.4 and briefly in this video at 25:18 and 31:19. HOWEVER, in this video, I imply that those formulas DO depend on the size of “n”… this is NOT true. My intention with the statements regarding when the Central Limit Theorem “works”, is so that you know WHEN a Sampling Distribution can be assumed to be Normally Distributed (So that a Z-score applies) and that the Mean of this Distribution is equal to the Population Mean and the Standard Deviation of this Distribution is equal to the Pop. Std. Dev / sqrt(n). This EVENTUALLY helps us prove claims about Population Parameters in Hypothesis Testing and Confidence Intervals. My sincere apologies for my mistake and lack of clarity.

[@tomaspianist]

I very much appreciate your update and your honest admission of a mistake. Thank you for that. I did suggest my students to watch your videos, but I did announce the error. Still, a very nice teaching!

[@turboleggy]

I've been watching all you stats videos to refresh my memory. They are easy to understand and the humour is on point; thank you for posting.

[@Regalert]

Professor Leonard Tanks, teacher. Greetings from Brazil.

[@bhaskardhariyal5952]

By giving this clarification, do you mean to categorically say. n>30 statement has nothing to do with "Standard Deviation of Sample Means = Pop. Std. Dev / sqrt(n)"? And in third case, formula will still be valid, but you can't say the sampling distribution will be normally distributed? 22:20

[@cherylpokemon6376]

Professor Leonard Hi Professor Leonard, I really enjoy watching your videos! Some of the other commenters have mentioned this but around 41:58, when you’re calculating the z-score, you wrote ‘3/10 = 0.33’. I think it’s meant to be ‘3/10 = 0.30’.

[@jakemiller6638]

Just wanted to take the time to say that what you are doing with these videos, in conjunction with your tremendous teaching skill, is incredible and I cannot thank you enough. You are partially responsible for my love of math, and you are an incredible Professor. Your students are incredibly lucky to have you, and so is everyone who watches your videos.

[@PhysicsThroughComputation]

Central Limit Theorem : Theory, Code, Visualization : All in one is here : kzbin.info/www/bejne/l3WYc6Gfdqaoes0

[@roberte.1379]

41:56 The Z-score is 0.30 (= 3/10) rather than 0.33 (=3/9). Therefore, the probability to the left of the Z-score is 0.61791, not 0.62930. 44:12 The probability to the right of the Z-score is 0.38209, not 0.37070. (44:05)

[@jeremylentz3907]

I was hoping he would catch it.

[@AnalystTosh]

That's what I thought..

[@stein5145]

I think he divided it inversely which is 10÷3 because the # 9 is not their

[@nursyasyazuhayra637]

​@@AnalystTosh 😊😊

[@elleskinner9515]

Dear Professor Leonard, Thank you so much for sharing your lectures! I am a graduate student taking online classes because I work 30hrs/wk in a professional job, writing the beginnings of my thesis and completing 3 graduate classes. I don't think you will realize how much you've truly saved my grade this semester!!

[@melissas6315]

I sit through three hours of lecture confused. Then come here and it all makes sense. Thanks Professor Leonard. I appreciate your help more than you know! Thanks for sharing these videos and saving this student!

[@teodyramon4456]

if this was the kind of instructor i had in college, i could surely grasped all the concepts. great job!

[@TheMika4576]

I'm currently taking Statistics and struggling, I'm grateful to have stumbled upon your video. Thank you immensely.

[@riaznapa]

after so many lectures, students still struggled to find out what n stood for. I really appreciate the patience of the professor!

[@PhysicsThroughComputation]

@Swagata Mukherjee Central Limit Theorem : Theory, Code, Visualization : All in one is here : kzbin.info/www/bejne/l3WYc6Gfdqaoes0

[@waseemqaffaf5715]

@Swagata Mukherjee I KNOW RIGHT!!!

