Introduction to the normal distribution | Probability and Statistics | Khan Academy

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Exploring the normal distribution
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[@outhmane6469]

Guys I found this amazing explanation in another video's comment section, I hope it helps! "It took me a while to understand this and had to rewatch this video a few times, but essentially there are ground rules for calculating standard deviation. Standard deviation is saying something like "Math test scores averaged 92 'give or take' 3 points". So 92=mean, deviation=3 (the give or take number), so 92-3 and 92+3 means that 68.26% of all students score fell within 89-95 (92-3=89 or 92+3=95). this is "1 standard deviation". This video is using the ground rules for calculating standard deviation. Since standard deviation is a mathematical formula, the existing ground rules dictate that you have a 68.26% of all test scores to be within 1 standard deviation (3 points in my example). So for 2 or 3 standard deviations, you use the same formula above, but instead of 3, use 6 (2 stDevs) or 9 (3 stDevs) 1 stDev = 68.26% probability (give or take 3 points) 2 stDev = 95.44% probability (give or take 6 points) 3 stDev = 99.7% probability (give or take 9 points) NOTE: The 68.26%, 95.44%, 99.7% are existing rules used for normal distribution and apply to any example."

[@azwell45]

so 1 std shows us how 68.26% of people deviated from the mean? and when std increases, i mean the points we add and subtract, we see more probability. ty

[@DendrocnideMoroides]

what video was it (at least which channel)

[@jkjoker777]

@@azwell45 not quite, the 68% is the probability of selecting one entity from a sample and that entity falling within 1 sd of the mean.

[@adnanhossain3152]

i dropped out of engineering and joined business school just so i dont ever have to see that GOD DAMN INTEGRATION SIGN... but... we meet again. my old foe!

[@Hertha_Ambambi]

Lol,,, u have to get used to it

[@blownspeakersss]

Real business is all about mathematics and statistics, especially finance!

[@GabrielleDaway]

I honestly feel your pain

[@leofelippe100]

I know what you are felling, failed calc 2 twice myself

[@anurag7058]

THAT INTERGRAL SIGN LITERALLY RUNS THE UNIVERSE SO NO SURPRISE THERE

[@MidnightFreefall]

I just wanted to say that I'm taking statistics while studying to be a veterinarian. The course at uni is very brief, nothing goes into too much detail on why certain things work the way they do. This playlist is a godsend. I love maths, and I love how passionate you are about these topics. Thank you for helping me study!

[@druca]

I find it very effective that you *repeat* the same ideas over and over again! It reinforces them.

[@evalsoftserver]

ITS NOT SO HARD , THE MATHAMATICINS USE WORD TO MAKE IT HARD! DONT PAY TOO MUCH ON THE EXACT WORD THE USE IT WILL JUST CONFUSE YOU MORE. JUST REMEMBER THING TO EQUAL THINGS LIKE HOW YOU MIND SEE IT EX. SET TO SETS , POINT TO POINTS IF ITS GEOMETIC IN FORM, NUMBER TO NUMBER , IF IT ARITHMETIC, OPERATION TO OPERATON EX. NULTIPLICATION CAN BE MATCHED TO ADDITION BY A EXTRA STEP! FOR EXAMPLE HE KEEPS SAYING " DISTRIBUTION" THATS JUST LIKE SAYING LIKE A FORM , OR SAME FORMULA ! '' SO IN YOUR MIND SUBSTIUTE DISTRIBUTION FOR LIKE FORM , NUMBER OR SHAPE IT WILL MAKE UNDERSTANDING ALL THIS A LOT EAZIER !

[@evalsoftserver]

HE IS REALLY VERY GOOD , BUT TO MOST PEOPLE ITS OVER THEIR HEADS, HE ASSUMES MOST VIEWERS CAN FOLLOW WHAT HE IS SAYING HE SHOULD HAVE STARTED THE VIDEO WITH A DISCLAIMER STATING SOME FOREKNOWLEDGE IS REQUIRED ABOUT DISTRIBUTION , AREAS UNDER CURVES , TRANSFORMATION THEORY LIKE MATRIX LINEAR ALGEBRA . AND INTERGRAL AND DERIVATIVE CALULUS , WITH EXTRA FOCUS ON MULTI-VARIATE CALCULUS , AND COMBINATORIAL MATH LIKE MIXING ALGEBRA , TRIG, CALCULUS TO SEE UNIQUE SOLUTIONS TO PROBLEMS !

