The Sampling Distribution of the Sample Mean

I discuss the sampling distribution of the sample mean, and work through an example of a probability calculation. (I only briefly mention the central limit theorem here, but discuss it in more detail in another video).
The mean and standard deviation of the amount of protein in a quarter pound patty of lean beef was found in the USDA nutrient database at:
ndb.nal.usda.gov/ndb/foods/sho...
For those using R, here is the R code to find the values in the examples:
The probability a randomly selected patty has at least 23.0 grams of protein (mu = 21.4, sigma = 1.9):
1-pnorm(23.0,21.4,1.9)
[1] 0.1998645
or, if we standardize:
1-pnorm((23.0-21.4)/1.9)
[1] 0.1998645
The probability that the mean of 4 randomly selected patties is at least 23.0 grams of protein (sampling from a normal distribution with mu = 21.4, sigma = 1.9):
1-pnorm(23.0,21.4,1.9/sqrt(4))
[1] 0.04607049
or, if we standardize:
1-pnorm((23.0-21.4)/(1.9/sqrt(4)))
[1] 0.04607049

Пікірлер: 123

  • @emem2060
    @emem20609 жыл бұрын

    thank god for these videos. I learned in 12 min what I didnt understand in 2 days worth of class

  • @jbstatistics
    @jbstatistics10 жыл бұрын

    Thanks, I'm glad you like my videos. Thanks for the suggestions. Gamma, Weibull, and lognormal are definitely somewhere on the horizon, but it will be a little while before I can get to them. (I've got a number of other videos lined up right now.) Cheers.

  • @dasgomezkanal
    @dasgomezkanal8 жыл бұрын

    My friend, I want to thank you a lot! for posting this videos. You are an excellent professor and communicator! My statistics profesor is the worse... I can't understand anything at all in class.. I don't even know why it is so expensive to pay for her class 4,500 a semester when all she does is talking and drawing numbers and we are all like 0_o ... Anyway, I really just wanted to thank you a lot for your help. This is my last semester as an undergrad and without these videos I don't think I could pass this subject and graduate. Thank you Thank you Thank you!

  • @jbstatistics

    @jbstatistics

    8 жыл бұрын

    +dasgomezkanal You are very welcome!

  • @MathsHistoryHelp
    @MathsHistoryHelp8 жыл бұрын

    best stats tutor on the internet . Well done !

  • @jbstatistics

    @jbstatistics

    8 жыл бұрын

    Thanks!

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    @os700710 жыл бұрын

    These are by far the best statistics videos on youtube! Thanks to you i now feel that i should be able to pass my stats exam. Thank you so much! Please keep up the good work!

  • @jbstatistics

    @jbstatistics

    10 жыл бұрын

    Thanks Arian. I'm very glad you've found my videos helpful. Best of luck on your stats exam!

  • @homeisthesailor

    @homeisthesailor

    2 жыл бұрын

    @@jbstatistics Having studied Stats at uni and having passed the exam but remained confused Well done! The very opposite of going from the notes of the lecturer to the notes of the student without going through the brain of either.

  • @homeisthesailor

    @homeisthesailor

    2 жыл бұрын

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    @rien20805 жыл бұрын

    CAN I JUST SAY THAT YOUR TUTORIALS ARE THE BEST AND EASIEST TO UNDERSTAND OUT THERE

  • @jbstatistics

    @jbstatistics

    5 жыл бұрын

    Thanks!

  • @alexandra-stefaniamoloiu2431
    @alexandra-stefaniamoloiu24318 жыл бұрын

    Those are simply the best explanations! I'll watch all your videos.

  • @jbstatistics

    @jbstatistics

    8 жыл бұрын

    +alexandra-stefania moloiu Thanks!

  • @radhikamamidi9070
    @radhikamamidi90706 жыл бұрын

    Very informative , crisp and clear.. Watched other videos but understood very well from your videos the concepts well. Thank you.

  • @jbstatistics

    @jbstatistics

    6 жыл бұрын

    Thanks for the compliments!

