# How can this regression graph be explained?

I am trying to understand this graph:

Why does explained deviation lies below the regression line? What else can concluded based on this chart?

• Well, it's only below because the slope is positive and you're looking at $x>\bar{x}$. If you changed one or the other, it would be above (if you changed both it would again be below). Apr 9, 2015 at 11:07

## 2 Answers

If you look at the regression model:

$Y = c_1 + c_2 \cdot X$

This model predicts that your dependent variable $Y$ can be predicted from your independent variable $X$ with a least-squared straight line.

Anything that the straight line tells you is the deviation explained by the model. This is the area below the line in your chart. Similar, the area above the chart is the deviation that your model can't explain. The total deviation is the sum of those deviations.

The present graph seems to be about modeling of a random variable $Y$. Two models are present here: $$Y=\bar{Y}+\epsilon$$ where $\bar{Y}$ is a reference and $\epsilon$ is a noise. The second model is $$Y=c_1 + c_2 X + \xi$$ where $c_1$ and $c_2$ are parameters, $X$ is an explanatory variable and $\xi$ is a noise.

The chart illustrates the comparison of both models: how much is better the regression (model 2) than simple mean (model 1). Observing a realization of $y$ and corresponding $x$, we can calculate the residual from the models - first model $$y-\bar{Y}$$ and second model $$y-(c_1 + c_2 x)$$ These residuals for all available realizations can be aggregated in the form of standard deviation. Numerically, this comparison between the first and second model correspond to the coefficient of determination. See the link to see a very similar picture to yours.