[@slowsatsuma3214]

KoDi-CIoner-TV u high bro ?

[@alexting827]

@@slowsatsuma3214 Lol

[@basschipper7258]

Ten years later, still relevant. Sal you are a gift from God

[@khanacademy]

the normal distribution is a class of distributions that have the same basic shape but different means and std deviations. Human height is probably close to a normal distribution with the mean around 5'6 (somewhere in between men and women average).

[@alexkuligowski4092]

The connection between the normal distribution and the cumulative distribution function just clicked! Thank you! I see the normal distribution as the distribution to explain most natural phenomenon if we use the law of large numbers to arrive at the central limit theorem. The cumulative distribution function is nothing more than the sum of the probability from negative infinity to a given X of interest. After seeing how this comes together the way we use the cdf to obtain an interval of probability along a continuous distribution is super slick. Last note: I've always wondered why we have an index of pages for the normal distribution. After the small note on how difficult the integral would be of the equation for a normal curve it makes so much more sense why these tables in all statistics books I've ever looked in the back of. Cheers!

[@hussainrangwala1078]

I wish I had watched this video earlier during my engineering.. Mr. Sal Khan take a bow!! this is how you teach something to someone..

[@donnajoe4442]

The only slight mistake was at 15:55. You were supposed to enter 10 and not 100 i.e e^-1/2(x+5/10)^2. This was a good explanation.

[@tseringangle21]

Thank you! It made so much more sense now. That graph representation and manipulating equation made it more easy to understand reason behind all.

[@randompotato258]

there is a small fault when you wrote down the fonction to integrate where the mean is -5 and the standard deviation is 10 you wrote ((x+5)/100)^2 the right expression would be ((x+5)/10)^2 or (x+5)^2/100) otherwise you have (x+5)^2/(the standard deviation)^4

[@spacetime_wanderer]

Yes. You are right. I had to scroll so much to find this comment. Wonder why it's not been noticed by so many.

[@wedeldylan]

Yep. Big boy made a mistake

[@McNoat]

I very much enjoyed the transformation of the formula into different "forms". I think this helps to reduce the fear that comes (for me) from looking at so many symbols and makes it a bit more palpable.

[@lobsterofficial1967]

11 years later and you made me understand what the normal distribution is all about for my Biomed Technology class thank you Khan

[@plaxen1]

Loved it! thank you..... different topics really helped to connect the dots!

[@JawadKhan1]

Thanks for explaining in detail.

[@TheMorttt]

12 minutes in and i'm more confused than I was 12 minutes earlier

[@Dingus7696]

skip this one video it's not that necessary, the rest of them are probably what you need

[@evalsoftserver]

ITS NOT SO HARD , THE MATHAMATICINS USE WORD TO MAKE IT HARD! DONT PAY TOO MUCH ON THE EXACT WORD THE USE IT WILL JUST CONFUSE YOU MORE. JUST REMEMBER THING TO EQUAL THINGS LIKE HOW YOU MIND SEE IT EX. SET TO SETS , POINT TO POINTS IF ITS GEOMETIC IN FORM, NUMBER TO NUMBER , IF IT ARITHMETIC, OPERATION TO OPERATON EX. NULTIPLICATION CAN BE MATCHED TO ADDITION BY A EXTRA STEP! FOR EXAMPLE HE KEEPS SAYING " DISTRIBUTION" THATS JUST LIKE SAYING LIKE A FORM , OR SAME FORMULA ! '' SO IN YOUR MIND SUBSTIUTE DISTRIBUTION FOR LIKE FORM , NUMBER OR SHAPE IT WILL MAKE UNDERSTANDING ALL THIS A LOT EAZIER !

[@KatoOnTheTrack1]

If you can't learn from Khan it's all bad for you

[@TiffTheTyrant]

moe.so.jiggy what is the probability of understanding after 12 minutes

[@shannonmiller1910]

Me too. Thanks. I just laughed out loud.

[@edithmambiri7089]

Thanks for clearing my confusion, well explained.

[@layallooloot9735]

Can't believe this video is 10 years ago 😭😂💔

[@akanshaw]

Glad I decided to watch this video before starting to study for test !! Helped a lot !

[@pradeepknal]

thanks a ton for all the teachers and staff team at Khan Academy!

[@Zartas17]

Very informative and well presented. Thanks Khan Academy!

