Statistics 101: Standard Error of the Mean

hello thanks for watching and welcome to the next video in my series on basic statistics now a few things before we get started number one if you're watching this video because you are struggling in a class right now I want you to stay positive and keep your head up if you're watching this it means you've accomplished quite a bit already you're very smart and talented and you may have just hit the temporary rough patch now I know with the right amount of hard work practice and patience you can get through it I have faith in you many other people around you have faith in you so so should you number two please feel free to follow me here on YouTube on Twitter or on LinkedIn that way when I upload a new video you know about it and on the topic of the video if you'd like it please give it a thumbs up share it with classmates or colleagues or put it on a playlist because that does encourage me to keep making them on the flipside if you think there is something I can do better please leave the constructive comment below the video on YouTube and I will try to take those ideas into account when I make new ones and finally just keep in mind that these videos are meant for individuals who are relatively new to stats so I'm just going over basic concepts and I will be doing so in a slow deliberate manner not only do I want you to understand what's going on but also why so all that being said let's go ahead and get started so this video is the next in the beginning series of inferential statistics now saying the beginning of inferential statistics is like saying the beach is the beginning of the ocean because so much of statistics and if you really think about it pretty much everything and statistics is inferential and again I talked about that idea in the previews the videos leading up to this one so I won't go into all that right now so we're going to talk about in this video is a concept that I unfortunately think is neglected in many stats classes so in previous videos I talked about how I think covariance and the covariance matrix is skipped over this is one of those concepts and it's called the standard error of the mean and when we are estimating a parameter about a population the standard error of the mean is fundamentally absolutely important to grasping that estimation so I really want you to understand what the standard error of the mean is so you can then apply it to these estimates estimators we are using in statistics now we'll say the general term standard error of the estimate the standard error of the mean is just one of those standard errors of the estimate so the mean is one of the estimates so if you ever hear standard error of the estimate the mean is just sort of a subset of that okay so let's go ahead and take a look at this topic so this problem is the same one we've been using in the previous video so I'm going to go over it very quickly I'm not going to do any ab living that's once you understand the problem so highway paving is a company specializing in residential Road surfacing many of its clients seek out its specialized product which is low noise pavement recycled rubber can be added to s fault mixtures to reduce road noise which appeals to environmentally conscious clients because the type of tires are recycled and it's good for people in the neighborhood and in the car because it reduces noise road noise for everyone however the viscosity of the asphalt it's resistance to flow okay think of it in sort of laypersons terms its thickness okay must be maintained within very tight limits otherwise it may be too thick or too sort of watery and thin to use so you can imagine having very thick asphalt that's hard to transport and then you try to get it out of the truck and it it's sticks to everything and you can't spread it around get this too thick on the other side it may be too thin so it comes out and you try to spread it but it's it won't hold its shape it is one that's wants to run everywhere so the viscosity of the asphalt is very important now for this company the goal okay is 3,200 and for the purpose of this video we're going to treat 3200 as the population mean so over several years of production in quality measurements the company has determined that the viscosity population mean and standard deviation for the low noise payment are as follows so the population mean is 3200 and the population standard deviation is 150 so remember when we use the Greek letter mu that's the population mean and the Greek letter Sigma is the population standard deviation so 3200 and 150 now that's a very important point down here below that I want you to really understand and you may have even come across it already in your class we seldom know the population parameters now if we knew the population parameters like the mean the population mean is their deviation why would we be trying to estimate them so we learn using examples like this first where we do know these parameters because it helps us better understand sort of the ugly reality which is most often we don't know these so again you can think about this another example so if I wanted to know the average income of everyone in the state of Ohio here in the US for example I can't go out there's no way that I could go out and ask every single income earner in the entire state of Ohio what their income is there's just no way so I would have to pull a sample of income earners okay the time and expense in the practical logistics of trying to ask everyone is just ridiculous okay so we very suddenly know a population Witter unless it's somehow kept in a database somewhere automatically okay but very suddenly do we know that so during the manufacturing of each batch of asphalt the quality control specialist takes 15 specimens or samples of the material and tests the viscosity and when I say takes 15 samples I meant to change that word samples I think of it as 15 measurements or 15 specimens I think of a sample as the collection of those 15 measurements now this ensures the batch has uniform viscosity as well because they test the viscosity in different parts of the batch so they want to make sure there's no areas that are sort of thicker and there are no areas in the Batchelor thinner so the the specimens come from all throughout the batch and there are 15 that are taken for