[@maintaineveryday4732]

Currently have prof dropping the ball for out stats class. He is depressed sounding 24/7 and complains about his life every class I just join zoom and put him on mute and watch your videos. You are the reason I am fucking finally understanding this. Glad I pay price for credits to have a professor making things harder than they already are. You literally are making my day to day life easier and more enjoyable I mean that. Cheers!

[@angelofdeath7341]

OMG....I’ve been confused for weeks in my dumb class, I watch this and totally get it. THANK YOU!!! Our instructor sits at his desk and reads the examples in the book to us, then tells us to go home and do the homework so that we understand. I’m a visual learner, and listening to him read is a colossal waste of time, not to mention how boring it is! I’ll go do my homework now, lol.

[@NelkaPerson]

These videos are STILL helping students. I'm another one who is just blown away by your ability to teach. You explain things in a way that my professor can't. It just makes sense. Thank you so much for making these, you're amazing!

[@seeunkim4185]

I am from South Korea and I am starting study psychology in the Netherlands, thank you so much for your lecture. You saved my life!!!!

[@nickerz81]

was wondering if anyone else caught the 3/10 error....we all make mistakes. Love your videos!! They make this learning process much easier, especially since your units tie into the pace of my class almost exactly! - Thank you!

[@YusifRefae]

ya i just caught it lol i was confused for a second. the answer is actually 38.2% doesn't matter tho, still an amazing teacher & lecture

[@thanasisconstantinou7442]

You are an awesome professor, sir. Being an analytical chemist, I have to deal with Z-scores and population distributions all the time, but, thanks to your excellent lecture, this is the first time I feel 100% confident how to use them correctly. I am grateful for that.

[@camvunguyen4670]

He is superman, he saves people's lives, i have such deep and profound respect for him

[@T181T]

I trust the Leonard so much that when he made a mistake with the .33 i thought my calculator was broken. For real i had to pull a second calculator to check the first calculator to make sure it was working. The Leonard is my hero no way he could be wrong. I f@#king ordered a new calculator from Amazon hopefully it calculates the Leonard way!

[@jestheall]

You just helped me go from low Ds on everything to a 100 on a quiz... I cannot say thank you enough!

[@ravineeshgoud8145]

Best explanation of Central Limit Theorem I've ever came across !!!!!

[@roseahmed9964]

You are talented in submit these difficult stuff honestly... I’m gonna tell my instructor to follow you and hopefully she can learn how to teach

[@drahmedadel-4873]

from EGYPT and we say to you are amazing THANKS

[@shivanisrivastava1567]

Having exam of data analysis Really you r a Saviour sir 🙏

[@banoocooks7890]

Thanks to you I got A in midterm.awesome explanation great job

[@Subboxx]

finally a teacher that makes stats loveble

[@PhysicsThroughComputation]

Central Limit Theorem : Theory, Code, Visualization : All in one is here : kzbin.info/www/bejne/l3WYc6Gfdqaoes0

[@ananyakapoor2677]

You changed my life Professor Leonard.

[@xiaya7345]

Thank you so much for sharing your lectures with us. I’m really struggling in my stats class right now, so this video has helped me enormously to understand the concepts better. Prof Leonard, you have a fantastic teaching style in presenting info that are easy to follow and grasp. A question I would like to ask u is- if u would consider using PowerPoint for the lecture materials? This would allow both prof & students to save time and carry on through lectures faster. Just a thought I had, but if PowerPoint isn’t your thing than please disregard my inquiry. Another big thanks for your help & generosity in sharing your knowledge with us. Well done 👍

[@perry9531]

I have a stats teacher that taught me less about the clt in the past two months than you did in the first 5 minutes of this video.

[@ainamukwiilongo]

From Namibia thank you so much prof💯

[@taekwondotime]

For the problem at 42:00: 3/10 is 30% or 0.3, not 0.33. The z-value to look up in the table is 0.3 which then produces the area .6179. Taking the reciprocal we get: 1 - .6179 = .3821 or 38.21%.

