100

An Introduction to the Normal Distribution


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.

Stephen Childs

100 Comments

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

  2. 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

  3. 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!

  4. 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.

  5. 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

  6. 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

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

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

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

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

    Thank you, very helpful! Better than any professor 🙂

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

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

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

  14. 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!

  15. 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?

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

  17. 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

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

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

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

  21. 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.

  22. I recommend this channel as supplementary to harvard statistics 110 or mit 6.041
    very clear explanation!

  23. 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?

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

  25. 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

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

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

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

  29. 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 🙂

  30. "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!!

  31. 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.

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

  33. 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

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

  35. 4:25, and I realized you were Canadian before you said Zed… Thanks for your help professor..

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

Leave a Reply

Your email address will not be published. Required fields are marked *