Measuring Tools

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Tools and practical methods for measuring and analyzing precision.

2-Shot Method

Brent Danielson described a no-frills method for accumulating arbitrarily large shot sample sets for statistical analysis. Since our statistics all depend on the sample radius:

  1. Fire two shots at a single point of aim.
  2. Measure their center-to-center distance. This produces two radius sample points, both equal to half the measured distance.
  3. Repeat this process until satisfied with the quantity of data points.

Calculation of sigma from Danielson's sample data, as well as confidence intervals and hypothesis testing, are shown in Media:DanielsonExample.xlsx.


Jeffrey Block's OnTarget Precision Calculator is the most convenient package for converting a target image into data points for analysis. It accounts for scale and distance and automatically calculates Mean Radius (called "Average to Center" in the software) and Extreme Spread (called "Max Spread").

The more expensive Target Data System can automatically identify and aggregate shots on scans of its specially-coded targets.


Taran (target analysis and shooting precision calculator) is a free online application to upload a target image, mark the points of impact, and download the coordinates of the points. Among others, it also calculates the Rayleigh CEP.

shotGroups Analysis Package

The free shotGroups package for the open-source statistical environment R provides functions to analyze target groups with respect to their shape, location (accuracy) and spread (precision). Among others, it provides implementions for many CEP estimators and descriptive precision measures. The package works with point data exported from OnTarget or Taran and includes functions to plot the group with precision indicators like the bounding box, maximum spread or minimum covering circle.

The main functionality of the package is also available as a set of web applications that do not require installing R or using R syntax.:

For more information, see the package description including a walk-through with sample diagrams and the complete manual for all functions. After having installed R and RStudio, open RStudio and install shotGroups by running: install.packages("shotGroups") . For a first introduction to R, see:


  • TargetScan (iOS) computes the unbiased estimate of Mean Radius for supported targets.

Spreadsheet Analysis

Given a target data set, whether compiled using the #2-Shot Method or #OnTarget, Closed Form Precision analysis can be performed using standard spreadsheet functions. See, for example Media:CCI 40gr HV 100yd.xlsx or any of the other workbooks linked in the Examples.

Media:RangeStatisticEstimation.xls is spreadsheet for calculating the statistical significance of Range Statistics estimates.