# Found an expression I haven't encountered before

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).

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

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

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