Let’s look at an introduction to the

normal distribution, also known as the Gaussian distribution. The normal distribution is an extremely

important continuous probability distribution. It arises very very frequently in

theory and practice. Here I’ve plotted out what is approximately the distribution of the height of adult Canadian males. And this, what we see here, is a normal

distribution. This is our variable down here and f(x) is what we call the probability

density function, which gives the height of the curve at point x. Here’s the equation for the probability

density function of the normal distribution. We’re not going to be working with this

directly very often, but there’s a few points we should note. First of all, if the random variable X

has a normal distribution, then it can take on any finite value. There are two parameters for this distribution. Mu is the mean of the distribution. And the mean can take on any finite

value as well. Sigma is the standard deviation and sigma squared, of course, is the variance. Sigma has to be some positive value and of course sigma squared would

have to be some positive value as well. Here’s a normal distribution plotted out. Mu, the mean, is also right smack dab in

the middle of the distribution. The distribution is symmetric about mu, so mu represents the mean of the

probability distribution, and it’s also the median of the probability distribution. No let’s try and get some perspective

for what sigma represents. Here I’ve plotted in one standard deviation below the mean to

one standard deviation above the mean. And as we’ll see a little bit later on this area between one standard deviation below

to one standard deviation above is approximately 0.68. 68% of the area lies within one standard

deviation of the mean Here I’ve plotted in mu minus two

standard deviations to mu plus two standard deviations. And approximately 95% percent of the area lies within two standard deviations

of the mean. And approximately 99.7% of the area lies within three standard deviations

of the mean. Here’s another plot to give us a little

perspective on what sigma represents. Over here this red line represents a

normal distribution with a standard deviation of sigma_2. And this white curve represents a normal

distribution with a standard deviation of sigma_1. But sigma_2 is double sigma_1, so the standard deviation on the red curve

is double the standard deviation of the white curve. And we can see that when the

standard deviation is greater there is more area in the tails and a lower peak. There are an infinite number of

different normal distributions corresponding to all the different

possible values of mu and sigma. If X is a random variable that has a

normal distribution with mean mu and variance sigma squared, we write this as the random variable X is distributed normally with a mean of mu

and a variance of sigma squared. X is distributed normally, a big N, first term is the mean,

second term is the variance. Be a bit careful here, as different

sources can have a different term in here. Sometimes people have the second term representing the standard deviation. I have the second term representing the variance. So just be a bit careful with that. By definition, the standard normal distribution is a normal distribution with a mean of zero and a variance of one. We often represent random variables

that have the standard normal distribution with the letter Z. And yes, I am Canadian. And this, we might say, if Z has a standard normal distribution, we might write this as Z is distributed normally with a mean of 0 and a variance of 1. That being mu, that being sigma squared. And if the variance is 1, of course sigma is 1 as well. As we will soon see, we very often need

to find areas under the standard normal curve. Probabilities are simply areas under the curve, and very often the question of interest

involves finding a probability. Finding probabilities and percentiles

for the normal distribution requires integrating the probability density function. There isn’t a closed-form solution

and it must be integrated numerically. Fortunately for us ,we’re going to use

software or a standard normal table to actually find these values in practice.

Thank you very much! I'll be getting back to adding videos soon!

"Z. Yes I am canadian."

great.

I confirm!

Great Videos! I am revising the subject through your videos, very helpful! 🙂

Thanks! I'm glad you find them helpful.

wow, u are doing some great karma in life. Very clear explanations. These videos are incredibly helpful. Thanks a lot sir.

wish you mental peace and physical well being

with metta loving kindness

Ravindra

You are welcome. And thanks for the well wishes!

I'm Australian and i say 'zed' 😀

powerful canadian accent!!!

you write it wrong Sir, It is 99.7% of the mean not 0.997 🙂 hehe

you're videos have helped me immensely, thanks so much for helping me understand concepts in 5 minutes instead of hours with a text book!

You are very welcome Connor. I'm glad to be of help! Cheers.

You're the man, JB

Thanks Brent!

LMAO

Sir, I am not even in your class, yet the day before the final midterm I'm viewing your videos. That itself should say enough, but just to reiterate: THANK YOU ON BEHALF OF ALL STAT*2040 STUDENTS IN ALL SECTIONS IN EVERY SEMESTER. You are the second prof @ guelph I have encountered that uses YouTube as a teaching tool, and I must say, it is probably one of the best ideas I have seen.