[@meligymohamed8980]

Privacy Matters none of his students comment on that May be there is something we don’t understand 🤔

[@taekwondotime]

@@meligymohamed8980 No he just made a mistake. 3/10 = 0.30, not 0.33. He has the wrong z-score.

[@taekwondotime]

@@paulkaczur791 No, he has the wrong z-score. It should be: z = (175 - 172) / 10 = 3/10 = 0.30

[@mikeb5957]

Right around 43:30- 3/10 is .3 not .33, it makes a difference in the Z score. Ten divided by three will give you 3.33 for sure, which would be easy to mix the two up. Only saying this to help anyone else, I surely wouldn't be watching this if I was smarter than Mr Leonard.

[@OMARYassin1]

42:00 3/10 = 0.3 Not 0.33 Thanks

[@hellmuth26]

Can't believe no one called him out on that.

[@TheFloral17]

Amazing teaching technique...never enjoyed anybody's lectures since junior school...you explain everything so clearly.... I wish i could meet you.. just love you for making me love statistics my actual school profs sucks...you are so Hot

[@keithhudson3138]

I'm staying with you the entire time in my Spring 2021 semester.

[@ericlenertz6967]

35 minutes thru and I gotta say, BICEPS....

[@observer698]

so distracting!

[@PhysicsThroughComputation]

@@observer698 Central Limit Theorem : Theory, Code, Visualization : All in one is here : kzbin.info/www/bejne/l3WYc6Gfdqaoes0

[@arsal16]

wasted 3 hours reading the book, this video was worth it

[@bindugokila3111]

Thanks, Professor. You teach very clear and it is easy to understand

[@johntindell9591]

Thank you so much from Thailand.

[@stevenmushaija2046]

I will start taking statistics and prob next sept, but i am really happy getng HIS lectures before i start taking ma course; hope they will be much of help. Though i will be using a different book.

[@nguyenminhtam1802]

So much passion in your lecture! Hope you can make some statistic exercises solution for revision.

[@elizabethvasquez1024]

What a great class . I’m getting it now 🧐😊

[@emelieokok]

Thank you for saving my life

[@slaozeren8742]

Thank you so much, you have just changed my destiny.

[@michealboyd161]

this man giving me hope to graduate :)

[@kayala667]

you are truly AMAZING at what you do! THANK YOU.

[@yenpham498]

great teacher, no matter what.

[@twomapunga5797]

very articulate with your stuff. might be too late but if you decide to do more, it may help to use slides and save us watching you writing everything, breaking them into small topics will also help as we struggle to find specific topics for quick reference. Cant see any1 failing after such lectures. thanks very much

[@rmcfar8818]

Which level is this course? Is this a community college? You may be saving my semester. Such an incredibly good tutorial.

[@DaDaDamage]

Doesn't really matter. I took a stats class at a university that is the same concept at a community college (for generals) the only difference is at a university is I was in a class of 30 people and where I am taking this same class again with only 5 people including myself with a way better instructor

[@juyifan7933]

@@DaDaDamage Depends, this is a stats class for "non-math" people. If you take a statistics class for STEM students at a university it will be very different. This class pretty much doesnt use any math other than basic operations and using a calculator, it is taught at a high school level. A typical stats class for STEM students at a university would assume knowledge of calculus, and in some cases would also require programing.

[@Beckyjo11]

Was having trouble with this but you explained great! Thank you!

[@sayaligaikwad8144]

I have fallen in love with mathematics because of your lectures.. I don't need calculas still felling to attend thodse

[@andreaalexandermccullough468]

I love this instructor!!!

[@theillustriousjohn]

Good job, Prof. Leonard.

[@Lena-wn2lg]

Fantastic prof!!

[@paulfynn6641]

Can you please clarify this for me ; The example that you gave ,about the '' population of men'' What if the question was framed as ; 1. Find the probability that a group of 20 men will have an average weight between 175 and 179. 2. Find the probability that a group of 20 men will have an average of less or equal to 175 .

[@JuiceBoxBoiii]

Thank you for this. You are a great teacher.

