# Why does this set of data have no covariance?

My understanding of how covariance works is that data that are correlated should have a somewhat high covariance. I've come across a situation where my data looks correlated (as shown in the scatter plot) but the covariance is near-zero. How can the covariance of the data be zero if they are correlated?

import numpy as np
x1 = np.array([ 0.03551153,  0.01656052,  0.03344669,  0.02551755,  0.02344788,
0.02904475,  0.03334179,  0.02683399,  0.02966126,  0.03947681,
0.02537157,  0.03015175,  0.02206443,  0.03590149,  0.03702152,
0.02697212,  0.03777607,  0.02468797,  0.03489873,  0.02167536])
x2 = np.array([ 0.0372599 ,  0.02398212,  0.03649548,  0.03145494,  0.02925334,
0.03328783,  0.03638871,  0.03196318,  0.03347346,  0.03874528,
0.03098697,  0.03357531,  0.02808358,  0.03747998,  0.03804655,
0.03213286,  0.03827639,  0.02999955,  0.0371424 ,  0.0279254 ])
print np.cov(x1, x2)

array([[  3.95773132e-05,   2.59159589e-05],
[  2.59159589e-05,   1.72006225e-05]])


• Hint: What happens when you look at the correlation? What's the difference between covariance and correlation? – aleshing Dec 22 '17 at 4:42
• If you are measuring numbers which appear small or close together on a particular scale, then differences between them will also seem small, and the products of differences seem even smaller. Try multiplying all your data by $1000$ and then redoing the calculations; the covariance should be $1000000$ times as large – Henry Dec 22 '17 at 8:38

The magnitude of covariance depends on the magnitude of the data and how close those data points are scattered around the mean of that data. It's easy to see when you look at the formula:

$cov_{x,y}= \frac{\sum(x_i-\bar{x})(y_i-\bar{y})}{n-1}$

In your case, the deviance of the x1 and x2 data points to the mean of x1 and x2 are:

x1-mean(x1)
[1]  0.006043341 -0.012907669  0.003978501 -0.003950639 -0.006020309 -0.000423439  0.003873601
[8] -0.002634199  0.000193071  0.010008621 -0.004096619  0.000683561 -0.007403759  0.006433301
[15]  0.007553331 -0.002496069  0.008307881 -0.004780219  0.005430541 -0.007792829

x2-mean(x2)
[1]  0.0039622385 -0.0093155415  0.0031978185 -0.0018427215 -0.0040443215 -0.0000098315
[7]  0.0030910485 -0.0013344815  0.0001757985  0.0054476185 -0.0023106915  0.0002776485
[13] -0.0052140815  0.0041823185  0.0047488885 -0.0011648015  0.0049787285 -0.0032981115
[19]  0.0038447385 -0.0053722615


Now if you multiply those two vectors with each other you obviously get quite small numbers:

(x1-mean(x1)) * (x2-mean(x2))
[1] 2.394516e-05 1.202419e-04 1.272252e-05 7.279927e-06 2.434807e-05 4.163041e-09 1.197349e-05
[8] 3.515290e-06 3.394159e-08 5.452315e-05 9.466023e-06 1.897897e-07 3.860380e-05 2.690611e-05
[15] 3.586993e-05 2.907425e-06 4.136268e-05 1.576570e-05 2.087901e-05 4.186512e-05


Now take the sum and devide by $n-1$ and you have the covariance:

sum((x1-mean(x1)) * (x2-mean(x2))) / (length(x1)-1)
[1] 2.591596e-05


That's the reason why the magnitude of the covariance doesn't say much about strength of how x1 and x2 co-vary. By standardizing (or normalizing) the covariance, that is dividing it by the product of the standard deviation of x1 and x2 (very similar to the covariance, i.e. 2.609127e-05),

$r=\frac{cov_{x,y}}{s_x s_y} = \frac{\sum(x_1-\bar{x})(y_i-\bar{y})}{(n-1) s_x s_y}$

you get the high correlation coefficient, of $r=0.99$, which confirms what you can see in your plot.

Let's talk about what can be seen from a quick glance at the plot and some reasonableness checks (these are the sort of things one can do as a matter of course when looking at data, simply being armed with a few basic facts):

However, first let's note that the $n$-denominator version of standard deviation can't exceed half the range (the $n-1$ denominator version can, but with more than a few observations not by much).

The ranges on both variables are on the order of 0.02 (roughly) so the variances should be no more than about half that, squared, or about $10^{-4}$.

Consequently, the observed values of the variances in your output make sense; they are both less than that, but more than a tenth of it.

The absolute value of the covariance must then be no more than the geometric mean of the two variances (otherwise the correlation could exceed 1). So the absolute value of the covariance should not exceed $\frac14$ of the product of the ranges.

So if the range of both variables were both close to $0.02$, we couldn't expect the absolute covariance to exceed $(0.02)^2/4=10^{-4}$.

From that very rough analysis, nothing looks surprising.

A more precise analysis comes from actually doing the calculations using more accurate ranges and then thinking about the shapes of the marginal distributions:
the ranges are just under $0.023$ and $0.015$ respectively, so the covariance should not exceed $8.6\times 10^{-5}$, but since the marginal distributions are not nearly-symmetric-two-point distributions it must be quite a bit less than that.

Indeed, if we say they're not so far from uniform, the covariance would be bounded by something nearer 1/12 the product, rather than 1/4 -- i.e. for roughly uniform variates with those ranges it would be less than about $2.9\times 10^{-5}$ -- but not a lot less because the correlation is high.
[These variates aren't uniform - they're left skew - but it's close enough for our present purposes.]

So just from looking at the range of each variable and the rough sense of the marginal distributions and correlation in the plot, I'd expect the covariance to be a bit less than $2.9\times 10^{-5}$. It is actually about $2.6\times 10^{-5}$.

(Not so bad for a quick back-of-the-envelope calculation starting with ranges to two significant figures!)