Standard Deviation Question

[686 Shooter]

Good day all. I have been reloading for years but have just recently gotten into trying to be as precise as I can for some target shooting and long range shooting. I am off to a pretty good start I think but I have a question regarding the calculation of standard deviation.

What is the correct way to calculate SD for bullets, is it the N or N-1 calculation?

 

[Papa Charlie]

N-1

If you are using Excel the equation is =STDEV.S(Range)

 

[686 Shooter]

N-1

If you are using Excel the equation is =STDEV.S(Range)

Thanks Papa Charlie. I think my Labradar uses the N calculation.

 

[Keith Glasscock]

It comes down to which standard deviation are you trying to derive, and for what purpose?

For determining the standard deviation of the sample taken (the actual shots fired), then it is N.

For estimating the standard deviation of the entire population of shots that are loaded to that same load, without having to fire them all, then it is N-1.

In the end analysis, we use SD to get a feel for the consistency of our load. Either one will serve that purpose as a lower number is more consistent than a larger one. So long as the same calculation method is used repeatedly, one can ignore the actual value, but, instead, look at the differences between loads.

That's enough math for one day...

 

[SGK]

Check your calcs against this handy calculator:

https://www.autotrickler.com/stats-calculator.html

The confidence intervals are important to interpreting results

 

[dcali]

You can read all about the theoretical differences in a stats book, but at the end of the day, n is pretty close to n-1 when you have a big enough n to make sense of. If it matters, you need to shoot more.

 

[Webster]

It comes down to which standard deviation are you trying to derive, and for what purpose?

For determining the standard deviation of the sample taken (the actual shots fired), then it is N.

For estimating the standard deviation of the entire population of shots that are loaded to that same load, without having to fire them all, then it is N-1.

In the end analysis, we use SD to get a feel for the consistency of our load. Either one will serve that purpose as a lower number is more consistent than a larger one. So long as the same calculation method is used repeatedly, one can ignore the actual value, but, instead, look at the differences between loads.

That's enough math for one day...

It's been to long since I have read up on this. It's obvious that the smaller the STDEV the better, isn't there also a calculation that shows how many N values you need to achieve a certain accuracy in the STDEV? Seems like I read somewhere that you may need at least a hundred N values or more in some instances to get good numbers. Determining the STDEV on 5 or 10 shoots may be meaningless regardless of what the numbers are.

 

[SGK]

That’s where confidence intervals come into play. Take a look at the calculator I linked to above and insert some numbers (real or made up). You can calculate the mean and SD of your sample set but how confident can you be that these calculations represent the true population you are sampling? You will see that you can only be, say, 95% confident that the true mean or SD is between a and b. It is these ranges that are meaningful statistically, not the simple sample set mean and SD. To have greater confidence you need more samples. Any chrono which outputs statistics should also output the confidence intervals associated with them in order to be helpful.

 

[dcali]

That’s where confidence intervals come into play. Take a look at the calculator I linked to above and insert some numbers (real or made up). You can calculate the mean and SD of your sample set but how confident can you be that these calculations represent the true population you are sampling? You will see that you can only be, say, 95% confident that the true mean or SD is between a and b. It is these ranges that are meaningful statistically, not the simple sample set mean and SD. To have greater confidence you need more samples. Any chrono which outputs statistics should also output the confidence intervals associated with them in order to be helpful.

It would be fun to see the reactions of shooters it chronographs *only* reported the 95% confidence intervals. Lots of lightbulbs would turn on when people figured out that their SD of 5 fps isn’t actually 5 fps.

 

[Ned Ludd]

We use velocity data such as "SD" and "ES" as readouts to gain some expectation of how a given load will perform over longer strings of fire. With the number of shots typically fired per group (maybe 5 to 10 shots total), there is a limit to the confidence of any predictions made using these outputs. Nonetheless, they provide a useful measure to evaluate load performance. People use them simply because they can serve a useful purpose, as long as the caveats associated with their use are understood.

Some people believe that ES data, which will only contain two values, no matter how large the individual shot string, is of little use. In fact, I generally find ES data to be of more use than SD in terms of predicting or estimating the vertical of Hi/Lo shots from a string at some distance.

 

[dcali]

We use velocity data such as "SD" and "ES" as readouts to gain some expectation of how a given load will perform over longer strings of fire. With the number of shots typically fired per group (maybe 5 to 10 shots total), there is a limit to the confidence of any predictions made using these outputs. Nonetheless, they provide a useful measure to evaluate load performance. People use them simply because they can serve a useful purpose, as long as the caveats associated with their use are understood.

Some people believe that ES data, which will only contain two values, no matter how large the individual shot string, is of little use. In fact, I generally find ES data to be of more use than SD in terms of predicting or estimating the vertical of Hi/Lo shots from a string at some distance.

ES contains more than two shots. It also contains the information that every other shot was between them. I think it’s a fine number to use.

 

[SGK]

It would be fun to see the reactions of shooters it chronographs *only* reported the 95% confidence intervals. Lots of lightbulbs would turn on when people figured out that their SD of 5 fps isn’t actually 5 fps.

