Transcript:

In this video, we’re gonna talk about how to calculate the standard error of the mean so. I’m going to give you two examples, so you could see some different ways on how to do it. Let’s start with this one number. One five students in a college were selected at random and their ages were found to be 18 21 19 20 and 26 Part A calculate the standard deviation of the ages in the sample. Now feel free to pause the video. If you want to try this first, go ahead and calculate the standard deviation. And then you can play the video to see if you have the same answer that I have. The first thing we need to do is calculate the mean or the average of the five ages that we have here, so we’re going to take the sum of those five values, so we have 18 plus 21 plus 19 plus 20 plus 26 And now let’s divide it by the number of the ages that we have, which is 518 plus 21 plus 19 plus 20 plus 26 That gives us a sum of 104 104 divided by five is twenty point eight. So this is the average age of the individuals in this sample. The sample size is five so now let’s calculate the sample standard deviation. The standard deviation of the sample is represented by the symbol S and it’s equal to the sum of all the differences between each value And the mean squared divided by N minus one. And then the whole thing is inside of the square root symbol, so we already have the mean X represents each of these individual values. This is going to take the whole page. So the first X value that we have is 18 We’re going to subtract it from X Bar. The mean, which is 20 point eight and then we’re going to take the square of that result. Now, let’s move on to the second observation that we have. The second age was 21 subtracted from the mean and then square the result and then repeat the process for the other ages as well. So I chose a sample size of five because anything more. This is just going to take too long not to mention that. I was going to run out of space too now. This five ages in the sample. So n is five. Now let’s take this one step at a time so first, let’s find the difference between eighteen and twenty point. Eight, eighteen minus twenty point eight, That’s negative. Two point, eight, twenty one minus twenty point eight. That’s going to be positive point – 19 – 20 point eight, That’s negative. One point, eight, twenty – twenty point eight is negative point eight and 26 – 20.8 That’s going to be five point. Two that 5 minus 1 is 4 so at this point. Go ahead and plug in everything you see here into your calculator. This is going to take a minute, so you can fast forward the video If you want to the value that I got is 3.11 four, so that is the standard deviation of the sample of the five inches that we have in this problem now. Let’s move on to Part B. Calculate the standard error. Now the formula that we need to calculate the standard error is this. The standard error is going to be the standard deviation of the sample divided by the square root of N. So, as we know, it’s 3.11 for the sample size is 5 There’s five ages in this sample, so we’re going to divide it by the square root of five, so this is going to be three point. One 1/4 divided by square root of 5 and that comes out to be one point three, nine three. So that is the standard error in this example. That’s how we can calculate it in this problem. So now let’s move on to our next problem. Number two in a certain. University, the mean age of students is twenty point five with a standard deviation of point eight Part. A calculate the standard error of the mean if a sample of 25 students were selected now. What type of spam deviation? Do we have in this problem? Because there’s two types. There is the standard deviation of the sample and a standard deviation of the population point. Eight is the standard deviation of all the students in this university, so that is the standard deviation of the population and is represented by the symbol. Sigma now to calculate the standard error and we need to use a formula that is very similar, but with different variables, the standard deviation. I mean, a standard error of the mean is going to be the stand deviation of the population divided by the square root of N. As you said before the standard deviation of the population is 0.8 and in Part. A we chose a sample size of 25 so n is 25 now. The square root of 25 is 5 so 0.8 divided by 5 will give us a standard error of 0.6 teen. So that’s the answer for Part. A now for Part B. It’s going to be very similar. What would the standard error of the mean be? If a sample of 100 students were selected, so let’s use the same formula, so we’re going to start with the population standard deviation of 0.8 and this time. The sample size is 100 The square root of a hundred is 10 because 10 times 10 is 100 so we have 0.8 divided by 10 and that gives us a standard error of 0.08 so notice that the standard error decreased from point 16 to point zero eight when N was increased notice that we increase N by a factor of four the standard error decrease by a factor of two. So anytime you increase if you quadruple the value of n, the standard error will decrease by half it’s an inverse relationship and the reason being it’s there’s a square root end on the bottom. The square root of 4 is 2 but nevertheless, though, as you increase the sample size, the standard error decreases, and so you’re going to get a more accurate answer That means the that you get for your sample has a higher accuracy. If the standard error is less now to represent this visually on a graph, let’s say if we were to take many samples and calculate the mean age of the students in those samples and plot it on a graph to create a distribution. This is known as the sampling distribution of the mean it’s going to have the shape of a normal distribution. So this is the mean of the sample means, and the random variable will be x-bar. Now, let’s say this shape corresponds to a sample size of 25 if we do the same thing. But with a sample size of 50 the graph is going to be more narrow and it’s going to be taller, so I need to redraw this one. It’s going to be narrow in shape, but the height is going to be greater, so that’s a visual representation that shows the effect of increase in the value of N. The standard error will decrease, creating a sampling distribution that is taller and more narrower, so there’s less spread, which means the mean of the sample means will approximate the population mean more accurately, But that’s it for this video. Thanks for watching.