[@KimNic]

amazing prof. Wish we had more like you at my school.

[@taekwondotime]

In my statistics textbook, the condition is *n >= 30* not n > 30. A minor detail. Maybe it's not consistent between texts. Source: *Probability and Statistics for Engineers and Scientists* by Walpole, Myers.

[@davidkeck1892]

Irrelevant

[@bryandesantiago14]

Ohhhhh man. I didn't know the god Leonard also taught statistics! I'm about to ace this class

[@ilovefredreika]

Very, very clear. Great teacher!! Thanks.

[@AuntyJasmine80]

Thank you for the explaination. It is very well explained. :)

[@TarunSingh-je9my]

Thank you for explaining the central limit theorem conceptually. Please upload some videos on Time series also :)

[@charlesw6429]

Thank you.

[@Cestuna]

thank you for saving my grade

[@aaronjenedwards3723]

it looks like this school is not in a gun free zone.

[@omarabu-khalaf9520]

Thank you Professor Leonard, I tried verifying the formula ( Sigma sub X bar) = sigma/sqrt(n) for the data values 1,2,5 by selecting all samples of size 2, namely, (1,2) , (1,5) and (2,5) but sigma for 1,2,5 is 1.699 while the other side of the equation I mentioned is 1.699/sqrt(2) which is approximately 1.2, but when I calculated sigma sub x bar directly it turned out to be approximately 0.849, I appreciate if you can help correcting any mistake in my work.

[@56348u]

He is good.

[@sumitsouravamohanty8935]

PROFESSOR U R JUST AWESOME

[@namra2184]

Again so GOOD!!!

[@IsilCaglaMorsunbulAlumni]

all in one ! Wish I can have you in my college

[@adarshparameswaran6722]

The Textbook he follows is Elementary Statistics by M. Triola

[@ZuhairJumaa]

Unbeatable prof :)

[@Pav9001]

1:11:08 Example: A sample of 106 people were selected... Professor: "Are we dealing with group or individual?" Dumb Students: "Individual".....hahaha are they sleeping? Don't they realize how freaking lucky they are to have such an amazing professor? My teacher right now told us to read a book and apply it to problems that he never explained to us...ugh

[@PhysicsThroughComputation]

Central Limit Theorem : Theory, Code, Visualization : All in one is here : kzbin.info/www/bejne/l3WYc6Gfdqaoes0

[@jamesleem.d.7442]

I think I heard you say in your lecture that a hypothesis can be "proved" or "disproved". You are half right. We can definitely refute (i.e. show to be invalid) any hypothesis given some set of empirical data that make that happen. However, the correct term for what you call "proving" a hypothesis is "Corroborating the hypothesis" or "Finding data that are consistent with the hypothesis". Read Karl Popper. Read Deborah Mayo. This has not been a contentious point at all for at least the past century of thought in science and the philosophy of science.

[@hyperklathos]

10:02:33 you could say 'minute' instead of 'really small'

[@ashleymtsac3679]

For 45:41 will the probability be .382 if we use the calculator instead of the normal distribution table

[@lancegabrieltanedo8614]

when I calcualted a z score of -6.6 I was confused if I was wrong or not because the probablity would basically be just 0% or 100%

[@Peter_1986]

1:00:53-1:00:59 What IS Professor Leonard's age, though? I have always assumed that he was born in 1980 or something like that.

[@knowledge9822]

3 divided by 10 is 0.3 not .33 :)

[@knowledge9822]

+Perfectionist00001 10001110011 3÷10 is just 0.3. You're just converting a fraction into a decimal. if you need 3 decimals, it'd be 0.300

[@LazyGates]

+Perfectionist00001 10001110011 You may be confusing 1/3 = 0.333... with 3/10 = 0.3.

[@lorilihime]

+Knowledge He does this so that they can use the wonderful little table of Z-Scores they are allowed to use on tests and everything. To eliminate guess work and stuff. 0.33 can be found on the table if I remember correctly. Then again I could be wrong... I dont have the table :P Yay calculator and precision!!!