[@JoshMolina]

Thanks to Sal and staff at Khan, I'm able to throw down with the quants at the bank. Thanks guys!!!

[@Yo-yx8wo]

great explanation, is always interesting try to see where the math formulas come from

[@mohit01985]

great work! thank you.

[@AmadeoArguelles]

Excellent, thank you for the explanation and the information. I referred the file to my students to make some tests related with Normal Distribution

[@sasamuraki]

wow, that part of linking normal distri to 'nothing is impossible, but the probability matters' is awesome.. deep...

[@Dermie]

Thanks. Well explained!

[@sajn0001]

Thank you very much Mr Khan. The best tutorial ever.

[@NGChannel97]

who else is watching these videos a day before the exam in 2019?

[@creaturesoul4469]

I’m doing this the period before the exam

[@theepicguy6575]

Yep that's me...tmrw exam

[@s.nithinutham5972]

Yo am doing this in 2020

[@_boris]

What about the day of the exam itself? 😂

[@matthew_natividad]

cramming for my 2020 quarantine final

[@achillesarmstrong9639]

the best video in this topic on youtube site

[@Jonny-uu7wf]

I find these videos really helpful. Got a statistics exam tomorrow, been revising off these for the past few weeks. This video is not very good though. Really confusing.

[@scienceblossom6197]

Really nice explanation. Thanks.

[@matthewa6881]

this is deep. How are pi and e and chance related in this way! My mind is blown right now. I discovered the norm dist. in uni in like 2007 but only now have a greater appreciation for it. Just number plugged in the past. This mathematical model is truly beautiful to me.

[@ravanabrahmarakshas4263]

very good. very clear. excellent. thanks.

[@muzic29]

thanks so much for this.

[@bambangeko221]

Excellent explanation of Normal distribution. Thank you very much.

[@Joe.0oo]

yay for watching in December of 2018... video is still relevant/helpful nearly a decade later lol I just wish my professor would let me use excel on my exam tomorrow hahaha

[@sumairakhan9608]

Thanks, you always save my life.

[@arunprakash4368]

lesser the time more the knowledge ,its very easy to understand ,,

[@artrocks8392]

+Arun Prakash Unless one requires more time to acquire the knowledge. Not all intellects function in the same way... thankfully.

[@harveyspecter3361]

I think for statistics in general we should make a big distinction between statistics for business, social sciences, medicine and statistics/ probability for engineering, mathematics, physics and econometrics. Clearly, many students here are the former and do not need to know most of what is being talked about here. (coming from a student with a bachelor in both business and econometrics)

[@rodrigocbv]

great video!

[@dckc3333]

very helpfull video!!

[@Timmytimmy123123]

15:54 There is a mistake here. Sal wasn't supposed to square the entire ((x+5)/100), he was just supposed to square (x+5) since the deviation was already squared from 10 to 100.

[@ankurm26]

That's right

[@Person514cs]

That's very interesting

[@AdemsPerspective]

You are the saver Khan! Saving my ass every time! Thanks dude

[@mmaking8664]

+Abdurrahman Köksal every time man this guy is the best

[@rp3291]

Thank you Sir!

[@MrAwesomeAsian22]

@dobromir2102 hello Dobromir, it is 2 years since you have seen this video, but i hope you still have your positive attitude for Statistics! anyways, to find the video you are looking for press Ctrl and f, and type in the keyword you wish to search for. In your case, i used standard deviation as my key words, and it brought me to the statistics section. In the section there is an intro to normal distribution. under that, you have z-score. Hope i helped! reply if you have any questions. -Wasfi

[@masoodghasemzadeh]

Excellent tutorial. Simple and clear. Just as feedback : It's gonna be perfect if you speak a bit slowly, specially for non-english viewers.

[@vdmur7952]

Thank you!

[@talentokie]

I wish i had youtube when I was in school...wud have learned a lot of things better.

[@norwayte]

You have the gift.

[@chrisaltec2992]

Lots of confused people on here. If you jump into this video without watching the videos leading up to this, and have a weak grasp of maths, of course you're not going to understand. I didn't understand every single part, but know why (because I haven't even learned integration in calculus and didn't watch all the preceding videos on the topic), so I accept that as a limitation and took away what I could. Don't skip to chapter 10 without reading the first 9 chapters and expect to know what's going on!

