Understanding Standard Deviation for Sample Means in Statistics

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Discussion Overview

The discussion revolves around the concept of standard deviation for sample means in statistics, specifically addressing the reasoning behind dividing the population standard deviation by the square root of the sample size (n). The scope includes conceptual clarification and technical explanation related to statistical formulas.

Discussion Character

  • Conceptual clarification, Technical explanation

Main Points Raised

  • One participant questions why the population standard deviation is divided by the square root of n when calculating the standard deviation of a sample mean.
  • Another participant provides a brief explanation, stating that the sample variance is defined in such a way that it involves averaging squared deviations across n observations.
  • A third participant points out that when estimating the sample variance using the sample mean, n-1 should be used instead of n, unless the true mean is known.

Areas of Agreement / Disagreement

Participants express different levels of understanding regarding the application of n versus n-1 in calculating variance, indicating some disagreement or uncertainty about the correct approach in this context.

Contextual Notes

There is a mention of the participant's AP statistics teacher being unavailable for clarification, which may indicate a lack of foundational knowledge in the class regarding variance and standard deviation concepts.

whitaleedee
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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:
 

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