[@yeshwantrajwade7396]

Rather than finding the Mean of sampling distribution isn't it easy to find the mean of population? If we take the example of the class with 27 students... rather than finding the sampling distribution with 80000 samples and find their means, isn't it lot easy to find the mean of the population? How does the concept of sampling distribution help in real life?

[@sauradeepbhattacharyya4117]

Could u pls tell me which book is being used for reference in this video.....The video has been extremely helpful

[@fliederblumen1843]

Hi Professor Leonard, I am thinking when n equal or smaller than 30, the formula of mean (that mean of sample mean equals original population mean) and std.dev. formula are correct when the population is normal distributed OR uniform distributed. (proved at the end of 6.4 actually)

[@carlosguilhermemenezesgarr9705]

Thank you very much !!!!!!!!!!

[@ericgourjon5285]

Great course... have a question though... my thinking process is probably wrong but I cant see where it fails... if I have a group with n=1 individual only, the 2 formulas (for data value vs. sample of data values) for the z calculation are matching exactly... but how come when we say that we are selecting 1 single random data value I can apply the formula w/o any assumption on the population distribution while if I say n=1 on a sample which is way lower than the threshold of n=30 items, even though the formula matches exactly at the end, I have to have the assumption that the pop distribution is normally distributed to be valid... any idea/clue around what I am missing here?

[@vainbow4632]

The central limit theorem doesn't state anything about a sample size of 30, or any finite number

[@isabelgoos6646]

When I build the histogram of bar(X) doing extractions of size n... I do that Nrep times. I mean, I need to repeat this experiment of extracting bar(X) in order to build a histogram. How does Nrep come into play? Does it have some effect?

[@leslieannemujica9912]

what college do you teach in? You're an amazing professor! My professor never really taught :/// Thank you so much for your videos, even if they are intended for your students.

[@roberte.1379]

I thought he said "Merced College" in his hypothetical example of sampling students for hair color.

[@Peter_1986]

@@roberte.1379 Yes, these videos were recorded at UC Merced, however I have heard that he no longer works there - but I am not sure about that.

[@rohank9292]

Thank you

[@yousseflahmar9036]

you are amazing thank you

[@anthonyvillarama6806]

@professor leonard, are there only 28 videos of this statistics course? Where can I find the rest of the section? Thank you.

[@adamel6354]

i would like to ask what if n is less than 30?then do we need to use the finite correction factor?

[@jamesleem.d.7442]

Great lectures and really good technique overall, but maybe you should decrease the asides and "double checking with class" which are very frequent and do seem to be a distraction. How many hours per week are you in the gym ?

[@organizedchaos4559]

Does he ever do multiple random variables?

[@ritankatha8981]

Prof at 43.00 the Z score is 0.3 not 0.33 (3/10) is 0.3 you know when you look at the table the answer will be different

[@mutlugundiler4458]

Dr. Leonard, please give the name of the book you used for this class.

[@purplebeard806]

In the rare event rule ex, how did you compute the area with z>3 ?

[@mertkocabali1095]

I need to study point estimation. Which videos should i watch?

[@robertpollock8617]

What text book is he using?

[@sisirbhattacharya338]

Which books is used by his class ??

[@manuelsojan9093]

do you use continuity correction for central limit theorem?

[@komalnagar7647]

great lectures my compliments please share the name and author of the Book the professor is referring to...

[@engaredwan]

elementary statistics , 11th Edition

[@komalnagar7647]

Thank you for the response... Elementary Statistics: Name of the author

[@engaredwan]

+komal nagar "triola" www.amazon.com/Elementary-Statistics-11th-Mario-Triola/dp/0321500245

[@vineethkrishnan8120]

Please help me clarify this Why we call average of sample mean or standard deviation of sample mean? We calculate the statistics(proportion, mean, std deviation etc) first and then the average of it right? so the terminology should be reversed.Someone please clarify.Thanks in advance