each batch so things to note there's no way to test every ounce of asphalt okay therefore the company must take samples from those samples the company must then make inferences about the entire batch and that's sort of the idea behind inferential statistics the inference is made using these samples are big net by definition incomplete therefore the sample characteristics will always have some error built in and again that's one of the fundamental ideas of inferential statistics there are at least things called error terms or margins of error on the end of everything because the samples are not perfect and they never really can be perfect unless you sample everything in the population but then that defeats the purpose the company took nine samples now each sample has 15 measurements in it so sample one had 15 measurements 15 specimens and that sample mean was thirty to ten point seven three then they took another sample sample number two that had 15 measurements in it that sample mean was 31 fifty point one three so you can see how these samples work okay so they took nine samples each of the same size 15 measurements and these are the sample means for all nine samples so we could do a curve for each one so sample 1 2 3 4 5 6 7 8 9 and you can see that's x-bar subscript 1 x-bar subscript 2 that means sample 1 sample 2 sample 3 etc and those are the measurements for each sample so again think about each sample mean as having its own distribution like this ok that's something I want you to get into your mind when you're taking multiple samples from a population each sample has its own sample characteristics now here are samples again now what I did is I created some ranges some you know most 100 because it's 49.99 is the end of the range of measurements so you can see that we had one measurement in the 29 52 30 49 range just want the very bottom we had one in the 30 50 231 49 range and then we had 4 in the next range etc of course we call this the sampling distribution this is a distribution of the sample means a distribution of the sample means of these 9 sample means now the way I figured this out the expected value of this overall sample the sample means was 32 17 point 0 8 now what how did I do that now in this case what I did is I just took the average the mean of the sample means that's because each sample was the same size so they are weighted the same so I went ahead and just took the average of the sample mean column and that gave me sort of an expected value of the overall distribution of 32 17 0.08 so basically another way of thinking about it is instead of treating these as nine different samples I kind of treated it as one large sample of a hundred and thirty-five specimens to get the expected value okay but again that's not really that important for what we're doing here but I want to show you where that came from now we can actually turn that on its side and we have an Instagram like this so you can see for each range we have a certain number so we had one measurement in the first range we had one measurement in the second for in the middle et cetera the thing is what does this shape look like well if you kind of look at this as a normal bell curve that's very important okay so this looks like a normal bell curve and that's actually related to something that you've probably heard of called the central limit theorem and that is if we take a large number of samples from basically any population of any shape we take many samples and we create a sampling distribution like this the sampling distribution will be normally distributed like the bell curve okay now there are many versions of the central limit theorem but that's the one that's most applicable to at least this part of introductory statistics okay so the expected value or the mean of our sampling distribution so you can see over here we have our graph now in this I want you to pause just for a second and look about what I'm saying here now the expected value of many samples okay so X bar sub M many samples we take we take their mean we take 30 samples and we have the mean as that number of samples becomes large okay in this case we did a nine but let's say we did a hundred now the mean of that sampling distribution is the population mean because you remember we're not in this case we know the population mean but in many cases we don't so basically what this is saying is that if we take a large number of samples okay and we put all those samples into a distribution like this here on the right the mean of that distribution is the same as the population mean we do not know expected value of X bar which is the sampling distribution mean that's the expected value of X bar so that's the expected value of that the sample of this distribution and of course mu is the population mean so the way this works is that if we take many samples we create this distribution then the mean of this distribution is the same as our population mean that's the idea now again I can splain this like I did in the previous video and sort of plain English but no matter how I simplified it it still sounds like a riddle but if we take many random samples from the population each sample will have its own sample mean so in this previous slide we did nine then we create a distribution based on all of those sample means which is what we have here in the upper right then the mean of that sampling distribution up here on the right is equal to the mean of the population so again that's a way of saying exactly it's everything on this slide okay so give me take your time to wrap your head around that but that's basically what we're talking about here let's remember this is an estimate at best the expected value of the sampling distribution is at best going to be an estimate of the population mean mu we would have to take every conceivable sample from the population to match the population mean perfectly but if we do that what is the point of sampling in the first place the best we're going to be able to do is to find an interval estimate for the population mean mu now our interval estimate will be influenced by sample size and that's really the heart of this video and the degree of confidence we are satisfied with so we're going to find this sample mean it is not going to be perfect but we can make it better through the sample size we take and