How do you DERIVE the BELL CURVE?

In this video, I'll derive the formula for the normal/Gaussian distribution. This argument is adapted from the work of the astronomer John Herschel in 1850 and the physicist James Clerk Maxwell in 1860.

References / Further Reading:

Erratum:
32:35 - the left-hand side should be positive, not negative.

[MomStillah]

Thank god someone cares to explain this equation that just floats around in the math realm with no explanation from teachers other than, "here!"

[HisMajesty]

This video is absolutely brilliant! I've always wanted to know how the normal distribution curve was derived, and your explanation was perfect! Thanks so much!!! Math is beautiful

[malignusvonbottershnike]

This video made me so happy, best 35 minutes of my day for certain, maybe my whole week. Cheers for this; if you see this comment 3 years later, you're a legend for taking the time to explain this so clearly.

[kerensahardesty]

This is fantastic! Everything is explained and paced so well; no other video online has derived the normal distribution so clearly as you have.

[youtubeyoutubian]

It’s beautiful to observe that the number of YouTube “likes” decrease as the video is of educational nature and not about useless make-up tutorials etc. This itself IS a proof that the number of curious people actually wanting to understand and therefore watching this helpful video completely are from the “other side” of the Gaussian Distribution. ;-) Thanks for the fantastic job!!

[nurulc]

What I love about this presentation, unlike some others I have seen, is that it does not skip over any steps. Each step is very clear and easy to follow. Wonderful job sir., thank you.

[marvinbcn]

This video is a brilliant illustration of Einstein's famous sentence: "Everything should be made as simple as possible, but not simpler". The derivation is beautiful, elegant and crystal clear. Thanks so much for sharing your knowledge!

[Jocularious]

Dude this was a great video, keep up the great work!! I love how at 3 am in the morning I am binging on your videos, goes to show that you have skill

[Barba_]

I'm so grateful that I found this absolute gem of YouTube, keep posting videos. You probably inspired thousands of people to be more interested in math/science

[viniciuscaina]

I'm from Brazil and i just found this class right here. I ask to my professor how to derivate this formula and she did not know it. This was one of the most impressive class that i ever saw, Thank you so much!!!!

[DawgKirkland]

Great vid. Cleared up much of the mystery of where normal distribution comes from. Have forgotten most of my calculus but was still able to follow along!

[Adam-Friended]

"We are more likely to find a dot near the bulls eye." You've oblivious never seen my wife play darts.

[HareSrinivasa]

I have always wondered about this formula. Your explanation is the most concise and understandable one even to a novice like me. Thanks a million.

[maxwall]

Wow, this was awesome. I'm reading E. T. Jayne's Probability Theory. In Chapter 7, he performs this derivation, but as is often the case, he assumes the reader is as fluent as he is with functional analysis. This video really helped me fill in the gaps. Can't wait to watch the rest of your videos. Thanks a bunch!

[andreweinhorn]

One of the best videos on the internet, love it!

[jacoboribilik]

I love this video, this derivation is spectacular, I love it when mathematics links various seemingly unrelated concepts with each other and yields this beauty. That being said, I also think Gauss' proof in the article you provide is far easier and more accesible to students like myself.

[spanishlanguageeducational]

Brilliant!!!! This is a MUST for any student of statistics. The way statistics are normally taught... many unanswered assumptions... which are answered by this video.

[rileyjeffries]

I am incredibly new to statistics and have never actually taken a course, but I have taken physics and engineering courses that apply physics; it's pretty neat to see that the dirac delta function is a limiting case of the normal distribution in which lambda approaches infinity, which I realized as soon as you showed how lambda transforms the shape of the graph. Very cool video!

[carlossp]

This is the most clear and complete derivation of the Normal distribution I've seen.
Thanks for sharing

[jackhuangly]

Fantastic explanation, step by step and does not assume the viewer knows any particular mathematical derivation. best video i've seen on the derivation of the normal distribution!