Sample and population variances: elementary question

In summary: I was trying to show that the two statements are contradictory, not that they are both true. Thanks for catching that.
  • #1
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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  • #3
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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  • #4
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.
 
  • #5
Sample variance is NOT equal to population variance divided by sample size.
 
  • #6
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.
 

What is the difference between sample and population variances?

Sample variance is calculated using a subset of data from the entire population, while population variance is calculated using data from the entire population. Sample variance is an estimate of the population variance.

How do you calculate sample and population variances?

To calculate sample variance, you take the sum of the squared differences between each data point and the sample mean, and then divide by the sample size minus one. To calculate population variance, you take the sum of the squared differences between each data point and the population mean, and then divide by the population size.

Why is it important to calculate sample and population variances?

Calculating sample and population variances allows us to understand the variability of data within a sample or population. This information is useful in making statistical inferences and drawing conclusions about the larger population.

What is the relationship between sample and population variances?

The sample variance is an estimate of the population variance. As the sample size increases, the sample variance becomes a better estimate of the population variance.

Can the sample variance ever be larger than the population variance?

No, it is not possible for the sample variance to be larger than the population variance. The sample variance is calculated using a smaller subset of data and will always be smaller than or equal to the population variance.

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