As discussed in the Introduction to Variation Write Up, variation is inherent in everything to do. It is up to use to determine what level of variation is acceptable and what is not. In a production setting it is not practical nor is it necessary to eliminate all variation. Nor is is practical or alway possible to measure every aspect of a product to ensure conformity, which is why statistics is used.
A classic example of this is the bullet seating operation. When seating the bullet the stem typically contacts the ogive of the bullet and presses it into the case. There are various mechanisms that can cause variation in how deeply the bullet is seated into the case. Someone seated at their reloading bench may begin to get frustrated if he sets his target coal at 2.188in and painstakingly gets the first round he loads to that Cartridge Overall Length (COL), only to find that the second round he loads is 2.192in.
What is the normal reaction here? Well if he’s a perfectionist, he’ll adjust the die in order to seat the bullet .004in deeper. However the next bullet he loads he gets 2.182in. Now he’s got to get his bullet puller out… this is the result of not understanding nor allowing for variation.
A better method is to load five rounds, and measure them, take the average COL and if needed make adjustments to the die. Then track the COL over time. Accept the fact that sometimes COL will come in above the mean, and sometimes it will fall below the mean. We can use basic statistics to determine when variation is a problem, and when it should be accepted.
In order to deal with variation tolerances are set. This tolerances should loose enough to allow for natural variation, but tight enough that they will not cause malfunctions or quality control issues. For ammunition a good place to start is the SAAMI Specs for the round, these are the loosest tolerances that should be set, it is fine to go tighter, but setting specs outside of SAAMI can cause quality and safety issues.
With the Tul Ammo, the measured max COL is 2.195in and the measured Min COL is 2.181in. This falls within SAAMI specs. However this is a small sample 40 rounds out of a lot that is likely 1,000’s of rounds in number. What are the odds we may see a round exceed the 2.200in spec? For this we need to look at the Standard Deviation for the set. In this case it is .0037in.
Remember standard deviation is a measurement of variation, the 68% – 95% -99.7% rule applies. 68% of the rounds should fall within 1 standard deviation of the mean, 95% will fall within two standard deviations of the mean and 99.7% will fall within 3 standard deviations of the mean.
Standard Deviations establish the “Upper and Lower” Control limit for the data set. Variation usually becomes interesting when it exceeds the upper or lower control limits. Typically these are set at two or three standard deviations from the mean. An upper control limit (UCL) is calculated by taking the Mean and adding 3x the Standard deviation.
Mathematically it looks like this:
UCL = Mean+3*St Dev. In this example the upper control limit is 2.1991in.
A lower control limit (LCL) for this data set would LCL = Mean -3* St. Dev. or 2.1769in.
What this tells us is that 99.7% of the rounds loaded should be under 2.1991in in length. Statistically there would be 3 rounds out of 1000 that could be above this limit, they still may fall below the 2.200 SAAMI max spec, but they are above the upper control limit. Variation becomes a problem when is exceeds the Control Limit and it exceeds the Spec Limits.
To go one step further and assess how many rounds will fall outside of the spec limits, in this case the SAAMI spec of 2.200, we need to take a look the Process Control Index often referred to as the Cpk #. This number takes into account the UCL and LCL and its relation to the Mix and Max specs to calculate how many defective parts a process may produce.
(Note: I have purposely left out the mathematical notation for Mean, and St. Dev to help those who aren’t as familiar with the notation)
The equation for Cpk is:
Cpk = min[USL-Mean / 3*Std. Dev , Mean – LSL / 3*Std Dev]
In our example the equation works out as follows:
Cpk = min[2.200-2.188 / 3*.0037 , 2.188-2.150 / 3*.0037]
Cpk = min[1.08 , 3.42]
Cpk = 1.08
A Cpk value on it’s own doesn’t tell you much, but it does represent how many defects can be expected on a part per million (PPM) basis. In our case we can expect 1,370 cartridges to exceed the 2.200in max COL Spec based on the chart below. This works out to ~99.73% of ammunition produced will be acceptable.
This write up is a crash course on using some basic statistical tools to look at ammunition quality. We primarily looked at COL, but it can be applied to just about any dimensional aspect of the round. We can apply it to powder charge, case weight, bullet weight, case length, production rates…if you can measure it, you can do this type of analysis.
If you want to find out how consistent your powder thrower is, throw 10 charges, and by using Standard Deviation and the 68%-95%-99.7% rule, you can look at how frequently the thrower may throw under or over charges. You can even set up specs and say you are ok if it is +/- .5gr of your mean. By using these methods you can determine the likelihood and the frequency you can expect to exceed that standard.
There is so much more that can be done with this, that I have not yet covered. There are Individual Moving Range (IMR) Charts that display the data in ways that will help to visually identify patterns or variation that is beyond the Upper or Lower control limits referred to as “Special Cause” variation. We will look at this stuff in future writes ups. For now I hope this provides a useful tool for those who want to assess the quality of their ammunition, and use some basic statistical tools to understand the occurrence of potential defects.