So, out of my curiosity, I wanted to find out if being religious has a correlation with IQ. I say that it does not. Time and time again people view data, look at the possible trend, and then assume that it's completely true. Some parts of this data are stupidly subjective; for example, one person's "contentment" ranking is not the same as what someone else would rate it as. It's a stupid measure, but I digress.
I did not check any of the data to see if it were accurate or how it was gathered, as it takes too much time and I don't feel like spending that much time trying to figure out how it was collected (even though I really should). Therefore, for lack of better data at hand, let us make a very, very loose assumption that all of the data are correct.
Let us start off with the big one: IQ vs. religiousness.
Here is the one-way ANOVA of the data:
Notice that the SSE (Sum of Squares error) takes up a HUGE majority of the total; it is over 3 times that of the SSM (Sum of Squares Model). We can calculate R^2 from this, but it is given to us anyway. Notice that R^2 = .2308. This says that only 23.08% of the data are described by a least squares regression line. That is very little data; it does not describe it very well in any way. We can safely say that IQ is not correlated with being religious. Notice that the p-value is 0.0002; it is statistically significant from this data that religiousness affects IQ (H0: mean=0 vs. Ha: mean does not equal 0); we can easily reject the null at an alpha of both 0.01 and 0.05; however, due to the fact that only 23.08% of the data are described by a trend line, it is difficult to accept that answer and a such we cannot say that IQ is not very affected by being religious. Remember, R^2 doesn't care what is x or y; I can swap religiousness and IQ on the axes and get the same R^2 value.
Let's look at this one in some more detail. Here is the residual plot of IQ vs. religiousness:
Poor. Look at the obvious linear relationship on the residuals plot. It is not evenly distributed in any way about the residuals=0 line; as a result, we cannot use a linear line to measure this. Let's look at the plot to get some more info:
Can you make a good approximation for any of that data with a trend line? Obviously not; the data are extremely random. There is virtually no relationship between IQ and being religious.
What about political affiliation? Is that affected by being religious, according to the data?
That F is huge. 62.38 is phenomenally large. We get a p-value of way, way less than 0.001. Note that in these one-way ANOVAs with one degree of freedom due to the single independent variable, the p value from t is exactly equal to the t value from the ANOVA F test. F of 62.38 is hugely statistically significant at 0.05, 0.01, and even 0.001, and with that R^2 value describing 55.61% of the data, we can pretty safely reject H0 and say that being religious tends to correlate with your political affiliation. In this case, it is conservative.
And what about theft?
Only 16.96% of the data are described, but the p-value is 0.0017. It's statistically significant at 0.05 and 0.01, but the data are practically randomly scattered and this isn't a good model. So no, your religiousness and amount of theft that you do is uncorrelated.
I will not go into detail with the others, as it will take too much time to perform the contrast analyses. So, instead, let's consider what happens if we put all of these variables together such that, mathematically, have an 8th dimension linear equation where we measure how religious you are vs. all those other factors.
Note that R^2 is 0.8099, and the F-test produces a value less than 0.0001. Let's look at the partial linear equations:
IQ is 0.5730, divorce is 0.5649, and the rest are all statistically significant at at least 0.05. The partials of IQ and divorce are incredibly poor indicators; so, they do not have much to do with being religious at all; this definitely follows with what we have seen thus far with IQ, and the divorce rate can be explained through its own analysis that I will not get into for reasons I have stated above.
This states only if those factors influence your religiousness. Now let's take a major factor, say, theft, and see if all of the factors affect it:
Here's a very interesting case of Simpson's Paradox! All of the variables themselves alone do not influence it very much, but the factors combined do! The R^2 is .4451, which is decent for describing it but not excellent. Now, let's see how much theft is affected if we remove religiousness, thus creating a 7th-dimensional linear equation (y=x1 + bx2 + bx3 + ... + bx6 + e, e -> N(0, sigma)).
R^2 = 0.4553, and the rest of the values are hardly affected. It, in fact, made it more accurate without religiousness being there. What does this say? Relgiousness has little to do with theft rates. Note how R^2 went up!
Let's check out IQ.
A very good R^2 of 0.7913, and good p-values for the most part. That is, except religion and a few others.
Let's take religion away:
Well now, the R^2 value is now 0.7952. A very slight increase, and a very slight change in the other p-values, too. That's hardly a difference at all. Interesting, is it not?
So, in conclusion, don't look at graphs that show pretty colors and assume there is correlation. People skew data all the time; you should ask yourself how biased the data are each time you see something. This entire graph proves itself wrong. It just puts on a pretty face to make it seem like there is a correlation when there really is not. Good game. Statistics 1, biased morons 0.