  • @WRONGTURN69
    @WRONGTURN699 жыл бұрын

    I love your videos! They're so informative and direct to the point! Plus, your voice is very clear and understandable! Thank you! +1 subscribe! :)

  • @jbstatistics

    @jbstatistics

    9 жыл бұрын

    Thanks for the compliments!

  • @lloydngnn

    @lloydngnn

    7 жыл бұрын

    excellent, extremely good

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    @catgarcia95278 жыл бұрын

    Thank you thank you thank you. My online university provides a link to your videos and I have been using them since the beginning of my stats course. I would be lost without them!

  • @jbstatistics

    @jbstatistics

    8 жыл бұрын

    +Cat Garcia You are very welcome! I'm glad you've found my videos helpful!

  • @hessaa3464
    @hessaa346410 жыл бұрын

    I really like your way in explaining its very simple and understandable. I hope that you create a video tutorial for Gamma, Weibull and Lognormal distributions because I didn't find any good video tutorials for these concepts. Thanks in advance =)

  • @physiciansmind
    @physiciansmind3 жыл бұрын

    Awesome video. This provided great help for my upcoming stats exam!

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    @Le_Parrikar5 жыл бұрын

    Your videos are a true life saver

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    @orca7194 жыл бұрын

    Incredibly well done, thank you

  • @BudskiiHD
    @BudskiiHD9 жыл бұрын

    Best stats videos on youtube!

  • @jbstatistics

    @jbstatistics

    9 жыл бұрын

    ***** Thanks!

  • @wiegehts50
    @wiegehts507 жыл бұрын

    Love these videos!!!!! You all are awesome

  • @jbstatistics

    @jbstatistics

    7 жыл бұрын

    Thanks! I'm glad to hear that!

  • @thecorgiman4118
    @thecorgiman41183 жыл бұрын

    Hey, dude, you're the best. I needed to pass AP stats or else I wasn't going to get into a UC. Thanks a lot for boosting my grade.

  • @jolantabenson9160
    @jolantabenson91604 жыл бұрын

    Thank you JB! Excellent tutorials :-)

  • @leahhall7504
    @leahhall7504 Жыл бұрын

    this is the video that made everything make sense in my head

  • @miguelese18
    @miguelese1810 жыл бұрын

    great video, great explanations!!! thank you

  • @gtudsd
    @gtudsd4 жыл бұрын

    Thanks for the amazing videos!!

  • @tongyulin6177
    @tongyulin61779 жыл бұрын

    Really useful! Thanks a lot!

  • @amandafalke7670
    @amandafalke7670 Жыл бұрын

    That was *GREAT.* Thank you!!

  • @muertaqueen
    @muertaqueen7 жыл бұрын

    the GOAT of stats tutor online

  • @jbstatistics

    @jbstatistics

    7 жыл бұрын

    Thanks!

  • @farlyn3
    @farlyn37 жыл бұрын

    You save lives my friend

  • @chumpapii
    @chumpapii4 жыл бұрын

    Hey man what if thre are 2 given sample mean like between 445 and 485?

  • @cesar.vasconcelos
    @cesar.vasconcelos7 жыл бұрын

    Can somebody please explain how to properly estimate the population parameters: true mu and true sigma, when we don't have them? Can I use the sample mean as a good point estimator for the population mean? Can I use the sample standard deviation as a good point estimator for the true sigma?

  • @XieQiu
    @XieQiu7 жыл бұрын

    Very clear explanation.

  • @jbstatistics

    @jbstatistics

    7 жыл бұрын

    Thanks!

  • @hiphopsocnroc
    @hiphopsocnroc10 жыл бұрын

    Great explanation.

  • @mutalemalama
    @mutalemalama9 жыл бұрын

    The terminology really fucks me over. I don't know when to implement certain formulas so that's a real hinderance although, this has sort of widened my perspective so cheers mate

  • @Un_Bison
    @Un_Bison8 жыл бұрын

    Very helpful, Thank you!

  • @jbstatistics

    @jbstatistics

    8 жыл бұрын

    +JustTheRickyshow You're welcome!

  • @legendarylioness1104
    @legendarylioness110410 жыл бұрын

    These are really great!

  • @jbstatistics

    @jbstatistics

    10 жыл бұрын

    Thanks! I'm glad you like them.