[@nv16123]

Good point :D

[@dissapoment5547]

Wow this is deep

[@indianarchangel]

The video was very good but please try to talk in a more coherent manner. It makes it hard to follow when you switch between concepts every 10 seconds.

[@kevinsurposa363]

+indianarchangel I like it that way :)

[@evalsoftserver]

ITS NOT SO HARD , THE MATHAMATICINS USE WORD TO MAKE IT HARD! DONT PAY TOO MUCH ON THE EXACT WORD THE USE IT WILL JUST CONFUSE YOU MORE. JUST REMEMBER THING TO EQUAL THINGS LIKE HOW YOU MIND SEE IT EX. SET TO SETS , POINT TO POINTS IF ITS GEOMETIC IN FORM, NUMBER TO NUMBER , IF IT ARITHMETIC, OPERATION TO OPERATON EX. NULTIPLICATION CAN BE MATCHED TO ADDITION BY A EXTRA STEP! FOR EXAMPLE HE KEEPS SAYING " DISTRIBUTION" THATS JUST LIKE SAYING LIKE A FORM , OR SAME FORMULA ! '' SO IN YOUR MIND SUBSTIUTE DISTRIBUTION FOR LIKE FORM , NUMBER OR SHAPE IT WILL MAKE UNDERSTANDING ALL THIS A LOT EAZIER !

[@archakasandeepiniacharya7106]

Terrific

[@tiff2794]

Hello! I was wondering what the y axes would be labelled on the normal distribution graphs. I understand the x axes would be the number of standard deviations, but i dont know what the y axis would be

[@123zozey]

+Tiffany K Hi Tiffany, First of all, the X-axis isn't really the number of standard deviations. It is whatever your variable is. (The standard deviations are just shown on the x-axis.) The Y axis would be basically whatever the other variable is that you are measuring. So if you were measuring how many finger taps a person can do in 20 seconds, then the time (in seconds) would be on the X-axis, and the finger tapping would be on the Y-axis. Hope that made sense! Zoha

[@enok82]

+Zoha Shoaib +Tiffany K Actually i would suspect that the X axis be the amount of taps and Y axis how many times that amount of taps acheived i.e. after 1000 trials (and sore finger tips) 120 of those trials ended up with 97 taps. Then a point or bar would be placed at X=97 and Y=120.

[@actuallyactuary2787]

+Tiffany K X-axis is the variable, whatever you wish to find probability for. Say you wanna find probability of range 5-15, then 5 & 15 would be on X-axis Y-axis represents the probability density function.

[@hung89341]

is it also means y-axis = height of the normal distribution?

[@vineethmenon2260]

The x axis will be variable that you are considering to sample and measure whereas the y axis is the probability distribtion for that particulat variable which is detected from the standard formula.

[@killjoy95590]

"for those of Y'all" heeeeeeeeeee haaaaaaaaaaa! Giddy up cowboy! 2:12

[@IsaacC20]

1:35-2:00: In a probability distribution of a continous variable, shown by the normal distribution curve, is it impossible, not correct, or simply not the job of the normal distribution curve to approximate the probability of obtaining a discrete integer value (i.e., a number on the x-axis)?

[@george4746]

Some tutorials show frequency on the Y-axis while some show frequency. Which one is correct? And how does the area come into the scene? I am all confused.

[@NaserNazary]

Also I find it very helpful thanks khanacademy. where can I fnd more videos?

[@nurulshazwani6807]

I watch these videos an hour before my exam.

[@aashijain6858]

Can you make a video explaining applications of lognormal distribution.

[@alexandardimitrov5064]

I am confused what is the difference and connection between Normal Distribution and the Probability Density function then?

[@victor_peral]

Probability density function shows the graphical distribution of outcomes , you can have probability density functions of both the binomial and normal distributions.

[@BeGunNer]

Is the Cumulative Distribution Function a Logistic Function with carrying capacity 100% or does it just look like one? I am not sure, because it seems to copy the typical shape, but I can conjure up no substantiation as for why it should be that way.

[@victor_peral]

It is preferable because of the difficulty of solving the integrals and exponents by the normal formulae , so we go by the CDF of negative infinity of a number to that number and the difference of both probabilities give a workable or logical probability

[@TheSupernastee]

ty

[@Poggle566]

I see this was 4 years ago but I feel compelled to say that the negative values are a deviation from the mean and not a negative height.

[@selenaburke2956]

What happened to the 2 around 8:50 when you rewrote the equation?