An example from yesterday. 308 Win. 4 shot string. Velocities as measured by Two Box Chrono.

2735
2734
2728
2729

Avg 2731.5
SD 3.51
ES 6

Pretty good...except we can only be 95% confident that the SD is somewhere between 2.0 and 13.1 (the latter suggesting an ES of at least 78). If I add another shot at 2730 (just made up) to take the sample set N to 5, we can only be 95% confident that the SD is somewhere between 1.9 and 9.0. A 10 shot string would increase confidence in a much tighter range.

 

[SGK]

ES contains more than two shots. It also contains the information that every other shot was between them. I think it’s a fine number to use.

One problem with ES is that it contains no information as to how likely the next shot is to be within that range of the average.

as long as the caveats associated with their use are understood.

Indeed

 

[wkdickinson]

When I took my Six-Sigma training, our Master Black Belt had a favorite saying, "the beauty of statistics is, you will never have to say - I'm sure". The whole point of the science of statistics is to attempt to draw conclusions about a population from a relatively small sample of that population. Obviously, the larger the sample, the higher confidence level (to a degree).

In my work I was often blessed with having the data from the entire population to analyze. Most of the time the statistical projection at 95% confidence level proved to be reasonable accurate. However, not always. The other thing you have to remember is that in order for your statistical projection to be valid, your sample has to be "random". So if you load a 100 rounds, you need to stir up the pail, then reach in and pull out 20, randomly.

So, the way I look at it is - if you have 100 test loads available, the SD from a 100 shots is much more representative of the "infinite" population than 20. But then again, I am not a statistical expert, by any means.

 

[Ned Ludd]

One problem with ES is that it contains no information as to how likely the next shot is to be within that range of the average.

I personally don't view that as a problem. I 'm more concerned about the extreme outliers that often cause the most problems, not the bulk of shot velocities that generally fall much closer to the mean. I only need a rough idea when developing the load of the extreme hi/lo to decide whether it's ready to go, or needs more work. ES is a very expedient way to do that, as I described more in detail in the following post.

 

[Ned Ludd]

ES contains more than two shots. It also contains the information that every other shot was between them. I think it’s a fine number to use.

Although by definition all the data points will lie within the ES, ES itself is determined solely by two values. The values that lie in between may not necessarily have much usefulness. As an example of this, consider the following velocity data set:

2840
2820
2820
2820
2820
2820
2820
2820
2820
2800

Avg. = 2820
ES = 40
SD = 9.4

This 10-shot string has a single digit SD, which some might at least consider adequate, if not quite optimal. I would not knowingly use such a load in a match, it needs work. The 8 numbers that fall in between the Hi/Lo determinants of ES contribute little to making such an evaluation.

Although I specifically chose this example data set to underscore the point, it is not that unrealistic. It is not at all uncommon to observe velocity data sets where the majority of individual values are clustered together somewhere near the mean, with only one or two outlying values contributing to excessive (unacceptable) ES. In a match, I am not concerned with the 17 or 18 shot velocities that fall close to the mean. If I point the rifle in the correct direction for whatever the wind is doing at that moment, they will generally stay inside the 10-ring. It is the velocity outliers that are likely to cause 9s out the "corners", which more often than not mean the difference between winning and not winning. I personally find it easier to think about the outliers from a sample distribution in terms of ES, rather than SD, because they will generally fall much closer to the extremes of the overall distribution. The same information can effectively be obtained using either value, but for me, the ES usually tells the tale at a glance.

 

[SGK]

Your most powerful statistical tool is confidence intervals. It gives you a range for underlying average and underlying SD and from that you can derive a range for underlying ES. In fact they are the only statistics that have any worth. It really is rather stupid that chronographs don't compute them for us.

Two shots can tell you the load is bad. If it is you may as well not shoot a third, except for practice. But it will take 10 or more shots to have statistical (rather than gut) confidence a load is good.

Once one has confidence intervals under their belt they can use them to make decisions about whether two sets of sample data actually depict different underlying behaviour rather than 'more of the same'. I highly recommend anyone reads Adam's blog post on the subject which helps put the use of statistics in a shooting/load development context.

 

[CharlieNC]

It is easy to get overly consumed with large sample sizes and 99.7% confidence, kind of like loading charges to .0001 grain, past the point of a meaningful return vs other factors which are practically more worthy of the effort . Usually n=10 provides an adequate estimate for the SD, and rarely beyond 20 is worth the effort.

 

[Webster]

It is easy to get overly consumed with large sample sizes and 99.7% confidence, kind of like loading charges to .0001 grain, past the point of a meaningful return vs other factors which are practically more worthy of the effort . Usually n=10 provides an adequate estimate for the SD, and rarely beyond 20 is worth the effort.

Every shoot is at a target.

 

[rsmithsr50]

just remember the formula /process is base on LARGE data sets and as a rule we DO NOT shoot large data sets(1000 or more).
its nice to know it is typically good when small, it is better when the group on the target is also small.
all uncontrolled due to weather