Thanks again,

a panicking gryphon.

99.7% = 0.997, notice he did not write the % symbol?

You are very welcome! Best of luck on your exam.

Is every data set following normal distribution ?

hi finley

"And yes, I am Canadian" hahaha! Great video thanks 🙂

So my guess was right with all those Canadian males questions.. you are one 😀

Great channel.. I wish to meet you. Words fail me. I just can't express how you turned a subject I hated to the one I enjoy the most!

Cheers from India

So my guess was right with all those Canadian males questions.. you are one 😀

Great channel.. I wish to meet you. Words fail me. I just can't express how you turned a subject I hated to the one I enjoy the most!

Cheers from India

Love your videos!

GUELPH REPRESENT

Thanks a lot. It is amazingly fine. Keep it up. Getahun from Ethiopia

eventhough, I am an arab(my english is not good),I understood this nicely.

Very Nice… Than you…

Very helpful. Thank you!

You have literally saved my grades with your videos. I'm so glad I found your channel! Thank you so much!

And YES I'M CANADIAN! (with that tone) :DDD

Thank you, very helpful! Better than any professor 🙂

LOLed at "Yes I am canadian". Ermmm we figured it already when you said "Out" in a few videos. Hahaha

Thanks for helping me out on my finals. I'm a fan! 😀 You make statistics bearable! I finally understand after all these years.

Zed and I am Canadian. Represent! From Quebec but Canadian nonetheless.

thank you….

Please Upload some videos of Triangular Distributions

Guelph, cool! I'm currently taking a stats course at the University of Waterloo and your videos have helped a lot. Nice to know you're so close!

I love Canada <3

great!!!

"And yes, I am Canadian."

thanks a lot!All your videos are helpfull.You never let me down.,,

this clarified a lot for me.

Thanks very much sir. Appreciated.

And making a step beck: How do I know that the data I collected are normally distributed? I do I came to the bell curve with my data?

+jbstatistics i do not understand why x-u/sigma =1 ?

you're from Toronto aren't u

Clear and simple…good job

4:25 😀 😀

canada much better than america

Your videos made statistics so much more bearable! Thank you so much, you Canadians are awesome!

wait what did being Canadian have to do with standard normal distribution and Z? just curious whether it was just randomly answering a question or something else more relevant xD

Cool

is there an exponential distribution? cant find it

"And yes I am a Canadian"……I am a Canadian too bro…..but gotta ask y don't u explain exponential distribution. 0.0

I'm in Ghana and I'm loving jb stats. awesome work !!! keep it up

Thanks men!. 😀

This was very well done.

how is that relevant,you being Canadian

go canada

Saved my ass. Really thankful.

I was recently introduced to your channel. Just wanted to say thank you for the clear presentations.

I think the normal convention is that it's represented in N(mu, sigma) and G(mu, sigma squared)

Thanks Mate you are a Legend. I have learned so much from my Canadian mates, now I am so eager to visit Canada to learn more. You guys are Awesome. Thanks Again.

I recommend this channel as supplementary to harvard statistics 110 or mit 6.041

very clear explanation!

I've got a Statistics exam tomorrow. Your videos are God sent.

Canada… You gave the world Drake and The Weeknd. You also gave us greats like JBStatistics. I ask, what has the world given to Canada?

Yes I am Canadian. Does the JB stand for Justin Beiber then? 😀

Thanks a lot sir 🙂

Thank you. I have a test in this tomorrow and I didn't understand all the concepts. You're a lifesaver, I already recommended your channel to my classmates

Tanks man best tutorial I've seen bout this on YouTube now wanna learn how to use it in data science

D Histogram of Simulations from a Mixture of 3 Bivariate Gaussian Distributions, grazie. Stefano Caser

bro, u cracked me up when u said "and yes I AM Canadian" . Thanks for making my day! xD

thanks I'm passing my stats exams because of u

I don`t usually comment on videos, but after this `yes, I`m a Canadian` haha. We`re missing some `eh` eh? By the way, I love these videos very well explained. Please keep it going 🙂

Yea, I figured you are a Canadian when you mentioned Toronto haha

"Yes I am canadian" loll…just killed it! although i knew that already. Great videos! only reason i am doing good in stats. So thank you!!

