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I'm doing my project about UFOs. I computed the correlation between population (X)~observation (Y) (UFO), and I got this graphic:

enter image description here

Afterward, I ran the cor.test() function in R which told me:

t = 12.92, df = 49, p-value < 2.2e-16,
95 percent confidence interval: 0.7966971 0.9295809,
sample estimates: cor = 0.8792381

Can you please help me understand what I can say about this correlation? Except for cor = 0.879, I don't have a clear understanding of the coefficients.

The main idea was to find the dependency between population of US states and the number of UFO reports per state. On the X axis - population by States, on Y axis- UFO observations for all time in States. As you can see, outliers are - California, Texas and Florida, which have the largest populations and the largest numbers of UFO reports.

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    $\begingroup$ Please provide some context about what you measured & what you're interested in finding out. $\endgroup$
    – Scortchi - Reinstate Monica ♦
    Commented Dec 19, 2016 at 14:02
  • $\begingroup$ We could answer the question you asked as you asked it but your graph raises other issues, at least for me, so it would be better if you provided the information that @Scortchi suggests. $\endgroup$
    – mdewey
    Commented Dec 19, 2016 at 14:05

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First the explanation of the numbers.

The value of $t$ is provided to enable the calculation of the later values. Then you have a $p$ value $<2.2e-16$ is how it prints $2.2 \times{} 10^{-16}$, that is a very small number. So it is saying your correlation is unlikely to have arisen by chance from a population with zero correlation. The confidence interval is such that 95% of the time it contains the true value.

Now some comments about your data

The variables you are using are skewed and it might be better to take logs of both before calculating the correlation. As it stands at least some of the correlation is being driven by the states which have large population and large UFO numbers. A further advantage is that if you carried out more advanced analysis you would probably want to use a model on the log scale.

You refer to outliers but what your graph shows on the present scale is some states which have high leverage (the best fitting line is being pulled to go near them) and one state which is quite some way from that line (the one at $x = 20$ with a low number of UFOs).

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Firstly, you should try your analysis on the log scale. I think your graph will look more smooth and linearly correlated.

cor.test tests whether your correlation is statistically significant away from zero. The result is highly significant at 5% or even 1% confidence level. This is also reflected in your t-statistic, which is quite high.

Overall, you can confidently state your variables are strongly linearly correlated (even more on the log scale).

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