Central Limit Theorem

[Agent Smith]

Central Limit Theorem: As the sample size increases, the distribution of the sample means approaches normal.
Is it also true that the sample mean of a larger sample is closer to the population mean than the sample mean of a smaller sample?

 

[Dr.Peterson]

Central Limit Theorem: As the sample size increases, the distribution of the sample means approaches normal.
Is it also true that the sample mean of a larger sample is closer to the population mean than the sample mean of a smaller sample?

If you mean that the standard deviation of the sample means is smaller, yes. Do you know the formula for that?

 

[Agent Smith]

If you mean that the standard deviation of the sample means is smaller, yes. Do you know the formula for that?

No, I don't know the formula, but I do know a little about kurtosis; very sketchy since the course I took didn't put too much of an emphasis on it.

 

[Dr.Peterson]

No, I don't know the formula

Here's the first thing I find on a quick search:

6.2: The Sampling Distribution of the Sample Mean

This phenomenon of the sampling distribution of the mean taking on a bell shape even though the population distribution is not bell-shaped happens in general. The importance of the Central …

stats.libretexts.org

​

That last line is the formula I referred to.

 

[Agent Smith]

@Dr.Peterson , yes I know that formula. The standard deviation of the sample means. I was also taught that the mean of the sampling distributions of the sample means = the population mean. Is there a reason why this is true? Law of large numbers? Yet, I don't recall that law being invoked as a justification.

From [imath]\sigma_{\overline X} = \frac{\sigma}{\sqrt n}[/imath], we can see that larger the [imath]n[/imath], smaller the [imath]\sigma_{\overline X}[/imath]. Would I be correct to say the sample means will begin to cluster around the true population mean with large samples?

Thank you so much.

 

[Agent Smith]

@Dr.Peterson I went through the linked document. Helped.


This is a question from the link you gave.

As regards this question ... the way it was answered, for question a. it seems we can talk directly about the population itself. The random variable [imath]X[/imath] represents values for members of the population.

For question b. what I got is that we're computing facts about [imath]\overline X[/imath], the random variable representing the means of samples (in a sampling distribution).

Is that correct?

 

[Dr.Peterson]

Yes, that's the point they are making.

 

[lookagain]

Is it also true that the sample mean of a larger sample is closer to the population mean than the sample mean of a smaller sample?

I want to answer that directly. I am using bold text for emphasis. The mean of one larger sample will
not necessarily be closer to the population mean compared to one smaller sample.

The chances increase that the mean of the one larger sample will be closer to the population mean
(compared to the one smaller sample) occurs when the size of the one larger sample becomes even
greater than the size of the one smaller sample than what it was previously.