I was reading the book an introduction to statistical learning with R (http://www-bcf.usc.edu/~gareth/ISL/data.html), and came across this expression that I haven't seen before. Can anyone tell me what this means:

$$ {\bar x^{2} \over \sum_{i=1}^n(x_i - \bar x)^2} $$

The full equation (where I saw it is:) $$ SE(\hat \beta)^2 = \sigma^2 [{1 \over n} + {\bar x^{2} \over \sum_{i=1}^n(x_i - \bar x)^2}] $$

FYI. this is the equation for calculating the standard error of the coefficients of linear regression (ML).

  • $\begingroup$ The bottom term is variance, but why is it being divided by mean squared? $\endgroup$ – Mehul Jangir Jul 4 at 11:58
  • $\begingroup$ What is exactly your doubt? Do you think there is something wrong about it? $\endgroup$ – David Jul 4 at 12:03
  • $\begingroup$ I wanted to know what this term means. As stated above, I've never seen it before. $\endgroup$ – Mehul Jangir Jul 4 at 12:06

Your expression is a formula for the standard estimation error for a given parameter $\beta$ in terms of the sample mean, the observations.

$\sigma$ stands for the standard deviation of the errors $\epsilon$, assuming a model of the form $y=\beta + \beta_1 x + \epsilon$

$\epsilon$ is assumed to follow a normal distribution $N(0, \sigma^2)$

The part you highlight is just the square of the sample mean divided by $n*Var(x)$

  • $\begingroup$ by 'it,' you mean the expression, right? $\endgroup$ – Mehul Jangir Jul 4 at 13:19
  • $\begingroup$ Sure! I mean "the expression" $\endgroup$ – David Jul 4 at 13:20

This is an expression that most statistics students are taught to derive when learning linear regression. I'm not necessarily sure that that term has a great interpretation, it's just derived. I would take a look at this thread to see how this term pops up: Derive Variance of regression coefficient in simple linear regression

  • $\begingroup$ Thanks a lot. I haven't done statistics at school yet, self-taught till now so that's probably why I didn't know. $\endgroup$ – Mehul Jangir Jul 4 at 20:45

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