Glossary

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Revision as of 03:21, 1 June 2015 by Herb (talk | contribs) (Mathematical Notation: messed up Rayleigh correction factor)
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Terms

Accuracy
Of a gun: The distance between the point of aim and point of bullet impact. Smaller differences reflect greater accuracy. See What is Precision?#Precision vs Accuracy
Colloquially interchanged with Precision.
Bivariate Distribution
Circular Error Probable (CEP)
CEP(p), for p ∈ [0, 1), is the radius of the smallest circle that covers proportion p of the shot group. When p is not indicated it is assumed to be 50%, so by default CEP is the median shot radius.
Chi-squared (\(\chi^{2}\)) distribution
Closed form
Confidence Interval
A range about an Estimate associated with a Confidence Level K. The correct way to read these values is to say, "If we repeatedly computed this Estimate using this same sampling method, we would expect the Estimate to fall within the Confidence Interval K% of the time." Therefore, the smaller the Confidence Interval the closer our Estimate is likely to be to the true value.
Confidence Level K
The probability associated with a Confidence Interval. K ∈ (0, 1), and we expect that in n' estimates n / K' will fall within the Confidence Interval.
Covering Circle Radius (CCR)
Degrees of freedom
Dispersion
A term to denote that shots are spread around the Center of Impact (CoI) without regard to any particular statistical model or measure. Shot groups that have a small spread around the CoI have low dispersion, and shots groups that have a large spread around the CoI have a high dispersion. Thus high precision implies low dispersion. In this mathematical sense precision and dispersion are inversely related.
Estimator
A formula or algorithm for estimating the true value of population parameter from samples of the population
Extreme Spread
When discussing precision, and throughout this site, this refers to Group Size, a Range Statistic defined as the maximum distance between any two shots on a target.
In shooting sports it can also refer to the difference between the highest and lowest value recorded for muzzle velocity.
Figure of Merit (FOM)
Gaussian Distribution
Synonym for Normal Distribution which is the preferred term.
Group size
Center-to-center distances between the two widest shots on a sample target. Because it is measured from center points it is independent of the caliber of the projectile. In practice this measurement is typically produced from the outside edge of projectile holes in a target and then the projectile caliber is subtracted.
Mean
the mathematical average.
The mean of a sample of n values from the population of values (i.e. the set of all values), \(\lbrace x_1, x_2, x_3, ..., x_n\rbrace\), is defined as \(\bar_x \equiv \frac{1}{n} \sum_{i=1}^n x_i \). In the limit as n approaches infinity, then \(\bar{x} approaches \)\(\mu_x\).
The mean value for the population, as opposed to the mean of a sample, is denoted by \(\mu_x\).
Mean Diameter (MD)
Twice the Mean Radius
Mean Radius (MR)
Mean value of the Radius of shots on target.
Median
The median is the value for which half the measures are smaller and half are larger. For example for a discrete set of five values ranked from samllest to largest, then the 3rd value would be the median. For a probability distribution function the median would be the value at the 50th percentile.
Minute of Angle (MOA)
Synonym for Minute of Arc, i.e., one arc minute
Mode
For a continuous distribution, the mode is the peak value. For a sample of values Mode is the value which occurs most frequently (typically from a histogram of the data).
Mil
One thousandth.
Milliradian, a.k.a. mrad or milrad
Normal distribution
POA
Point of Aim
POI
Point of Impact
Probability density function (pdf)
Precision
A term to denote how shots are spread around the Center of Impact (CoI), generally in reference to a particular statistical model or measure. Shot groups that have a small spread around the CoI have high precision, and shots groups that have a large spread around the CoI have a low precision. Thus low precision implies high dispersion. In this mathematical sense precision and dispersion are inversely related.
Radial Standard Deviation (RSD)
Rayleigh Distribution
Sample
Sample is used in two different contexts, so careful reading may be required. First a sample could mean a set of shots on a target. Second it could mean the measurements made on a set of such targets.
Shooter's MOA (SMOA)
Angular measure defined as one inch at a hundred yards.
Target
In this wiki target has the implicit notion of one or more shots taken at the same POA. There are of course paper targets with multiple bulls-eyes per paper sheet. For such a paper target each bulls-eye would be a different target.