  • @cameellesuarez298
    @cameellesuarez2987 жыл бұрын

    A great help! God bless you!

  • @jbstatistics

    @jbstatistics

    7 жыл бұрын

    Thanks!

  • @pomegranate8593
    @pomegranate85932 жыл бұрын

    good explaining!!!!!!!!!

  • @jbstatistics
    @jbstatistics10 жыл бұрын

    Thanks!

  • @jeeruajay2111
    @jeeruajay21115 жыл бұрын

    I have one doubt , x' mean is equal to population mean when we take mean of the sampling means otherwise not ? but your told that only one sample mean is equal to the population mean how tell me ?

  • @user-cm4tu5jj5n
    @user-cm4tu5jj5n5 жыл бұрын

    Thanks a lot. It is helpful to solve my problems.

  • @jbstatistics

    @jbstatistics

    5 жыл бұрын

    I'm glad to be of help!

  • @rosemarieguira1042
    @rosemarieguira10423 жыл бұрын

    It helps me a lot. Thank you so much po.

  • @Oingoboingo710
    @Oingoboingo7103 жыл бұрын

    Can someone please refer me to the video wherein Mr jbstats prove that E({x bar}) is equal to the population mean?

  • @ItsASoccerTing
    @ItsASoccerTing7 жыл бұрын

    how did he get from 0.8 to 0.2 in the first example

  • @overlordfury996

    @overlordfury996

    6 жыл бұрын

    search for statistical table for normal distribution and see 0.8 exact in the table

  • @ilose21
    @ilose219 жыл бұрын

    Thank you, apparently you are a better instructor than my online applied statistics instructor...

  • @jbstatistics

    @jbstatistics

    9 жыл бұрын

    You are very welcome.

  • @jietang118
    @jietang1187 жыл бұрын

    Great video!

  • @jbstatistics

    @jbstatistics

    7 жыл бұрын

    Thanks!

  • @hakuyihnew499
    @hakuyihnew4993 жыл бұрын

    Thank you, very interesting

  • @shefalisingh4675
    @shefalisingh46752 жыл бұрын

    I really like your way of explaining concepts. But if in a real case scenario where we don't have mu value(suppose a very large sample) how do we even calculate the probability??

  • @jbstatistics

    @jbstatistics

    2 жыл бұрын

    Thanks for the compliment. As you know, we don't (generally) know mu in real life. So then we can't calculate a probability this way. The primary purpose of the discussion in this video is that we later turn this problem on its head, and use a known value of X bar to say something about the unknown value of mu. Working with the mathematical concepts discussed in this video (and some others), we can come up with an appropriate formula for a confidence interval for mu, and an appropriate test statistic to use in hypothesis testing. Statistical inference is built on concepts related to the sampling distribution of the estimator of a parameter. Edit: Which is what I allude to at the end of the video.

  • @mrx42
    @mrx424 жыл бұрын

    Thank you master

  • @katieandrews3005
    @katieandrews30059 жыл бұрын

    Thanks for the videos! Like another commenter, I can't seem to figure out how you got 0.200 at the time mark of 7:20 in your video.

  • @jbstatistics

    @jbstatistics

    9 жыл бұрын

    Jenny FromDaBloc It's an area under the standard normal curve, and it can be found with software or a standard normal table. I have videos that show how to use the standard normal table. Cheers.

  • @CorginShep

    @CorginShep

    9 жыл бұрын

    Take the Z value (at 0.842) and subtract it from 0.5. Because we want the probability above 23grams (at least) and the Z value 0.842 represents the area from 0 to 0.842 you need to subtract it from the whole area (which is 0.5 on the normal distribution table).

  • @superkamiguru6510
    @superkamiguru65104 жыл бұрын

    My teacher never teaches in class so i'm forced to scour the internet and search for problems similar to the work he gives out to help me understand it better. After watching this video, I still am at a lost for what to do

  • @xitrumkc
    @xitrumkc10 жыл бұрын

    So Good!

  • @letseconomics2938
    @letseconomics2938 Жыл бұрын

    @8:39 why do we choose the middle data? We have to draw a sample of size 4 so we can take four values from the left or right as well? And will the mean will be the same as 21.4 for all samples we draw from this distribution?

