Sample and population variances: elementary question

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Discussion Overview

The discussion revolves around the relationship between sample variance and population variance, particularly in the context of estimating population parameters from a sample. Participants explore concepts related to statistical variance, including the differences between sample variance and the variance of the sample mean.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant suggests that the sample variance is approximately the population variance divided by the sample size, raising a question about the implications when the sample size equals the population size.
  • Another participant clarifies that the goal in statistics is to estimate population parameters from a sample, noting that population parameters are constants while sample parameters are random variables.
  • A participant points out a potential confusion between sample variance and the variance of the sample mean, indicating that the variance of the sample mean converges to zero while the population variance does not.
  • One participant asserts that sample variance is not equal to population variance divided by sample size, which aligns with the concerns raised in the initial post.
  • A later reply acknowledges the confusion and expresses understanding after receiving clarification from another participant.
  • Another participant reiterates that the assertion about sample variance leading to absurd conclusions indicates a misunderstanding of the original assumptions.

Areas of Agreement / Disagreement

Participants express disagreement regarding the relationship between sample variance and population variance, with some asserting that the initial claim is incorrect. There is no consensus on the implications of the relationship discussed.

Contextual Notes

Participants highlight the importance of distinguishing between sample variance and the variance of the sample mean, as well as the assumptions underlying their discussions. Some statements reflect confusion about the definitions and relationships involved.

nomadreid
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Given a sample of a normally distributed population, then the sample variance ≈the population variance divided by the sample size. Nice. However, if one now increases the sample size to the population, this becomes that the population variance ≈ the population variance divided by the population size, which is absurd. What elementary concept am I missing here? Thanks in advance
 
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nomadreid said:
Given a sample of a normally distributed population, then the sample variance ≈the population variance divided by the sample size. Nice. However, if one now increases the sample size to the population, this becomes that the population variance ≈ the population variance divided by the population size, which is absurd. What elementary concept am I missing here? Thanks in advance

In statistics the idea is that you have a population and are trying to figure out its parameters. So you take a sample to get an estimate.

The population parameters are assumed to be constants. The sample parameters are random variables, because they will vary from sample to sample.

You are also confusing the sample variance with the variance of the sample mean.

The variance of the sample mean (usually) converges to zero, while of course the population variance does not. The sample variance converges to the population variance.

This stuff is confusing, but it is important to get it straight or you will never understand statistics. So good for you for asking.
 
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From ImaLooser
You are also confusing the sample variance with the variance of the sample mean.
You hit the nail on the head! Perfect. I now understand. Thanks very much, ImaLooser.
.
And thanks also to Simon Bridge for replying.
 
Sample variance is NOT equal to population variance divided by sample size.
 
ssd:
Sample variance is NOT equal to population variance divided by sample size.
Yes, I know, that was the absurdity in my mini-proof that something was wrong with the original assumptions. That is, if I make a point that X is wrong because it leads to 1=0, then saying that 1≠0 is missing the point.
 

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