Mathematical Notation

Variable Name Definition Formula
\(\bar{ }\) as \(\bar{h}\) "Bar" Denotes a sample average
\(\hat{ }\) as \(\hat{\sigma}\) "Hat" Denotes an statistical estimator. For example \(\sigma\) would be calculated directly from the set of data in the usual way, but \(\hat{\sigma}\) would calculated some other unusual way.
\(\mu_h\) Mean horizontal position of the population Mean horizontal position for all possible shots as opposed to a sample of n shots. In essence the mean horizontal position of the population is a theoretical abstraction which is unknowable. \(\frac{1}{n-1} \sum_{i=1}^n h_i \) converges to \(\mu_h\) , as n approaches \(\infty\)
\(\mu_v\) Mean vertical position of the population. Mean vertical position for all possible shots as opposed to a sample of n shots. In essence the mean vertical position of the population is a theoretical abstraction which is unknowable. \(\frac{1}{n-1} \sum_{i=1}^n v_i \) converges to \(\mu_v\) , as n approaches \(\infty\)
σ Greek letter sigma, Population Standard Deviation The true population's standard deviation. For a set of n values, \(\lbrace x_1, x_2, x_3, ..., x_n\rbrace\), as n approaches infinity, then sample standard deviation s approaches the population standard deviation, σ. \(\sigma = \sqrt{ \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2 }\), as n approaches \(\infty\)
\(c_B(n)\) Bessel Correction Factor Multiplicative factor which adjusts the sample variance to use (n-1) instead of (n) which reduces the bias of the estimate. \(c_B(n) = \frac{n}{n-1}\)
\(c_G(n)\) Gaussian Correction Factor Multiplicative factor which adjusts the standard deviation to correct bias due to taking the square root of the variance. Thus the Bessel correction corrects the variance itself, but taking the square root introduces a different bias.
\(c_R(n)\) Rayleigh Correction Factor Multiplicative factor which adjusts the Rayleigh shape parameter to reduce the bias of the estimate. \(\frac {4^n n!(n-1)!\sqrt{n}} {(2n)!\sqrt{\pi}}\)

where n is the number of shots on the target

h Horizontal position or axis of a target. Typically a target would be mounted vertically and the line of fire would be horizontal. Thus the horizontal axis of the target would be synonymous with the X axis when using X-Y Cartesian coordinates for the target.
\(\bar{h}\) Average horizontal position of a sample of shots Average horizontal position of n shots. The n shots are a sample of shots from the infinite population of possible shots. \(\bar{v} = \frac{1}{n-1} \sum_{i=1}^n h_i \)
\(h_i\) Horizontal position of shot i The shot is the ith shot of a sample of shots. Typically n shots in the sample.
\(\mathcal{N}(\mu,\,\sigma^2)\) Normal Distribution This denotes the normal distribution with mean, \(\mu\), and variance \(\sigma^2\).
r Radius Here this almost always refers to the distance of a shot on target from the center, or sample center, of a target group.
\(\bar{r}\) Mean Radius Mean radius of a sample of n shots. \(\bar{r} = \frac{1}{n-1} \sum_{i=1}^n r_i \)
\(r_i\) Radius of ith shot Unless otherwise stated it is implicit that the polar coordinate center is at the mean POI.
  • If the target is analyzed with polar coordinates this is just the polar radius.
  • If the shots are recorded with Cartesian coordinates {x, y}, then:

    \(r_i = \sqrt{(X_i - \bar{X})^2 + (Y_i - \bar{Y})^2}\),

where \((\bar{X}, \bar{Y})\) is the Cartesian Coordinate mean POI.
s Sample Standard deviation An experimental estimate of the population's standard deviation which is calculated from a of a set of n values, \(\lbrace x_1, x_2, x_3, ..., x_n\rbrace\). In the limit as n approaches \(\infty\), then s approaches the population standard deviation, σ. \(s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2}\),

where \(\bar{x}\) is the sample mean.

\(s^2\) Sample Variance The square of the sample standard deviation, typically as done with Bessel's correction of (n-1) instead of (n) to get a more unbiased estimator.
v Vertical position or axis of a target. Typically a target would be mounted vertically and the line of fire would be horizontal. Thus the horizontal axis of the target would be synonymous with the Y axis when using X-Y Cartesian coordinates for the target.
\(\bar{v}\) Average vertical position of a sample of shots An experimental estimate of the population's mean vertical position, \(\mu_h\), which is calculated from a of a set of n values \(\lbrace (h_1,v_1), (h_2,v_2), (h_3,v_3), ... (h_n,v_n)\rbrace\). In the limit as n approaches infinity, then \(\bar{v}\) converges to the population's mean vertical position, \(\mu_h\). \(\bar{v} = \frac{1}{n-1} \sum_{i=1}^n v_i \)
\(v_i\) Vertical position of shot i The shot is the ith shot of a sample of shots. Typically n shots in the sample.
x variable x probably a undefined variable. X will be used for the X-axis.
X X-axis Xi will be used for X position of the ith shot.
y variable y probably a undefined variable. Y will be used for the Y-axis.
Y Y-axis Yi will be used for Y position of the ith shot.