  • @jbstatistics

    @jbstatistics

    Жыл бұрын

    I don't know what you mean by "why do we choose the middle data". We're randomly picking 4 values from that distribution; they'll be whatever they'll be. It's 4 random draws from that distribution. The distribution in green is the distribution of the mean of 4 randomly picked values from the distribution in white. Yes, each of the 4 has exactly the same distribution (the distribution in white).

  • @letseconomics2938

    @letseconomics2938

    Жыл бұрын

    @@jbstatistics I mean if we have to choose randomly 4 values. Why didn't we choose the extreme 4 values of the distribution?

  • @jbstatistics

    @jbstatistics

    Жыл бұрын

    @@letseconomics2938 You're not understanding what's happening. We're not choosing any specific values, or letting our judgement decide which side we want to grab them from. We're randomly picking 4 values from the distribution in white. If we do that, then the distribution of the mean of those 4 randomly picked values is the green distribution. That's just how the math works out. The green distribution has a lower variance for the reasons I discuss in the earlier part of the video.

  • @jbstatistics

    @jbstatistics

    Жыл бұрын

    I'll just add that sure, we might end up randomly picking 4 extreme values from one tail of the distribution from which we're sampling, that will have non-zero probability of occurring. But the probability of that is very small and is reflected in the sampling distribution of the sample mean.

  • @ShuusakuSama
    @ShuusakuSama9 жыл бұрын

    Is this only for for n>=30? I remember you need to use the t distribution is n

  • @jbstatistics

    @jbstatistics

    9 жыл бұрын

    I talk about this in detail in my other videos. When sampling from a normally distributed population, the random variable Z = (X bar - mu)/(sigma/sqrt(n)) has the standard normal distribution. When sigma is replaced with the sample standard deviation S, the quantity T = (X bar - mu)/(S/sqrt(n)) has a t distribution with n-1 degrees of freedom. This is always true when we are sampling from a normally distributed population, regardless of the sample size.

  • @edwardtification
    @edwardtification9 жыл бұрын

    which is the other video you derived the formula?

  • @ezquerzelaya

    @ezquerzelaya

    8 жыл бұрын

    +e t wondering the same thing

  • @CarlosAugusto-yr3bn

    @CarlosAugusto-yr3bn

    3 жыл бұрын

    Same

  • @MannonCannon
    @MannonCannon7 жыл бұрын

    I get that the SD decreases as N increase. But I'm stuck on the idea that variance increases as sample size increases. I dont get it. Please explain

  • @jbstatistics

    @jbstatistics

    7 жыл бұрын

    I am not sure what you are asking me. The standard deviation of the sampling distribution of the sample mean is just the square root of the variance of the sampling distribution of the sample mean. The standard deviation of the sample mean and the variance of the sample mean both decrease as the sample size increases.

  • @MannonCannon

    @MannonCannon

    7 жыл бұрын

    That's ok I realised I read my lecture slides wrong. Was confused as to why they were saying the SD and variance increased with sample size, however they were saying the opposite. Thanks for actually replying though :)

  • @beabittenbender8373
    @beabittenbender83736 жыл бұрын

    super helpful thanks

  • @jbstatistics

    @jbstatistics

    6 жыл бұрын

    You are very welcome.

  • @himayunrashid2864
    @himayunrashid28649 жыл бұрын

    thankyou sir..............

  • @ThePeacefulResistor
    @ThePeacefulResistor9 жыл бұрын

    thank you so much!

  • @jbstatistics

    @jbstatistics

    9 жыл бұрын

    You are very welcome Nunya!

  • @lucaslau8379
    @lucaslau83797 жыл бұрын

    perfection!

  • @jbstatistics

    @jbstatistics

    7 жыл бұрын

    Thanks!

  • @bushranahvi2263
    @bushranahvi22637 жыл бұрын

    how did you calculate P(z>= 1.648)=0.046?

  • @jbstatistics

    @jbstatistics

    7 жыл бұрын

    That area can be found by using software or a standard normal table.