[@panos4386]

what do you mean i cant use the diagram for a simple number...for example for x=5 ...p(5)= is on y axis following the diagram...it gives you the relative frequency of your measured samples.....why i need to take an area from 4.9 to 5.1 acording to your example?

[@Tehcarp]

If you did statistics in high school, this might come up. Don't worry though, if you spend more than half an hour on it, or work backwards in the maths you need to understand (try an elementary stats textbook from your local library) this will quickly come under your power to manipulate. Watch some more videos!

[@witheredways4198]

using the free program u have made i can add all the info in i need but this doesnt take intocount the z chart i have been given so i am not sure if i am missing something

[@Jakers2009]

Sal... super video by thw way... Dont mean to get involved but maybe this guy should do some research into the example you said about your atoms falling through a chair if he wants an extreme example... What I am interested in is you continuing this idea into the central limit theorem and also some of the financial examples you said you may do.... I am starting to see how this could be used to value options or the intrisic value of an financial instrument....

[@TheFunkiGuy]

More Confused? Yes More Research? Yes

[@THE16THPHANTOM]

this is still super difficult and its the most important important one.

[@anhaonguyen912]

If somebody find he's speaking too fast. Just adjust the playback speed to 0.75 (or whatever that suit you), you would find it more easier to follow up.

[@Kieranlovesunicorns]

dude kinda sounds like Paul Scheer....also kinda love his voice #asmr

[@AdrianWildgoose]

I have a question that i need help with. The question goes...."The scores on a statistics examination are normally distributed with a mean of 60 and a standard deviation of 10. Calculate the following probabilities: A. The lielihood that scores will be less than 50. My question is for the z- score ...would it be (z--1) thus P(z

[@victor_peral]

Since the question is on probability and not the z score , calculate for the probability between -0.1 and 50 then subtract the probability of 50 from the probability of -0.1 to get the answer

[@odaneswaby8714]

at 15:50 you should divide by the standard deviation and not the variance....... And Good video by the way

[@bpirat]

would have been so good if you just kept focus on the topic and did not deviate constantly

[@thermie]

*standardly

[@suuup4711]

variably*

[@evalsoftserver]

ITS NOT SO HARD , THE MATHAMATICINS USE WORD TO MAKE IT HARD! DONT PAY TOO MUCH ON THE EXACT WORD THE USE IT WILL JUST CONFUSE YOU MORE. JUST REMEMBER THING TO EQUAL THINGS LIKE HOW YOU MIND SEE IT EX. SET TO SETS , POINT TO POINTS IF ITS GEOMETIC IN FORM, NUMBER TO NUMBER , IF IT ARITHMETIC, OPERATION TO OPERATON EX. MULTIPLICATION CAN BE MATCHED TO ADDITION BY A EXTRA STEP! FOR EXAMPLE HE KEEPS SAYING " DISTRIBUTION" THATS JUST LIKE SAYING LIKE A FORM , OR SAME FORMULA ! '' SO IN YOUR MIND SUBSTIUTE DISTRIBUTION FOR LIKE FORM , NUMBER OR SHAPE IT WILL MAKE UNDERSTANDING ALL THIS A LOT EAZIER !

[@jhonatanhernandez3568]

you could say, his deviation was pretty... standard

[@harveyspecter3361]

He is not doing that, it is just that he is explaining stuff you don't need to know when you do a bachelor without advanced mathematics.

[@drewmetra]

This is a great video, y'all haters weren't paying attention.

[@debendragurung3033]

I have been here a 100 times but still find myself alienated. I went all the way to find what gaussian function is, and finally come back to find it is actually e^-(sqrt x) and not e^-square x, that I should look for. Also, Does P(x) here means (mean + sigma.x)......

[@theproofessayist8441]

The Gaussian Integral which actually requires polar coordinate change of variables double integral integration to solve is integral(e^(-x^2)). Double integrals are learned in Calculus 3/Multivariable Calculus or Vector Calculus.

[@indiansoftwareengineer4899]

Why this introduction part is after exel exercise in playlist order. This must be put before exel video. Videos in playlist are not sorted according to time. Introduction to normal distribution would have come before normal distribution exel video.

[@MissNORY22]

when you go to his channel you will find playlists based by subjects so go to statistic playlist

[@mandylu6558]

Can't believe I'm learning this at 8th grade

[@cloud-tutorials]

If many process in world approximate to Normal Distribution, how do before hand we know mean and variance?

