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Today is the day we finally talk about the normal distribution! The normal distribution is incredibly important in statistics because distributions of means are normally distributed even if populations aren't. We'll get into why this is so - due to the Central Limit Theorem - but it's useful because it allows us to make comparisons between different groups even if we don't know the underlying distribution of the population being studied.
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Text Comments (119)
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Ramanuja Rao (7 days ago)

This is counter intuitive.

Shawn Loewenberg (7 days ago)

😱

JAMES PURKS (8 days ago)

With all due respect, I must take issue with your comment that the mean, medium and mode are the same. To use your illustration of income, if you have a few people who make very high incomes, or very low incomes, ie CEO's, neurosurgeons, for example, or people who make minimum wage, and you have a larger number of people who earn less than very wages or very low wages, these will skew the mean highly in the first example, or lower in the second example. Needless to say, the standard deviation will be skewed as well. If you use the median, this is less influenced by extreme values and the standard deviation is less skewed as well. This statistic will give a much better average income. The mode, on the other hand, give a clumps of incomes that occur > 2 times. Just for the record, I am a sociologist who concentrated on methodology, lots of stats.

dev 17 (9 days ago)

algebra 2 regents?

Soojong Ha (11 days ago)

This series is becoming my second least favorite CC series (after sociology)😭

Joy JIN (12 days ago)

This’s exactly what I need

TheyCalledMeT (12 days ago)

i have to join other commentors, a very interesting video but a couple of concepts or arguments require more background intel or a little deeper dive into.
keep up the good work!

Liz C (13 days ago)

This is a lot of concepts in one video. They're related, but I feel like some of the topics should maybe have been split up to go into more depth.

Scartkabel (13 days ago)

This is like five weeks of content crammed into 11 minutes. Slow down plz.

DerUnbekannte (14 days ago)

who is the audience for this? someone like me that's already understood all of this at some point? it better not be for someone who's new to it, cause this is how you kill interest and make people feel dumb

Arnie Nelson (14 days ago)

Excellent video as always

Daniel Houck (14 days ago)

The sample distribution can't still be perfectly normal, though. To use your example, if you measure gross income instead of net income, it is never negative no matter how many samples you take. For a true normal distribution, for any arbitrarily extreme value, if you keep drawing samples you will almost surely get something at least that extreme; for gross incomes, you will never find a sample with a negative mean no matter how big it is.
I haven't done the math to see how exactly that fails to contradict the central limit theorem, but my guess is just that the standard deviation gets smaller faster than the sample distribution approaches normal, or something like that.

Alec Warden (14 days ago)

It's so beautiful :'D

drew sykes (15 days ago)

i would like to see a video describing the pareto distribution

Brian (15 days ago)

Repeating words heavily makes me dizzy when I'm not quite following.

Amanjeet Singh Bhatia (15 days ago)

Amazing

Terence Wang (15 days ago)

Thought Cafe always on point. I love their animations so much! Thank you for the upload :D

Isaac Liu (15 days ago)

Curious if there will be a video on order statistics, because that has always been the most fun part of that class for me!

Isaac Liu (15 days ago)

Lol how was there a video on z scores before normal distribution?

Osiris Malkovich (15 days ago)

Mean sample mean distribution sample mean distribution normal distribution normal mean normal distribution normal sample means standard deviation sample mean sample mean normal distribution sample blue jay sample mean mean standard deviation sample means...
That's what this video sounded like to me. I feel dumb.

Galakyllz (15 days ago)

Why didn't this episode explain more about the shape of this graph as the values change? That would make it clear to everyone.

markanixon77 (15 days ago)

I made it to the end!...🙃🧐🙃... need a lie down now! 🤯😂🙊🙈🙉

markanixon77 (15 days ago)

My brain is beginning to itch! 🙃😬😂🙊🙈🙉

markanixon77 (15 days ago)

Did anybody else pause the intro to read the fact bubbles?!?!.... gotta get all of that knowledge! Lol. 😂🙊🙉🙈

Mohammed Shafei (15 days ago)

I still don't understand what a sample meanS is in practical terms other than a pair of dice. Where do we in real life ONLY analyse for sample meanS not sample meaN? Makes no sense the way it was presented.

Mohammed Shafei (15 days ago)

Thank you, that makes it so much clearer.

