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            "59": {
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                "title": "References",
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                        "*": "'''[[Prior Art]]''' details previous work on the problem of estimating shooting statistics.\n\n'''[[CEP literature]]''' focuses on the broader body of work related to characterizing Circular Error Probable, which is applicable not only to ballistics but also to fields like navigation and signal processing.\n\nFollowing is a complete list of useful References and Prior Art:\n\n* Bookstaber, David (2014).  [https://web.archive.org/web/20171120094420/http://www.thetruthaboutguns.com/2014/12/daniel-zimmerman/understanding-rifle-precision/ '''Understanding Rifle Precision'''].\n\n* Danielson, Brent J. (2005).  [[Prior_Art#Danielson.2C_2005.2C_Testing_loads|'''Testing Loads''' &ndash; ''detailed in Prior Art'']].\n\n* Gammon, W. J. (2017), [[Prior_Art#Gammon.2C_2017.2C_Shot_Group_Statistics_for_Small_Arms_Applications|'''Shot Group Statistics for Small Arms Applications''' &ndash; ''detailed in Prior Art]].\n\n* Grubbs, Frank E. (1964).  [[Prior_Art#Grubbs.2C_1964.2C_Statistical_Measures_of_Accuracy_for_Riflemen_and_Missile_Engineers|'''Statistical Measures of Accuracy for Riflemen and Missile Engineers''' &ndash; ''detailed in Prior Art'']].\n\n* Hogema, Jeroen (2005).  [[Prior_Art#Hogema.2C_2005.2C_Shot_group_statistics|'''Shot group statistics''' &ndash; ''detailed in Prior Art'']].\n\n* Hogema, Jeroen (2006).  [[Prior_Art#Hogema.2C_2006.2C_Measuring_Precision|'''Measuring Precision''' &ndash; ''detailed in Prior Art'']].\n\n* Hornady Podcast (2022).  [https://www.youtube.com/watch?v=QwumAGRmz2I '''Episode 050 \u2013 Your Groups Are Too Small'''].  [https://www.snipershide.com/shooting/threads/ill-post-this-here-hornadys-podcast-50-i-thought-it-was-one-of-their-best-but-some-reloaders-might-not-like-what-they-see.7152297/post-10620383 Summary here].\n\n* Kolbe, Geoffrey (2010).  [[Prior_Art#Kolbe.2C_2010.2C_Group_Statistics|'''Group Statistics''' &ndash; ''detailed in Prior Art'']].\n\n* Leslia, John E. III (1993).  [[Prior_Art#Leslie.2C_1993.2C_Is_.22Group_Size.22_the_Best_Measure_of_Accuracy.3F|'''Is \"Group Size\" the Best Measure of Accuracy?''' &ndash; ''detailed in Prior Art'']].\n\n* MacDonald, Adam (2017). [https://www.autotrickler.com/blog/thinking-statistically '''Thinking Statistically'''].\n\n* Molon (2006). [[Prior_Art#Molon.2C_2006.2C_The_Trouble_With_3-Shot_Groups|'''The Trouble With 3-Shot Groups''' &ndash; ''detailed in Prior Art'']].\n\n* Precision Rifle Blog (2020). [https://precisionrifleblog.com/2020/12/12/measuring-group-size-statistics-for-shooters/ '''Statistics for Shooters'''].\n\n* Rifleslinger (2014). [http://artoftherifleblog.com/on-zeroing/2014/02/on-zeroing.html '''On Zeroing'''].\n\n* Saleh, A. K. Md. Ehsanes (1967). [[Media:Order_Statistics_of_Exponential_Distribution_in_Censored_Samples.pdf|'''Determination of the Exact Optimum Order Statistics for Estimating the Parameters of the Exponential Distribution from Censored Samples''']]. Technometrics 9, no. 2.\n\n* Sarhan, A. E., Greenberg, B. G., & Ogawa, J. (1963). [[Media:Simplified_Estimates_for_the_Exponential_Distribution.pdf|'''Simplified Estimates for the Exponential Distribution''']]. The Annals of Mathematical Statistics, 34(1), 102\u2013116.\n\n* Siddiqui, M. M. (1961). [[Media:Some Problems Connected With Rayleigh Distributions - Siddiqui 1961.pdf|'''Some Problems Connected With Rayleigh Distributions''']].  The Journal of Research of the National Bureau of Standards, Sec. D: Radio Science, Vol. 68D, No. 9.\n\n* Siddiqui, M. M. (1964). [[Media:Statistical Inference for Rayleigh Distributions - Siddiqui, 1964.pdf|'''Statistical Inference for Rayleigh Distributions''']].  