Sample mean

Sample mean

Five samples from this probability distribution function, plotted it right there. I got four instances of this random variable, I average them, I have another sample mean.

I could write it like that, or I could write the standard deviation there.

Sample mean symbol

Now let's see what happens if we do the same thing with a larger sample size. So this is a more normal distribution. What I can do is take samples-- in the last video of say, size four-- that means I take literally four instances of this random variable, this is one example. So let me make something kind of crazy. So if something has no skew, that means that it's nice and symmetrical around its mean. So I'm already getting pretty close to the mean there. So my sample size is going to be five. I got four instances of this random variable, I average them, I have another sample mean. And the skew and kurtosis, these are things that help us measure how normal a distribution is. It'll take some time.

What I can do is take samples-- in the last video of say, size four-- that means I take literally four instances of this random variable, this is one example. Let's say that's a perfect normal distribution.

Population mean formula

So something like this would have negative kurtosis. Just let me clear them. And negative kurtosis has smaller tails, but it's smoother near the middle. We're going to talk a little bit more about in the future. Take 25, get the mean, and then plot it down there. Add together the numbers in the data set. If you have a positive skew, that means you have a larger right tail than you would otherwise expect. So when our sample size was n equal 5 and we did 10, trials, we got pretty close to a normal distribution. If you want to figure out the average distance from your coworkers' birthplaces to the workplace, you'll first collect the data.

The standard deviation is the square root of the variance, 9. So that is a negative skew. So let's do that. I take their mean, and I consider this the sample mean from my first trial, or you could almost say for my first sample.

sample mean definition

Through statistical analysis using the sample mean, statisticians can calculate items such as standard deviation and variance. This is their app, which I think is a really neat app, because it really helps you to visualize what a sampling distribution of the sample mean is.

Standard deviation of sample mean

If you want to figure out the average distance from your coworkers' birthplaces to the workplace, you'll first collect the data. So here the sample size is 25, here it's five. So here's n equal 5. For example, the median is the value exactly halfway between the lowest and highest in the dataset. And then I'm going to click it again, and it's going to do it again. And just so we have a little terminology on our belt, this frequency distribution right here that I've plotted out, or here, or up here that I started plotting out, that is called-- and it's kind of confusing, because we use the word sample so much-- that is called the sampling distribution of the sample mean. But sampling distribution of the sample mean is the most common one. This result follows from the fact that any linear combination of independent normal random variables is also normally distributed. Through statistical analysis using the sample mean, statisticians can calculate items such as standard deviation and variance. Our mean is now the exact same number, but we still have a little bit of skew, and a little bit of kurtosis. And negative kurtosis has smaller tails, but it's smoother near the middle.

Sample means are important in that they can give an idea of central tendency-- that is, an idea of the general tendency of a set of numbers. Another measure is the mode. We'll talk about that in a future video.

So this is a more normal distribution.

sample mean vs population mean
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Sampling distribution of the sample mean (video)