4:22 "with the letter Z, and yes I am Canadian" LOL 😂😂💔

What happpens to a normal distribution if n =infinite ?

I use your video to learn English. Thinks!

0:00 Let's look at an introduction to the normal distribution also known as the Gaussian distribution.

The normal distribution is an extremely important continuous probability distribution.

It arises very very frequently in theory and practice.

0:18 Here I've plotted out what is approximately the distribution of the height of adult Canadian males and this what we see here is a normal distribution.

This is our variable down here and f of X is what we call the probability density function which gives the height of the curve at point X.

0:40 Here's the equation for the probability density function of the normal distribution.

We're not going to be working with this directly very often but there's a few points we should note.

0:49 First of all if the random variable X has a normal distribution then it can take on any finite value.

0:57 There are two parameters to this distribution.

Mu is the mean of the distribution and the mean can take on any finite value as well.

Sigma is the standard deviation and Sigma squared of course is the variance.

Sigma has to be some positive value and of course Sigma squared would have to be some positive value as well.

1:24 Here's a normal distribution plotted out.

Mu the mean is also right smack dab in the middle of the distribution.

The distribution is symmetric about mu.

So mu represents the mean of the probability distribution and it's also the median of the probability distribution.

Now let's try and get some perspective for what Sigma represents.

1:49 Here I've plotted in one standard deviation below the mean to one standard deviation above the mean and as we'll see a little bit later on this area between one standard deviation below to one standard deviation above is approximately zero point six eight 68% of the area lies within one standard deviation of the mean.

2:13 Here I've plotted in mu minus 2 standard deviations 2 mu plus 2 standard deviations and approximately 95% of the area lies within two standard deviations of the mean and approximately 99.7% of the area lies within three standard deviations of the mean.

2:41 Here's another plot to give us a little perspective on what Sigma represents.

Over here this red line represents a normal distribution with a standard deviation of Sigma 2 and this white curve represents a normal distribution with the standard deviation of Sigma 1 but Sigma 2 is double Sigma 1 so the standard deviation on the red curve is double the standard deviation of the white curve and we can see that when the standard deviation is greater there is more area in the tails and a lower peak.

3:16 There are an infinite number of different normal distributions corresponding to all the different possible values for MU and Sigma.

If X is a random variable that has a normal distribution with mean mu and variance Sigma squared we write this as the random variable X is distributed normally with a mean of mu and a variance of Sigma squared.

X is distributed normally a big N first term is the mean second term is the variance.

Be a bit careful here as different sources can have a different term in here sometimes people have the second term representing the standard deviation I have the second term representing the variance.

So just be a bit careful with that.

4:07 By definition the standard normal distribution is a normal distribution with the mean of 0 and a variance of 1.

We often represent random variables that have the standard normal distribution with the letter Z and yes I am Canadian and we might say if Zedd has a standard normal distribution we might write this as Zedd is distributed normally with a mean of 0 and a variance of 1 that being mu that being Sigma squared and if the variance of is 1 of course Sigma is 1 as well.

4:46 As we will soon see we very often need to find areas under the standard normal curve.

Probabilities are simply areas under the curve and very often the question of interest involves finding a probability.

5:01 Finding probabilities and percentiles for the normal distribution requires integrating the probability density function.

There isn't a closed form solution and it must be integrated numerically.

Fortunately for us we're going to use software or a standard normal table to actually find these values in practice.

Thanks…

thanks

This graph is the cause of all my pain and sadness.

Wonderful ……your videos are really helpful ….Iam kindly requesting more calculation …,..specially part of poison and binomial

"And yes, I'm canadian" haha

I have a stats midterm test tmr, still dont know whether i can make it through but i grasp a lot from your channel and am more confident about the test. Thank you so much

Amazing!!!!

Things got so much clear now.. Thanks a lot 🙂

And Yes I'm Canadian! That was hilarious.

That was perfect

thank you for saving my life

Just a russian passing through… can anyone explain the joke with "I am canadian"?)))

Is that not how everyone pronounces Z ?

4:25, and I realized you were Canadian before you said Zed… Thanks for your help professor..

Thanks mate! I'm Iranian lol

very good explanation and very helpful. thank you!

Currently trying to create a normal curve by popping popcorn, this helped a lot for understanding the normal distribution, thanks!

Great explanation

what does 'Z' have any thing to do with being a Canadian ?!

Thank you Sir.