Examples of (classical) measurements that are not independent? What are some simple examples of measurements that are not statistically independent, i.e. with nonzero covariance?  I'm looking for real examples that might reasonably come up in an undergraduate laboratory, for the purpose of illustrating experimental covariance.
To clarify:  if N measurements of quantities X and Y are $x_i$ and $y_i$, with means $\bar{x}$ and $\bar{y}$, then the deviations are $x_i-\bar{x}$ and $y_i-\bar{y}$, and the covariance is the correlation of the deviations:  $\lim_{N\rightarrow\infty}\left[\frac{1}{N}\sum (x_i-\bar{x})(y_i-\bar{y}) \right]$.  The two measurements are statistically independent if this is 0.  I'm looking for examples of X and Y where this might not be 0.
 A: Super simple generic example
Think of any quantity which you can measure by more than one technique.
Get one sample where that quantity has value $X$ and another sample where that quantity has value $Y$.
Have the students measure the parameter in a few different ways.
The measurements for sample $X$ will show correlation with one another, and the measurements for sample $Y$ will show correlation with one another.
Electrical example
Get an approximately white noise source (i.e. a hot resistor) and capture a time trace on the oscilloscope.
The low pass filtering of the oscilloscope's analog input causes the measured voltages to be correlated in time.
You can use different filters to show the students that the correlation time of the voltage increases as the filter upper cutoff frequency is lowered.$^{[a]}$
If you have a spectrum analyzer you can relate the frequency domain spectral density to the time domain correlation time.
This is a great demo because the students can visually see the correlation of the measurements on the scope screen.
In this example the mean voltage measured at any particular time is zero.
You can establish this by simply computing the mean voltage from a series of samples on the scope.
With the same dataset you can see that samples close in time are correlated.
I really like this because it shows that a single set of data contains interesting statistical structure.
Mechanical example
Start a pendulum swinging.
Without stopping the pendulum, have the students measure the pendulum's position at equally spaced times.
They will find that the measurements are indeed correlated.
In fact, they can compute the correlation function and will find that it is sinusoidal.
This may seem like a trivial example but it really isn't, especially if you then go to large displacements where the motion is not harmonic!
Like the electrical example, this shows a nice contrast between mean and correlation.
If the students plot a histogram of their measured values they'll find a zero mean (assuming they don't measure with frequency commensurate with the oscillation!), but then if they look at measurements with various time differences they see nonzero correlation.
If you want a correlation which isn't in the same variable at different times (i.e. not an autocorrelation) you can measure the position of the pendulum and the tension in the string (use a spring scale as a tension-ometer).
Something which isn't an autocorrelation
Build an emitter-follower circuit with a BJT.
Measure the base current and collector current on the oscillscope.
Their fluctuations will be correlated.
You could in principle watch the gain change as temperature is varied.
Whether you regard that as correlated drift (which wouldn't really be a cross-correlation about a mean value as described by the formula in the OP) or as a real fluctuation about the mean depends on how you like to think about the time scales in the experiment.
This could actually be a good lesson for the students.
$[a]$: You can build different filters with just an $RC$ circuit. You can also probably just change the scope's input bandwidth, which requires a lot less effort :)
A: Astronomy example
Starting the day after a full moon, observe the time the Moon rises and estimate the fraction of the Moon that is illuminated. The time and illumination will be highly correlated, but the correlation won't be perfect because the relationship is not quite linear. A circular orbit would give a nice linear relationship. The Moon's orbit isn't quite circular, and even the Keplerian approximation of an ellipse isn't perfect because of solar and other perturbations. Correlation only looks for linear relationships.