Life Happens (15 days ago)

In real life we nearly always take one sample, which has a single mean. But theoretically, if we were to re-sample the same population ten times, we'd have ten samples, each with its own mean (hence "sample means", plural).
The Central Limit Theorem tells us these sample means are normally distributed around the true population mean; the standard deviation of this distribution of sample means is what we call the standard error. The standard error can be estimated from the variation in our sample, and estimating it gives us a measure of how certain we are that our sample mean represents the true mean. If the standard error is small relative to the sample mean, that tells us we've got a decent estimate. If the standard error is large, we might need a larger sample.

Yvan Roy (16 days ago)

I'll give you an upvote because you tried to use a French word "wallah"! :)

Michael n. (16 days ago)

What stops a population from having a nonstandard distribution?

Dammedman (16 days ago)

Whats everyone's favourite normally distributed graph? also post links

Erwin van Dooren (16 days ago)

I teach this stuff to 16 year olds and i think they could have been able to follow everything up to here, but why pick up the pace like this here? Its too much info with too few 'quirky' examples imho. There are lots of people who can do math, but very few that can make videos like you can. Remember what you're good at!

Morgan Larson (12 days ago)

Mohammed Shafei sample means I’m referring to the sampling distribution of means. Where you take the mean of a sample, plot that point and do that for all your samples, then you take the data from that distribution.

Mohammed Shafei (15 days ago)

Perhaps you can make me understand then what she means by "sample means" and why not sample mean? I use statistics in real life if not deeply and I never came across a plural sample means.

Aditya Nihal Kumar Singh (16 days ago)

Information overflow

michele bagaglio (16 days ago)

i find this woman quite attractive.

DeusExAstra (16 days ago)

Very confusing video. Was there even an explanation of what the "normal distribution" even is? The video starts off talking about it and why it's useful, but I kept waiting for what it actually is mathematically. Also, many other terms and ideas are thrown out with no explanation at all. I feel like instead of 11 minutes, this topic needed to be much longer, and deal with these topics a bit slower and more thoroughly.

Carmeops (16 days ago)

this episode is mean

erick contreras (16 days ago)

Just in case you ARE comparing individual kill scores....I had 17 kills as jungle Yi last night. Just sayin

Matt Whitby (16 days ago)

Nope. Didn't understand a word of that.

Ansh Gupta (16 days ago)

Could not understand a single thing. What is the point of crash course if it just like any other boring lecture. You keep introducing new concepts but fail to explain ongoing ones. CrashCourse Statistics will not do well like this

Ishita Bajpai (16 days ago)

The Normal Distribution......
shape of a Normal Distribution.......
follows a Normal Distribution.....
in a Normal Distribution......
Normal Distribution into.....
data is Normally Distributed .....
finally Normal Distribution............!

Mohammed Shafei (15 days ago)

The sample means will fall near the sample means and therefore the sample means will be mean to to poor sample means

Daniel Astillero (16 days ago)

I remember Derek's video about regression to the mean. :)

Hemakumar Gantepalli (16 days ago)

awesome

Hugo Iwata (16 days ago)

Catan!

clearlybehind (16 days ago)

My head is spinning so I am jumping to the video about centripetal force. Repeat words are a no no especially at the pace that crash course videos go.

Mohammed Shafei (15 days ago)

Sample means are sample means, and this is why the sample means are the used in the sample means... blah blah blah

Aryan Divyanshu (16 days ago)

You guys have worst views and subscribers ratio.

Raymond K Petry (11 days ago)

*_...so in practice, if 100 voters cast 'randomly' (e.g. uninformed) they'll pass any Bill 50% of the time without meaning-to (i.e. uninformed)—and,—to reduce that to 5%, requires a Vote minimum of 58—but also, we can estimate that any Vote within the ±7 of their mean 50, is indistinguishable from 'random' (the 'drunk-walk' vote however-much they're informed)..._*
*_...so if statistics is worth anything it is that it tells us there is no game won by a majority..._*
*_...if athletes are drug-tested to prove they're not-'drunk-walking', Senators should be too..._*

TheSaxRunner05 (16 days ago)

I think it's far more likely that politician simply vote along party lines rather than randomly. The deviation amount from the party line would be very small.