depending on the degree of students we want to have whether or not that is representative of our population mean so that's at the heart of what we're doing okay let's go ahead and actually talk about what the standard error of the mean is so everything up to now was precursor to understand what we're actually wanting to talk about here okay but I had to make sure you have that sort of pre knowledge before we like to get to this so up to now we have talked about the mean of this sampling distribution over here on the right but now we're interested not in the mean of the sampling distribution but it's standard deviation so that is what the standard error of the mean is the standard error of the mean is the standard deviation of this sampling distribution so standard error of the mean now it has this formula here okay now we're going to use this formula here in a second but that's one of the sort of you decode what it is so what is this saying sigma sub x-bar okay it's the standard deviation of the sampling distribution over here on the right so that's what we're trying to find now Sigma is the standard deviation of the population now in this case we know it because we are given it at the beginning but in most situations we don't know it which will save that for a different video but in this case we do know that so Sigma there on the top is the standard deviation of the population of course n is the sample size so we to find the standard deviation of the sampling distribution which is another way of saying find the standard error of the mean we really need to know two things the center deviation of the population and its sample size now it's important to point out again that this first example assumes we know the population standard deviation Sigma there at the top but this is really the case we will talk about what to do when Sigma is unknown in a later video so just keep that in mind okay so let's go ahead and do a simple calculation so to find the standard error of the mean we have our formula here now we know that the Sigma sub X that is the centered deviation of this distribution over here on the right is what we're looking for now we are told at the beginning of this problem that the population standard deviation was 150 remember our sample size for our samples was 15 okay so n is 15 now if we go ahead and calculate that out we have 150 divided by the square root of 15 which equals 38.7 so what we are saying here is that the standard deviation of this distribution over here on the right hand side is thirty-eight point one seven now note how the sample size changes the standard error of the mean and I'm going to show you a real example here in the next slide but notice that the sample size is in the denominator of this fraction so as sample size gets larger as n becomes larger what happens to this overall fraction well remember Sigma stays constant okay because that's the population standard deviation but as n becomes larger that denominator becomes larger therefore the overall fraction becomes smaller okay and this is extremely important in thinking about the influence of sample size okay so let's go ahead and take a look at this influence of sample size now our sample in the first case was 15 so we've figured out a standard error of the mean was thirty eight point seven now what happens if we change that sample size to 135 look what happens to the standard error of the mean now it goes down to twelve point nine so the standard error of the mean which is the standard deviation of the sampling distribution which isn't the same way of saying the variation of the sampling distribution goes from thirty-eight point seven to twelve point nine by increasing the sample size what about we increase it to 500 well now the standard error of the mean is six point seven one so as the sample size increases what happens to the standard error well it decreases a better way to think about it is that it narrows when the standard deviation decreases the distribution around the mean literally narrows or squishes in around the mean a larger sample size reduces the standard error and which actually means it reduces the variability it reduces the centered deviation of the sampling distribution so if we have three curves like this now with a sample size of 15 it may look like the green curve if you notice the green curve is shorter in the middle and it has wider or fatter tails so there is more variation in that curve because the standard deviation is larger then we move up to a sample size of 135 well that curve is narrower and taller in the middle and the tails are shorter or thinner so more probability is pushed in towards that mean okay of 3,200 it's the probability is pushed inward then with a sample size say of maybe 500 the probability is pushed in even further because the standard deviation is so the curve is squished in towards the mean so a larger sample size decreases or you can think of it as narrowing this curve which is the standard error which again is the standard deviation of the sampling distribution so the values of x-bar and the sampling distribution will have less variation and therefore will be closer to the actual population mean so larger sample size reduces the standard error that gives us a better approximation of our actual population mean now pause here for a second this should make intuitive sense if I do want to know the average income of every person in the state of Ohio here in the United States what do you think is a better technique I go out and ask for people their income or I go out and ask ten thousand which do you think is going to give a closer approximation to the actual mean income for the entire state well it's going to be the survey that asks ten thousand people and why is it going to be better well we just showed that because the sample size is in the denominator of the standard error it is going to decrease the error okay it's going to decrease that standard deviation of the sampling distribution so that's how sample size comes to be so important because it reduces the standard error in these estimates now a larger sample size up to a point generates a better approximation of the underlying population because it minimizes this error now there is a point and there's a way I could maybe show you this in another video we're increasing the sample size further