1
$\begingroup$

In my statistics book, it says that the variance of sampling distribution of sample mean become smaller as the sample size increases. But it doesn't say why it tends to be smaller than the population distribution. Are there any theories or conditions for the tendency?

  • 0
    is narrow a technical term? what do you mean 'under what theory'? Did you mean under what conditions? – 2017-01-26
  • 0
    Also, it doesn't always become smaller. Usually, but not always. – 2017-01-26
  • 0
    When does the variance of sampling distribution of sample means become smaller and when not? I am just reading my statistics book but I want to understand why it becomes smaller. – 2017-01-26
  • 0
    @spaceisdarkgreen: are you say half of $0$ is not smaller than $0$? or half of infinity is not smaller than infinity? or something else? – 2017-01-26
  • 0
    @Henry Yes, half of infinity not smaller than infinity. e.g. the sample mean of the cauchy distribution having the same distribution regardless of $n$. – 2017-01-26
  • 0
    @spaceisdarkgreen: I do see see that as the best example: the Cauchy distribution does not have a mean so cannot have a variance, though its second moment is indeed infinite – 2017-01-26
  • 0
    @Henry I think it's a great example in the spirit of the question (well, before they edited it, anyway). They asked about situations when the sample mean's distribution doesn't narrow. Who cares about the technicality of the word 'variance'. (And to that end I think saying the variance of the cauchy is infinite isn't a bad thing although you're right that it is technically undefined, not infinite) – 2017-01-26

1 Answers 1

2

Two basic properties of variances:

  • The variance of the sum of independent random variables is the sum of their individual variances

  • The variance of a constant $k$ times a random variable is $k^2$ times the variance of a random variable

So if your sample values are the random variables $X_1,X_2,\ldots,X_n$, independent each with variance $\sigma^2$ then

  • $\text{var}\left(\sum X_i\right) =\sum\text{var}\left( X_i\right)= \sum \sigma^2 = n\sigma^2$
  • $\text{var}\left(\overline{X}\right) =\text{var}\left(\frac1n\sum X_i\right) =\frac{1}{n^2}\text{var}\left(\sum X_i\right) =\frac{1}{n^2}(n \sigma^2)= \frac1 n\sigma^2$
  • 0
    According to the two basic properties, the way of getting the variance of sampling distribution of sample means makes the variance of sampling distribution of sample means smaller because the original variance is divided by the sample size? – 2017-01-26
  • 0
    Assuming independent samples, then yes. But I might not the word *because* – 2017-01-26