This tutorial with cover some measurements of the distribution of sample means. You’ll learn about:
In this tutorial, you’ll use the mean to measure center and the standard deviation to measure the variation.
Suppose you have a spinner with three sectors:
It's pretty easy to find the mean of the spinner from one spin. You just add up all the values, and divide by 8 because they're all equally likely sectors, and end up with 2.375 as your mean. The standard deviation of the spinner is 1.218.
Now suppose you spun it four times to obtain an average. And then suppose you did it again and obtained another average, and then another average. You could eventually consider every possible set of four outcomes. If you consider every possible scenario, and plot each scenario on a graph like the one below, you can create what's called a sampling distribution of sample means.
So this distribution has a sample means of 1, for instance, where you got 1 on all four of the spins. It looks like the data average is somewhere around the 2.25, or 2.5 region. Those are the most likely averages from four spins.
This distribution of sample means itself has a mean.
In the case of this distribution of sample means, the mean is somewhere in the center. In fact, the number is 2.375, which is the same as the mean of the original spinner.
So the mean of the sampling distribution of sample means is the same as the mean of the original distribution for the spinner. Symbolically it looks like this:
The mean of all of the sample means, x bar, is the same as the mean of the original distribution. Sometimes we call this the parent distribution or the grand mean.
Recall that the standard deviation of the original distribution for the spinner was 1.218. This distribution also has a standard deviation. It's 0.609. How does 0.609 relate to 1.218?
0.609 is half of 1.218. The standard deviation on the distribution for four spins is only half as large as the original standard deviation was.
What is the distribution of sample means when you spin nine times?
First look at the shape. You can see that the extreme values are much less likely now, and things start moving towards the center. This might be indicative of the standard deviation getting smaller yet again. And notice that the mean of the distribution for nine spins is the same as the other means: 2.375.
The standard deviation is again smaller, just 0.406 on either side of the mean.
As the number of spins increases, the standard deviation goes down. But it's not linear; it's proportional to the inverse of the square root of n. So you divide the original standard deviation by the square root of sample size.
The mean of a sampling distribution, from samples of size n, is always going to be the same as the mean of the original distribution it came from. The parent distribution. The mean of all the x bars is the same as the original mean from the parent distribution.
The standard deviation, on the other hand, gets smaller as the sample size increases. The larger the sample size, the more likely it is that the extreme values will get evened out and pulled back towards the mean. Thus, the standard deviation decreases, and you can quantify the decrease in standard deviation. Standard deviation for the sampling distribution is the standard deviation of the parent distribution, divided by the square root of sample size.
Thank you and good luck!
Source: THIS WORK IS ADAPTED FROM SOPHIA AUTHOR JONATHAN OSTERS
The average of all possible means from all possible samples of a given size. It will be equal to the mean of the original population.
The standard deviation of all possible means from all possible samples of a given size. It will be equal to the standard deviation of the original population, divided by the square root of the sample size.