[@TheSunshineRequiem]

okay, for those ppl who are confused with this, you need to understand about pdf and cdf first, the normal distribution is just a bell curve pdf, it happens to be very common in our universe.

[@TheSunshineRequiem]

standard normal dist. is just normal dist. where the mean is set to 0, and variance set to 1.

[@vincenzoimparato7077]

So, why it's not possible to calculate a precise value x? Analitically I have only to put the value x in the formula, but geometrically I can see whay value y take the curve for the value x. Why I'm oblied to take an interval?

[@saalank]

You can but any exact value you may pick would have the probability of zero. That is the difference between discrete and continuous functions. In a discrete function there is (for example) 2 and only 2, no 2.0000000000000001 or 2.0000000000000000000000000000001. But those silly numbers do exist in a continuous function. There are in fact more than infinity(aleph one) of them between any two intervals.

[@watsonholmes1750]

I really dont understand why a med student needs to know this

[@michaelruderman6544]

So you can make better decisions.

[@rakeshk6799]

We are making use of Cumulative Distribution Function so that we don't have to evaluate the integral of p(x), but how are we finding CDF? Isn't that the same integral with limits changed?

[@victor_peral]

He didn’t dig into finding CDF analytically because there is an excel function for it

[@ramzi45]

I have a question: the P(x) reperesents the probability density/probability and P(x) goes from 0 -> 1 and we have a normal distribution/gaussian distribution where every x have a p(x) The question is why do we have in the p(x) axis values higher than one??? (I even use the G.D formuala and get values greater than 1) Thank you

[@victor_peral]

It doesn’t go from 0 to -1 it goes from -infinity to any value of x

[@saurabhmahajan4459]

So a normal distribution is the one whose pdf is symmetric about its mean ! Is that a correct simplification of the definition?

[@drasifkhan2575]

❤️

[@DivinityFaith]

I believe you are somewhere to the right of the normal distribution graph, probably somewhere near 6.9 billion, if we were to make such a graph for "helpful people".

[@bambinarm4071]

...because not EVERYONE understands one way, hence the ENTIRE point of the Khan approach. You're asking him to do what classrooms do. If that's the case, then why watch a redo of what you experienced in class -__-

[@bnglr]

what is P(x) in the excel?

[@YouPokerAcademy]

Salman, I'm working through some math for my own channel and I was wondering if you could help me understand this: If I am 1% to win, and I do 50 trials, am i 50% to win one of them? It seems like a normal distribution question so I'm posting here.

[@adenpower249]

Yes but it's not a normal distribution question. The probability * the number of trials is the number of trials is the number at which you can expect to succeed.

[@victor_peral]

You have 50 percent of winning all the trials not one of the trials . For I trial you will choose 1 out of the 50 and do a binomial for it

[@liamhoward2208]

What does the height (y-variable) of the probability density function measure?

[@ghost_2]

y axis shows probability

[@victor_peral]

It measures probability of a certain outcome of x axis

[@liamhoward2208]

I thought the y-axis showed likelihood which is different from probability. Also, isn’t the probability of one specific value equal to zero? That’s why for continuous distributions like the normal we have to use a range.

[@victor_peral]

@@liamhoward2208 likelihood of occurrence is same as probability .the probability of an exact occurrence is zero only for a continuous outcomes it isn’t zero when an outcome is discreet like the case of a dice or fair coin . Most natural events don’t have predictable outcomes that’s why a range around the expected outcomes is solved for

[@liamhoward2208]

@@victor_peral I am speaking in terms of continuous data. I just always thought that the height of a normal distribution technically would be the probability of a single x-value occurring. However, that idea doesn’t line up with having to find probabilities with ranges. For example, if I plug a single x-value into the normal distribution formula I will get the height of the curve which is a single point on a continuous line. There would be no area under that curve because a 2 -dimensional line doesn’t have area. That’s where I’m confused. Probability is area. Therefore likelihood and probability cannot be the same

[@sherridonlyons-verse172]

I usually love your examples and look for them, but you thoroughly confused me on this one.

[@alphaena1]

with what program do you work that allows you to "write" into excel?

[@mohsinali236]

Give a normal distribution which Mean is 40 SD is 6 Find the X assume a value 1 below 42 2 above 27 3 between 42 and 51 Plz tell me

[@dauletbukanov9703]

How is it that the standard deviation is not the same as the average distance from the mean? my brain is just not able to accept it