Thread: A Note on what ES and SD are

Extreme spread is the difference between the highest and lowest velocities observed in a string of shot, usually 5.
The number of observations is usually called n. eg n=5 for a 5 shot string.
Each string of 5 will of course give a slightly different ES result.
Longer strings eg n=10 will usually give a higher ES; shorter strings smaller eg n=1 ES = 0 !
ES is a similar concept to group diameter.

Standard deviation is a characteristic ("parameter") of variables with a "Normal Distribution" which defines the variability.
It is the limiting value of the square root of the sum of the squares of the difference between each value and the mean. (a mouthful, but fairly straightforward when you see it written as a formula. Schoolboy statistics) See wikipedia : https://en.wikipedia.org/wiki/Normal_distribution
Normally distributed variables have more values close to the mean and fewer further away - the classical bell shaped distribution. About 68% of values fall within 1 SD of the mean, 95% with 2 SD and 99% within 3 SD.
The underlying mathematical assumption is that the variability is made up of the sum of many independent factors like neck tension, powder weight, bullet weight, barrel fouling, primer flash and that these are all varying independently from shot to shot. It's probably fairly true for MV and you could check by making a histogram of your MV over a long string.
Anyway, you can estimate the long term "true SD" (technically called "sigma") from a small sample of shots by dividing the sum of squares by n-1 instead of n. This is because you also have to estimate the true mean (called "mu") from the average of the actual shots you've fired, so the difference of each shot's MV from the true long term mean is underestimated. The more shots in your string, the better you can estimate the true SD but the "n-1" estimate is unbiased and doesn't grow or shrink as you fire more or fewer shots. This "Root Sum of squares of differences from observed mean/n-1" idea is called the Student's T distribution. Your observed T SD will vary between one string and the next but is more stable than the ES, which is determined by just two of the shots fired.

Since 95% of shots will fall within +/- 2 SD of the mean, typically 19/20 shots will be in this range. About 80% (4/5) will fall within 1.3 SD. So you can use the SD to easily estimate how often you will have a "wild shot" outside a given desirable range. If your SD is 10 then every 5th shot is likely to be outside a range of +/- 13 f/s. Of course, its not so likely to have the other extreme shot also 13 f/s in the other direction so ES is likely to be somewhat less than 26 f/s in this example. The location of each shot on target is also a normally distributed variable with different SD for vertical and horizontal errors. For hunting we are usually interested in how far away a single flyer will be from your established zero, so this SD gives a very good idea of how likely you are to miss a given sized target at long range due to variation in MV.

Bryan Litz uses this to generate hit probabilities for his Weapon Employment zone analysis. The SD has a great advantage mathematically in that you can add together errors due to other factors like shooter/rifle grouping, wind and so on by using "root mean square" addition and get the SD of all the errors added together. He finds that for ranges like 1000 yd, MV SD is very important to your hit probability. At 300m MV contributes very little and the main thing is how well the rifle and shooter can group.

For a shooter simply trying to reduced MV errors, measuring either ES or SD from the chronograph is OK as they are closely related and the smaller the better for long range.

Interesting topic this

Litz mentioned on a similar thread on one of the yank forums that ES is not necessarily repeatable from string to string, where SD is more likely to be consistent.

I think it is easy to fall into the trap of seeing MV consistency as the be-all, but as several people have pointed out on the other thread, you can have strings with low ES or SD numbers still group like crap. Harmonic etc play an important part as well.

In a pretty simplistic way I look at it like this:

ES or SD give you an indication of reloading quality rather than accuracy/prescision.
Barrel timing and harmonics, along with seating depth will give you increased accuracy once you have consistent loads.

Originally Posted by ebf

Interesting topic this


Barrel timing and harmonics, along with seating depth will give you increased accuracy once you have consistent loads.

@ebf Have you read Chris Long's OBT paper? Optimal Barrel Time Paper

Originally Posted by zimmer

@ebf Have you read Chris Long's OBT paper? Optimal Barrel Time Paper

Yup, read it a while ago when I did some extra reading on OCW, not sure I've completely digested it yet

I know this is an old thread but I'm curious about SD, what is considered to be passable SD for target shooting?

A single figure SD is the aim for long range eg 1000yds target shooting. For 100yds Benchrest where charges are thrown not weighed SD becomes irrelevant.

Interesting subject - could explain the 'frustration' suffered by benchresters and ELR shooters.
For hunters - shooting 5-100M in bush, 'bang,flop' will be fine.

I think a single digit SD is something reasonable to aim for, from memory, the last time I chronographed handloads I had a SD of 6 fps for a 5 round string, which I was pretty happy with. Assuming muzzle velocity variation can be plotted with a normal distribution (a pretty reasonable assumption) then if a SD of 6 fps is accurate then we would expect 64% of shots to fall between 2744 and 2756 fps (2750 being the average), 95% of shots between 2738 and 2762 fps and 99% between 2732 and 2768 fps; one, two, and three SD's respectively. However If I'm zeroed at 2000m for ammo at 2750 fps and one shot is actually 2732 and the next 2768 then one is going to land 32 inches below POI and the other 32 above POI. So even though 6 fps is a fairly small SD, all it tells me is that I can expect I am going to hit within 32" of my POI, 99% of the time, and within 21" 95% of the time, if my aim is perfect, at 2k. So even with perfect aim and perfect wind reading I can still expect 1 in every 20 rounds (5%) to be at least 21" high or low of POI. Hopefully my maths is correct. I thought one SD was 64% not 68%.