# What does “dependency between variables” mean (graphically)?

For an exercise I'm asked to resolve the "dependency" of two variables (data sets) using x-y plots. So I plot to the x-axis the variable that I infer as being the predictor and to the y-axis the one that I infer as being the response.

However,

how am I supposed to infer whether and what kind of dependency there is by just looking at the following plot?

To me it seems like there's some correlation, that increasing mom_iq also increases kid_score, but that this correlation is not "clean", but rather there's a lot of variation in as well. I.e. in some cases kid_score is high (or higher than it should be) even if the mom_iq wouldn't be.

So in this case, where do I strike a line between "there's strong correlation" and "there's weak correlation"?

• Is the question about ''dependency'' or about ''correlation'' ? – user83346 Sep 9 '16 at 12:12
• @fcop Dependency, but I've seen the word correlation being used (also in the Pearson correlation coefficient), when also talking about dependency. – mavavilj Sep 9 '16 at 12:15
• Depency means that, given a value for $x$, the probability of getting a certain value for $y$ depends on $x$. So if you look at $x=80$ for that value of $x$ you look at the fraction of y-values with $y=60$, would that fraction be different for $x=100$ , so for $x=100$ what fraction has $y=60$ , for $x=80$ what fraction of that $y$ has $y=60$ ?? – user83346 Sep 9 '16 at 12:18

Dependence between two variables in general is whenever the following equation does not hold:

$P(x,y)\neq p(x)p(y)$

There are quite a few tests that test this hypothesis HHG, Hoffdings Test and more.

If you are interested in the different types of dependence, you can use Pearson correlation which evaluates the linear relationship between two variables, or Spearman correlation which evaluates the monotonic relationship between two variables.

In the case you presented it seems that you want to say something on the nature of relationship. In order to do so we use Regression. If the assumptions of the regression hold, you can test the Pearson correlation and see if you reject the null that the Pearson correlation equals zero. You can also estimate the coefficients and test them.

In the plot you presented it seems that there is dependence, but is not linear. You could try to fit a polynomial regression.

$Y = \beta_0 + \beta_1 x_1 + \beta_2 x_1^2$

$y$ being the kids score and $x_1$ the mom iq. Then you can evaluate your adjusted $R^2$ which in the one dimensional case coincides with the Pearson correlation. Check if the assumption hold and then see the size and significance of the coefficients.

For a strong correlation to be present, the data points should fall on a line or close to it. In your case, the data points resemble a cloud, therefor the correlation is weak, around 0.4 is my guess by just looking at it. How to interpret a scatter plot visually is explained in most elementary textbooks on statistics.

• Yeah I can get the Pearson correlation coefficient as well and it's around 0.45. But since the range of this coefficient (as well as the number of graphical interpretation) is large, then how can I judge more than just "there's strong correlation" or "there's weak correlation"? Like is 0.45 correlation value totally meaningless in terms of correlation or should it rather be interpreted as "in some cases there's correlation"/something else? – mavavilj Sep 9 '16 at 11:15
• I'm not sure I understand what you are looking for. You say that "how can I judge more than just 'there's strong correlation' or 'there's weak correlation'?". What else is it that you want to know or judge? Knowing that the correlation is 0.45 is certainly valuable information. The Pearson correlation r is a measure of statistical linear relationship for the data set as a whole, not individual data points. – jsb Sep 9 '16 at 11:49
• Well how would I interpret the 0.45? Or the graphical plot for that matter? Is it correct to say that: there's moderate correlation, not too weak nor too strong. How much of correlation is meaningful for talking about there being correlation? – mavavilj Sep 9 '16 at 12:02
• I think it is more accurate to say that there is weak correlation. The word "moderate" I would reserve to correlations above 0.5, and "strong" perhaps above 0.85. I think most textbooks do the same or similar. The Pearson correlation r is just a number between -1 and 1, so even if the correlation is tiny, there is still correlation. – jsb Sep 9 '16 at 12:06
• Note that ''correlation'' and ''dependence'' are not the same, the OP's question is about ''dependency'', correlation implies dependency but the inverse is not true ''in general''. – user83346 Sep 9 '16 at 12:11