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Co-authored-by: Jonas Landsgesell <[email protected]>

doc/tutorials/error_analysis/error_analysis_part1.ipynb

@@ -128,9 +128,9 @@
     "\n",
     "where $\\hat\\sigma_\\overline{X}$ is the estimated SEM, $\\hat\\sigma$ is the estimated standard deviation (see eq. 1) and $N$ is the number of samples. But what does it mean for samples to be uncorrelated?\n",
     "\n",
-    "An example for uncorrelated samples would be the rolling of a dice. The outcome of each trial is completely independent to the previous trials. We might guess any number from 1 to 6, regardless of what has been the last result. The same could be true if we ran an experiment many times independently from one another and measured a quantity each time. By looking at one experimental value, we would'nt be able to predict the next one. The best guess would be simply the mean value of the entire series.\n",
+    "An example for uncorrelated samples would be the rolling of a dice. The outcome of each trial is completely independent to the previous trials. We might guess any number from 1 to 6, regardless of what has been the last result. The same could be true if we ran an experiment many times independently from one another and measured a quantity each time. By looking at one experimental value, we would'nt be able to predict the next one. The best guess would be simply the mean value of the entire series. In the case of rolling a dice, correlations could for example be observed if it was more probable to obtain the same result as in the previous dice roll rather than another result.\n",
     "\n",
-    "However, usually, when you run a molecular dynamics simulation, the particles will only move by a tiny amount during a time step. Consequently, most observables also change only by a small amount during a time step. If we were to sample an observable in every time step, we would get a lot of samples with very similar values. It is said that the samples are *correlated*. Only if we wait for a sufficiently long time, the system will eventually have evolved to a completely different configuration, and we can expect the observable to assume a truly independent, *uncorrelated* value.\n",
+    "Usually, when you run a molecular dynamics simulation, the particles will only move by a tiny amount during a time step. Consequently, most observables also change only by a small amount during a time step and it is, therefore, more probable to obtain a similar result rather than a completely different result. If we were to sample an observable in every time step, we would get a lot of samples with very similar values. It is said that the samples are *correlated*. Only if we wait for a sufficiently long time, the system will eventually have evolved to a completely different configuration, and we can expect the observable to assume a truly independent, *uncorrelated* value.\n",
     "\n",
     "It is often easy to see when samples are correlated. Execute the code cell below for an example, where a small part of `time_series_1` is plotted."
    ]