Statisticians have developed a standard distribution of shots fired based on probability analysis. The measurements for circular error probability are 50 percent within "n" distance from the target, 43 percent within "2n" from the target, 7 percent within "3n" from the target and .2 percent of the remaining shots more than "3n" from the target. These numbers fall within the normal distribution and would be expected under typical conditions.
The root mean square is an operation in statistics used to help calculate the probability for the circular error. It is found by taking a series of numbers, squaring them, finding the average of the squares, then the square root of that number as follows:
3, -2, 4, -4 --> 9, 4, 16, 16
Average of the squares is 11.25
Square root is 3.35
This technique was used to determine the normal distribution of the radius using large datasets.
Circular error was actually first noticed by Galileo when he was trying to measure the swinging of a pendulum. He noticed the microscopic differences in the path of the ball and found that there was a regular probability of the flight path. He realized that the reason for the slight change in the pendulum was actually due to the rotation of the Earth.
Precision-guided guns or missiles are excluded from the circular error probability analysis because their shots do not have a normal distribution pattern. In fact, precision-guided shots are able to self-correct in mid-flight, so they do not act according to expected patterns. As more weapons become precision-guided, some aspects of circular-error probability will become less useful to analyzing ammunition results.