Kelly Kurt (16 days ago)

Normal? I have heard it described as taking both ends of a spectrum, and dividing it accordingly. Subjective bias often skews perfectly good data.

Максим Ряховский (16 days ago)

Gauss for the win!

Scerttle (16 days ago)

A lot of concepts in this video that probably could have used some more room to breathe...

R3Testa (13 days ago)

Statistics is the subject they chose. This one should have said more about standard error, and how 15 is close enough to 16 that it doesn't matter.
15 is way too far from 16 in this example, BTW.

Meris (14 days ago)

Statistics is not a simple subject. I have watched entire classrooms grind to a halt over tiny details for half an hour. CC does not have the luxury of being able to address every issue every viewer has with it. This series was always going to suffer from such a feeling.

Scerttle (15 days ago)

By "more room to breathe" I don't mean more pauses. I mean there is too much stuff crammed in and glossed over.

Scerttle (15 days ago)

Nah I didn't need to pause. I've just heard better explanations elsewhere. The point of Crash Course (as far as I was aware) was to give you an overview of a concept so you can grasp the important foundational concepts. This video doesn't really do that. It's the weakest video in this series.

Paulina Salcedo (15 days ago)

just pause whenever you need to! .. I know that's what I did. xD

Adaginy (16 days ago)

"Normal" "sample" "distribution" and "mean" are all getting to that point where the word feels like it isn't real anymore.

Mohammed Shafei (15 days ago)

I wish it were "sample mean", that's understandable. But sample means, as in the sample of many means of many pairs of dice... which applies to nothing else but dice... It's silly and meaningless.

Aldo Velez (16 days ago)

PARETO!!!

The Savant (16 days ago)

I was lost at 0sec

Sonia Rodrigues (16 days ago)

a little confused explanation, don t look crash course at all

absolute moist (16 days ago)

What the fuck😂😂😂

Mirko Di Benedetto (16 days ago)

That "Voilà" was wonderful

bowie brewster (16 days ago)

I kinda wanted the general formula for a normal distribution and its explanation. But hey

Aadi Bhavsar (16 days ago)

...that's just a theory. A statistical theory.

Asad Bilal (16 days ago)

Wtf she keep talking n talking n talking.

Patrick Ormerod (16 days ago)

please next the pareto distribution and Kumaraswamy distributions

Zhu Bajie (16 days ago)

Still, I have a problem with using standard deviation where the standard deviation is large compared to the mean and you get a large probability that some quantities that cannot be less than zero are. Example: if have a mean height of 2 m and a standard deviation of 5 m then there is a significant portion of the curve is less than zero meters in height. What is negative height? It is nonsense.

hmata3 (13 days ago)

fatsquirrel75 Here is where u need to assess your data to see if you have an outlier. A series of 1's and an 11 suggests something the 11 may be an accident and may be thrown out with consensus.

pw (13 days ago)

fatsquirrel75 "It wouldn't be possible to have a mean near zero and large variance and still be normally distributed." this is simply not true.

fatsquirrel75 (14 days ago)

pw where did I say otherwise? I said they will typically normalize the data. This is the process by which you shift and shape the data for easier comparison. Only way for length to end up in the negative would be to shift it there.
Picture a sample 1,1,1,1,1,1,1,1,1,11 mean =2, Variance 10.
Does that look very normal to you? It wouldn't be possible to have a mean near zero and large variance and still be normally distributed.
If it isn't normal you don't have to worry about the existence of a symmetrical tail on the negative side.

DerUnbekannte (14 days ago)

as fatsquirrel75 said, your distribution isn't normal. problem solved

hmata3 (15 days ago)

Just like when solving for polynomial roots, you plug in the answer to see if it makes sense.
Sometimes you get a negative polynomial root that would not be a the answer in real life. So you just throw it out.
Same with your example. If negative lengths don't make sense, throw them out.

Rand Huso (16 days ago)

Why should a hiring manager hire single mothers? Single mothers don't spend enough time at work, and are a bad investment.

Len Arends (16 days ago)

For a cordial society, either have a requirement to hire indiscriminately, or extract taxes from the owners of capital to support universal basic services. Leaving everyone to their own devices trends toward feudalism, balanced on a revolutionary knife edge.

