Honestly trying to understand the difference here, and admittedly need to read more into ES and MR... but if I shoot 100 rounds in a 2moa group, my ES is 2moa, and I know I have a 100% chance of hitting a 2moa target (excluding outliers like weather, ammo lots, etc). If I shot 100 rounds with 95 of them inside 2moa, and 5 of them outside of, say 2.5moa... then I have a 95% chance of hitting a 2moa target.
Isn't the ES telling me what I can hit with 100% certainty the more valuable metric in the field as a hunter?
Just trying to see both sides here.
Both ES and MR are metrics that aim to quantify the dispersion of your system. As Form points out, ES is better for getting a quick measurement in the field to have a simple and practical way to gauge the limits of your system given a big enough group. MR, in terms of what it tells you about your system, is better than ES. It is using all the information from your group to represent the underlying distribution (the Rayleigh probability density function), so you can do a lot more with it than ES and have more confidence in your estimates.
For practical purposes, if you did what you state in your example on a big group, you’d be fine (even though the true percentages are different from what you state). But if you really want to evaluate your system well and be able to answer questions like, “With what probability will I hit an A inch target at B yards?”, you use MR, not ES (you can approximate with ES but it’s noisier and throws out a lot of data).
I just ran some simulations and plotted them to explain better. All we have to assume for this is that the the x and y coordinates of each impact follow a Normal distribution. We get to choose the parameters of this distribution for the simulation, so here I chose the “truth” to be that this system will put a shot in a 1.5 MOA target with 99% probability. Now we run random simulations of varying group sizes (each different group size is a new row; each column is a repeated trial with that same group size). Imagine each plot here is your paper target after shooting the same rifle at each target to get a group. And for the sake of the example, imagine the target is a 100 yards away and that’s a 1.5 inch diameter black circle you’re shooting at.
Because we designed the simulation, we know the true diameter of the circle we’ll impact within at 99% probability (1.5 inches). We can calculate the mean radius from the actual shots on target and then use those to estimate a 99% probability circle (we don’t know the “truth” about our system as a shooter, we only know our shots on paper, and we do our best to estimate the truth). We can also measure the extreme spread.
You can see a few things: 1) predicting via MR gets you much closer to the truth much faster. At only 10 shots, the predictions are very close to 1.5 inches. It takes ES more shots to get close. 2) MR converges. It approaches the truth and with more shots only gets more precise in its prediction. ES is not stable, and artificially inflates (starts too small and keeps getting bigger). 3) When ES is wrong (which it often is at common group sizes <30), it is usually being overly optimistic about how accurate the system is. So the argument of “prepare for the worst” actually favors MR.
There’s some other neat stuff in the plots, but those are the main ones. Overall, yes bring a ruler to the field and get a quick practical ES measurement on a big group. But understanding what’s happening under the hood and how ES can mislead you is helpful. It’s all manifestations of probabilities and randomness. A “1 MOA” rifle statement begs the question of “at what confidence level and across how many shots”. You can take shortcuts with ES and get by just fine if you do it right, but using MR will yield higher quality probability estimates. It comes down to defining your objective and understanding the pros/cons of the statistical tools you have.
Lastly, after all this math dorking and egg heading, if you want to know how you’ll shoot at distance, nothing beats shooting at distance.
