Understanding Standard Deviation for Sample Means in Statistics

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When calculating the standard deviation of a sample mean, the population standard deviation is divided by the square root of n to account for the average squared deviation from the mean across n observations. This approach reflects the concept of sample variance, which is defined in this manner. It's important to note that when estimating sample variance using the sample mean, n-1 should be used instead of n, unless the true mean is known. The discussion highlights a lack of clarity in understanding variance due to instructional gaps. Overall, the explanation clarifies the rationale behind the formula for standard deviation of sample means.
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Hiya guys, I just have what I'm sure is a simple question about statistics, but I can't seem to find it anywhere ...

I was wondering, when finding the standard deviation of a sample mean, why do you divide the population standard deviation by the square root of n? I'm not really sure why they took the square root ... if anyone could help, I'd really appreciate it :biggrin:

Thanks so much!
 
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The short answer is because sample variance is defined as this. And the heuristic explanation for the sample variance formula is that the sample variance is the average squared deviation from the mean, so it is being averaged across n observations.
 
Slight nitpicking: When you use the sample mean to estimate the sample variance, you should use n-1 not n. If you happen to know the true mean, then you can use n.
 
Thanks!

Thank you so much! I was really curious and my AP stats teacher is kind of confused as to what's going on because she hurt herself at the beginning of the year and has been pretty much gone since then, and then refuses to explain why of anything ... I don't think we've actually learned about variance yet, but thanks a lot anyway, I really appreciate it :smile:
 

Thread 'Hypothesis testing: Defining H0, HA hypotheses so that ( H_A)_A' makes sense'

The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense. Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with ## (M^{T})^{T}=M##, and just wanted to see if it made sense to talk...
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