The Journal of Research of the National Bureau of Standards, Sec. D: Radio Propagation, Vol. 66D, No. 2.  (''Summarizes and extends Siddiqui, 1961.'')\n\n'''''Important Note on Siddiqui''': Siddiqui parameterizes the Rayleigh distribution with <math>\\frac{\\sigma}{\\sqrt{2}}</math>.  Therefore, should you endeavor to relate Siddiqui's work to that referenced here and in more modern usage, remember that <math>\\sigma_{modern} = \\sqrt{2} \\sigma_{Siddiqui}</math>.''\n\n* Strohm, Luke (2013).  [https://apps.dtic.mil/sti/tr/pdf/ADA588846.pdf '''An Introduction to the Sources of Delivery Error for Direct-Fire Ballistic Projectiles''' (ARL-TR-6494)].\n\n* Taylor, M. S. & Grubbs, Frank E. (1975).  [[Prior_Art#Taylor_.26_Grubbs.2C_1975.2C_Approximate_Probability_Distributions_for_the_Extreme_Spread|'''Approximate Probability Distributions for the Extreme Spread''' &ndash; ''detailed in Prior Art'']].\n\n* Triplett, Ben (2019). [http://www.bisonops.com/2019/08/17/rifle-ammunition-load-workup '''Rifle Ammunition Load Workup'''].\n\n\n= Reference Data =\n\n* [[File:Confidence Interval Convergence.xlsx]]: Shows how precision confidence intervals shrink as sample size increases.\n\n* [[File:Sigma1RangeStatistics.xls]]: Simulated median, 50%, 80%, and 95% quantiles, plus first four sample moments, for shot groups containing 2 to 100 shots, of: Extreme Spread, Diagonal, Figure of Merit.\n\n* [[File:SymmetricBivariateSigma1.xls]]: Monte Carlo simulation results validating the [[Closed Form Precision]] math.\n\n[https://github.com/dbookstaber/ballistipedia/blob/main/BallisticSimulations.ipynb BallisticSimulations.ipynb] is a Jupyter notebook containing extensive illustrations and validation of the math used throughout this site."
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            "160": {
                "pageid": 160,
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                "title": "Sight Zeroing Example",
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                        "*": "[[Category: Examples]]\n<p style=\"text-align:right\">... back to [[Closed Form Precision#How many sighter shots do you need?]]</p>\n\nThis example shows how confidence in the center of impact increases with the number of shots used to compute it.  We're showing the 90% Confidence region, or CEP(90%), which is the area (around the sample center) that should contain the true center 90% of the time.  [[Closed_Form_Precision#Summary_Probabilities|As detailed here]], CEP(x) = <math>\\sigma \\sqrt{-2 \\ln(1-x)}</math>, and for purposes of estimating the true center the radius of that region scales with the square root of the number of shots ''n'' used to compute it.  I.e., 90% of the time the true center is inside the circle of radius <math>\\frac{CEP(90\\%)}{\\sqrt{n}}</math> about the sample center.\n\nThe following diagrams are based on a simulation with \u03c3=1.  The first group was simulated with the sights misaligned to center on (1, 1).  '''''Nota Bene:''' In real life we don't get to know the true center of impact.  We can only estimate it by looking at where shots hit.''  In this example we know the true center because it was specified in the simulation.\n\n[[File:SighterGroup1.png|center]]\n\nBased on the first group's center hitting near that point, the sights are adjusted down 1 MOA and left 1 MOA.  The record of those first three shots is also adjusted by that amount so that they can continue to contribute to the estimate of the gun's zero.\n\nShots on all of the following targets were simulated with a center at (0,0).\n\n[[File:SighterGroup2.png|center]]\n\n[[File:SighterGroup3.png|center]]\n\n[[File:SighterGroup4.png|center]]"
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