8.2 Distribution of sample means and the standard error of the mean

Once again I turned to the distribution of samples of marbles and used their masses to demonstrate the distribution of sample means. 


I distributed 60 marbles with a mean marble mass of 5.1417 grams per marble (determined post hoc). I gave five marbles each to 12 students in the class. 

Spreadsheet

I then went around and obtained the mass of every marble. For each of the 12 students I had a spreadsheet that calculated their sample mean. I did not preset the names, those were entered during the class. The twelve students were those that were on time, late arrivals were not given marbles. 12 students and 60 marbles was two students and ten marbles more than the prior term, but the measurements were still finished at 9:18. 


Post hoc I added the above section to reinforce that the point estimate of a population mean is whatever the sample mean is for a sample. And that will be wrong. The fifth line is for Friday, those are 95% confidence interval lower and upper bounds. A function determines if the known population mean was captured.


None of the sample means were 5.1417 grams. But 11 of the 12 were close. Those sample means distribute closer to the population mean than the individual marble masses. Remember that the sample standard deviation tells us about the spread in the data. The standard error of the mean estimates the theoretical spread in the sample means. The standard error of the mean is less than the standard deviation by a factor of the square root of n. 

11 of the 12 samples captured the population mean in their 95% confidence interval. One sample did not. That is a success rate of 91%. Why wasn't the success rate 95%? Because we would have to do many more samples. As in hundreds. Then we would get close to a 95% capture rate for the population mean. And 5% would not capture the population mean. 


The narrower nature of the distribution of sample means was perhaps not as clear this term as prior terms, but the effect was enough to use the SMARTboard markers to sketch out the standard deviation of the data and the standard error of the sample means. This system still well demonstrates how multiple sample means distribute around a known population mean. 

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