  • @bushranahvi2263

    @bushranahvi2263

    7 жыл бұрын

    thankyou!

  • @nickyclement

    @nickyclement

    7 жыл бұрын

    P(Z>1.648) can be found by looking up Z of 1.648 on standard normal table. Area under the standard normal curve to the RIGHT of Z is ~0.954. Area to the LEFT of Z is what we are looking for, so P(Z>1.648)= 1(total area under std. norm. curve) - 0.954 = 0.046

  • @bigfatjuice
    @bigfatjuice6 жыл бұрын

    Where is the video where you derive the properties of X bar?

  • @jbstatistics

    @jbstatistics

    6 жыл бұрын

    I derive the mean and variance of the sample mean in this video: kzread.info/dash/bejne/aaGNpqqboq-rgdI.html. I discuss the central limit theorem in this video: kzread.info/dash/bejne/gqme0c5qqaaXj6Q.html

  • @trishayvonnefabular7119
    @trishayvonnefabular71196 жыл бұрын

    im confused about the square root of n.. i didnt get the answer.. pls. help

  • @jbstatistics

    @jbstatistics

    6 жыл бұрын

    What part were you having trouble with? What value were you getting?

  • @sadiaafrin6240
    @sadiaafrin62406 жыл бұрын

    Thank you so much

  • @jbstatistics

    @jbstatistics

    5 жыл бұрын

    You are very welcome!

  • @elvinjafarli6257
    @elvinjafarli62576 жыл бұрын

    Man, I wish you was my stats professor

  • @jbstatistics

    @jbstatistics

    6 жыл бұрын

    I'm glad I can still be of help!

  • @randallkim5903
    @randallkim59036 жыл бұрын

    If sigma is not given, is it possible to do any probability calculations?

  • @jbstatistics

    @jbstatistics

    6 жыл бұрын

    In short, no. We might conceivably have an estimate of sigma, which could give us a rough estimate of the probabilities, but without knowledge of the value of sigma the exact probabilities can't be calculated.

  • @randallkim5903

    @randallkim5903

    6 жыл бұрын

    Thank you!

  • @nutritionbyelsa
    @nutritionbyelsa Жыл бұрын

    wooohoooooo thank you

  • @2foolish
    @2foolish5 жыл бұрын

    beautiful

  • @jbstatistics

    @jbstatistics

    5 жыл бұрын

    Thanks!

  • @corinalohrman5764
    @corinalohrman576410 жыл бұрын

    why is there no sound

  • @jbstatistics

    @jbstatistics

    10 жыл бұрын

    Hi Corina. There is sound when I play the video. I'm not sure what the problem is, but I believe it's something on your end. Cheers.

  • @corinalohrman5764

    @corinalohrman5764

    10 жыл бұрын

    Yup it was thanks. I appreciate your good work

  • @nimas1638
    @nimas16385 жыл бұрын

    i am the 1000th like.... what was the probability of that?

  • @etlogfactory6253
    @etlogfactory6253 Жыл бұрын

    I hope someone still sees this, but can someone explain how 0.842 becomes 2 😭

  • @jbstatistics

    @jbstatistics

    Жыл бұрын

    The area to the right of 0.842 under the standard normal curve is 0.200. That's found with any statistical software, such as R, or by using a standard normal table.

  • @jimallysonnevado3973
    @jimallysonnevado39736 жыл бұрын

    Wheres the proof

  • @jbstatistics

    @jbstatistics

    6 жыл бұрын

    Of what part? The mean and variance? The fact that if we're sampling from a normally distributed population, then the sample mean is normally distributed? Of the central limit theorem? I derive the mean and variance of the sampling distribution here: kzread.info/dash/bejne/aaGNpqqboq-rgdI.html. I don't have video proofs of the other parts yet.

  • @musicmanxii
    @musicmanxii8 ай бұрын

    This is like chinese to me. God help me.

  • @randomrandom316
    @randomrandom3168 жыл бұрын

    Really useful, thanks a lot!

  • @jbstatistics

    @jbstatistics

    8 жыл бұрын

    +Shailendra Pandey You are very welcome!

  • @jbstatistics
    @jbstatistics10 жыл бұрын

    Thanks!