Teresa White (16 days ago)

Great explanation. Thanks!

Oliver Bayley (16 days ago)

Their fate will be in their own hands as they decide whether to share or to shaft.

Andy Brice (16 days ago)

Crash Course Game Theory?

Lit Crit (16 days ago)

Not at all clear -- especially for a Crash Course video.

Isaac Liu (15 days ago)

Yes, but you can rewatch it again and again and at your own pace which is still better than a lecture lol

Mohammed Shafei (15 days ago)

All I could hear is "sample means, sample means." Why sample means not sample mean?

tygonmaster (15 days ago)

They seem to be doing alright with examples and visual cues to make the information a bit easier to relate to. The issue I am having is the video is just drowning in jargon to the point of sickness. Take a drink every time "mean", "median", or "normal" are said without much levity. Except do not do that as you will be dead by 2 minutes in from alcohol poisoning. It is a fact that people just filter jargon, which is why much of traditional "boot camp" style lectures like this simply do not work. People chime in and out naturally in listening and when you have something this information dense using terminology that people are naturally less likely to listen too...well you get drop off and people stop caring. CC is about making people care about things and introducing them to subjects unfamiliar to them in an entertaining well. Nothing here is particularly entertaining or interesting. It is just being talked at. Ms. Hill is obviously knowledgeable and seems to be doing her best with the script, but holy hell...the script needed to be re-done with the audience in mind.

Life Happens (16 days ago)

The Statistics series has been suffering from confusing writing in many places, I think. Concepts are introduced in obtuse ways, or applied without explanation; complex examples are shown first and simpler examples later; mathematical notation is avoided altogether, and when it *is* shown it's barely explained.
There's a concept hiding just behind the examples and concepts discussed in this video that never *quite* comes across: that the normal distribution is the result of the sum of many small contributing processes. Your height is the result of many small genetic and environmental factors; these factors might have all sorts of distributions, but sum them all together, and you get a trait that's normally distributed. (This is true for many or most traits you can measure in nature, which is why the distribution is called "normal".) You can easily simulate this by imagining, for example, each gene that contributes to height as a coin toss (where heads is an allele that makes you taller, and tails is an allele that doesn't). Have each person in a large group toss a coin ten times and sum their tosses, and you'll end up with a fairly normal distribution of sums (with a mean of 5, since that's the most likely result of ten coin tosses). This is what the Central Limit Theorem says.
We see this again in the die rolls: the die results have a uniform distribution, but once you start adding dice together you approximate a normal distribution (the more dice, the more normal). But for some reason the video foregoes this simpler perspective and dives straight into distributions of sample means, which is a higher level of abstraction that makes *no* sense to a viewer who hasn't already grasped what a normal distribution is and why it happens. It's frustrating.
They also mistakenly said, again, that the standard deviation represents the "average distance from the mean". It doesn't. :(

Kyle Ward (16 days ago)

I was wondering if I missed an episode or three. I feel like we need an episode 18.5.

Owen Grimm (16 days ago)

13th@

Von Schmidemayer (16 days ago)

UPDATE THAT WEAKARSE LEGACY CLIENT LMFAO

Your gf is Ricci flat lol (16 days ago)

Was always interested in why the gauß function was so special

Farhana Yeasmin (16 days ago)

Your videos are so good ..i love it .....

Andrew Johnston (16 days ago)

If only I had the patience to roll a die 20 times!

Andy Brice (16 days ago)

If you're in a hurry, just roll 20 dice one time.

Mika Waldmann (16 days ago)

Is this college or high school level?

Mika Waldmann (14 days ago)

Moritz Ernst Jacob ah du bist von Deutschland. Dann ist es ja genau wie bei mir

Mika Waldmann (14 days ago)

Moritz Ernst Jacob theres a 13th grade in US?

Moritz Ernst Jacob (16 days ago)

Highschool 12th and 13th grade. If you have a higher maths course in highschool you might even integrate the polar coordinate gauss curve. At least we did that...

Graham Milligan (17 days ago)

Notification squads

hassan nada (17 days ago)

second

Raheeb Rahman (17 days ago)

First

Numan haris (17 days ago)

First Comment!!!

Lord Ankylos (17 days ago)

first

Chuey V (17 days ago)

1st

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