doesn't really get you more and really to tell you the honest truth it's simple way to show that you can go into any graphing calculator and a graphing calculator online and you can actually graph the function that we had in the previous slide so we could graph 150 divided by the square root of x and of course X is our sample size so we would have a graph for every conceivable sample size given the population standard deviation that we could come up with in the next video maybe I'll actually do that for you so but there's a point where a larger sample just doesn't give us anything more okay so it's not infinite now this also has one more employment implication so if we look at the standard error of the mean over here on the left this is everything we've had before so their population standard deviation was 150 and our sample size was 15 so we put that into our formula and we have a standard error of the mean of 38.7 now if you're sort of intuitive you may have sense to this already but there's a really important thing here as far as the sample sizes we pick and that is the standard error of the mean Sigma X bar is the same for all samples of the same size so in this case any sample of size 15 will have a standard error of the mean of 38.7 assuming that population standard deviation of 150 remains the same so if you look in the formula there there are only two pieces the population center deviation Sigma there on the top which were given and the sample size in the denominator so what that means is that any sample size that's to you the same sample size will have the exact same standard error of the mean and that's a really important thing when we're talking about the sample sizes we choose from the population and things like that so what would this actually look like in terms of our samples so here where are nine samples we took from the asphalt batch so we have sample 1 sample 2 sample 3 symbol for now they all have different sample means but remember the sample size for each one was 15 so what does that mean for their standard errors of the mean well they are all 38.7 so even though each sample has its own sample mean the standard error of the mean is the exact same and that's because the sample size is the exact same now another question so my students have when I work with them is they get this the Sigma is confused so they ask well why is the standard error of the mean the Sigma X bar so much smaller than Sigma which is the population standard deviation and the quick answer is is that they're measuring the standard deviations of two completely different distributions so just Sigma is the standard deviation of the population were interested in Sigma X bar is the standard deviation of the sampling distribution so it's the distribution of all these samples okay so the Sigma's are measuring two different things it's really easy to get them confused but just keep in mind that they are the standard deviations of two completely different things but I do want to point out here the fact that any sample that's the same size the next sample the next sample an example in this case they're all 15 no matter what the sample mean is they will all have the same standard error of the mean okay so remember standard error summary the standard error of the mean is another name for the standard deviation of the sampling distribution standard deviations allow us to calculate z-scores and therefore the area or probability under the curve for certain regions now any point estimator is just that it's an estimation therefore it will contain error this error term can be minimized by selecting a large sample from the population up to a point the error component means we cannot nail down a parameter perfectly we can only provide a range or an interval that may cover the parameter we call it as confidence intervals now most often we do not know the population standard deviation therefore we have to estimate it and make necessary adjustments and that's going to be the topic of one of our next videos okay so we have covered a very important concept and that is the standard error of the mean so when we do not necessarily know the mean of the population but maybe we do know the population standard deviation which we must often don't know that either but what we're doing here is we're taking sample data and we are estimating some parameter in the population usually the mean and the standard deviation so the standard error of the mean tells us sort of how close we can get to the actual population mean and of course we learned that that is influenced by sample size up to a point after which a larger sample does not do us any good so it's all about inferring or estimating a population parameter from a sample that's one of the fundamental ideas in basic statistics okay so if you're watching this video because you are struggling in a class stay positive and keep your chin up your smart and talented I know it everyone's around you knows it so you should think that - I do like the video please give it a thumbs up share it with classmates or colleagues or put it on a playlist that does encourage me to keep making them on the flipside if you think there is something I can do better please leave a constructive comment below the video and I will try to take those ideas into account when I make new ones and finally just remember that the fact that you're on here trying to learn trying to improve yourself as a student or as a business person that is what really matters I'm a firm believer that who have a good learning process in place the results will take care of themselves so thank you very much for watching I wish you all the best of luck in your studies and in your work and I do look forward to seeing you again next time you

Channel: Brandon Foltz

Views: 174,106

Rating: 4.9560566 out of 5

Keywords: standard error of the mean, standard error statistics, standard error, standard error of mean, standard error of the estimate, sample distribution, sampling distribution, brandon foltz, brandon c foltz, brandon c. foltz, statistics 101, standard, mean, anova, linear regression, logistic regression, standard deviation, statistics, 101, confidence interval, stats, sample size, sample mean, margin of error, statistics 101 brandon foltz

Id: uIHFbMn8SBc

Length: 32min 3sec (1923 seconds)

Published: Thu Jan 31 2013