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Over the past 3 articles, we covered a lot of ground! Before I jump to other topics, I wanted to simply provide a bullet-point recap of key points from the end of each of those articles and publish it here as an executive summary. This might be a good overview or reminder for those that read each article or hits the highlights in case you missed one.

Statistics for Shooters – Executive Summary

Over the past 3 articles, we covered a lot of ground! I realize this series was deeper than what most people cover in blog posts. I may tend to get closer to a technical whitepaper than traditional blog posts! 😉

Before I jump to other topics, I wanted to simply provide the bullet-point recap of key points from the end of each of those articles and publish it here as an executive summary. This might be a good overview or reminder for those that read each article or could hit the highlights if you missed one. (If I’m honest, part of this is simply to make it easier for myself to quickly to reference!)

Part 1: Statistics Fundamentals & How To Predict The Future

  • Often the ammo performance we measure at the range isn’t what we actually care about. Whether we realize it or not, we’re often collecting a sample of data and using that to predict the future performance of our rifle or ammo. That’s when statistics can help!
  • Descriptive statistics (like average, median, extreme spread, standard deviation) are very good at summing up a bunch of data points into a single number. They provide a manageable and meaningful summary, but because they exist to simplify that implies some loss of detail or nuance. An over-reliance on any descriptive statistic can lead to misleading conclusions.
  • Average and median are both measures of the “middle” of a set of numbers. Neither is wrong. The key is determining what is more accurate for a particular situation: Average is more affected by outliers. Median is less sensitive to outliers.
  • Standard deviation (SD) is a number that communicates how spread out values are from the average.
  • Because there are so many independent factors that play into how a rifle and ammo performs, it can be an ideal application for a normal distribution. The power of a normal distribution comes from the fact that we know how the data will be spread out by only having to know one stat: standard deviation. In a normal distribution, we know precisely what proportion of the observations will lie within one standard deviation of the average (68%), within two standard deviations (95%), and within three standard deviations (99.7%).
  • How many shots do we need to fire to have a “good” sample size? The answer depends on how minor the differences are that we’re trying to detect and how much confidence we want to have in the results being predictive of the future.
  • The more samples used in a calculation, the more confidence we can have in the results.
  • A very important key is to understand that average and SD have confidence levels associated with them in the first place. Just because we fire 10 shots and measure an SD doesn’t mean we will get the same SD next time. In fact, it’s unlikely we’d get the same number. Just because we can measure or calculate something to the 2nd decimal place doesn’t mean we have that level of accuracy or insight into the future! We can only speak in terms of absolute, precise values about shots fired in the past. When we’re trying to predict the future, we can only speak in terms of ranges and probabilities.
  • The confidence level you are comfortable with is a personal trade-off between accepting some risk that your results are not accurate vs. investing more time and money to keep testing.

Read the full article: How To Predict The Future – Statistics For Shooters Part 1

Part 2: Practical Statistics Related To Muzzle Velocity

  • It’s probably a bad idea to be completely dismissive of either ES or SD. Both provide some form of insight. An over-reliance on any descriptive statistic can lead to misleading conclusions.
  • SD is a more reliable and effective stat when it comes to quantifying muzzle velocity variations. ES is easier to measure but is a weaker statistical indicator in general because it is entirely based on the two most extreme events.
  • ES will grow with sample size, but average and SD will begin to converge on the true value and don’t simply continue to grow the more shots we fire.
  • While it’s easy to get close to the average muzzle velocity with 10 shots or less, it’s exceedingly more difficult to measure variation and SD with precision. There is a tendency for SD to be understated in small samples. To have much confidence that our SD is accurate, we need a larger sample size than many would think – likely 20-30 shots or more. The more the better!
  • It is very difficult to determine minor differences in velocity variation between two loads without a ridiculously large sample size (like 50+ rounds or more of each load to differentiate even a 20% difference). Often we make decisions based on truly insufficient data because the measured performance difference between two loads is simply a result of the natural variation we can expect in small sample sizes.
  • ES can help to eliminate bad loads early but use SD to prove a good load with confidence.

Read the full article: Muzzle Velocity Stats – Statistics for Shooters Part 2

Part 3: Practical Statistics Related To Precision & Group Size

  • Accuracy and precision are not the same things. Accuracy refers to how well a group of shots is centered on a target. Precision describes the spread of individual shots about the center point of a group of shots. Unlike accuracy, precision can’t be adjusted by dialing a knob on the weapon.
  • Extreme Spread (ES) is not a very good measure of dispersion. Range statistics like ES are statistically far weaker because they virtually ignore inner data points. They are the least efficient statistics but are also the most commonly used because they are so easy to measure in the field and so familiar to shooters.
  • Mean radius, also known as average to center, is the average distance from each shot to the center of the group.
  • Mean radius uses information from every shot in a group, not just the two most extreme points. Because of this, mean radius can provide a higher confidence measure of precision than ES. However, mean radius is a little harder to measure than ES.
  • Mean radius allows us to resolve smaller differences in precision with fewer shots. If you are comparing two loads or two rifles and trying to decide which is superior in terms of precision, comparing the mean radius of the groups fired will lead to more reliable conclusions than comparing ES.
  • Don’t exclude outliers in your groups unless one was undeniably a result of human error.
  • 10-shot groups are a more reliable indicator when it comes to predicting what a load is likely to do in the future.
  • By making statistically sound judgments, you may find that many loads produce statistically similar results and loads, in general, are not as finicky as conventional wisdom would lead us to believe.
  • As humans, we naturally tend to see patterns everywhere, but that can often lead to us finding meaningful patterns in meaningless noise. Statistics is a tool that can help us differentiate between true patterns and meaningless noise.
  • It’s not just about firing more shots. Plan your tests and analyze your targets in a way that you’ll be able to walk away with confidence in your decisions.

Read the full article: Precision & Group Size – Statistics for Shooters Part 3

If you’re interested in learning more about this topic, I’d invite you to read the full articles or visit my works cited page for this Statistics For Shooters series. There I provide links to many other books and online resources that dive deeper into related topics.

Note: I didn’t enable comments on this post, because it’s simply excerpts from other posts. If you have comments or questions, go add those on one of those original posts.

About Cal

Cal Zant is the shooter/author behind PrecisionRifleBlog.com. Cal is a life-long learner, and loves to help others get into this sport he's so passionate about. Cal has an engineering background, unique data-driven approach, and the ability to present technical information in an unbiased and straight-forward fashion. For more info, check out PrecisionRifleBlog.com/About.

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Precision & Group Size – Statistics for Shooters Part 3

This is the final article in my Statistics for Shooters series, and it focuses on the application of statistics when it comes to quantifying precision and group size. It provides practical answers to some age-old questions: How many shots per group do I need? Should I exclude fliers? Is extreme spread the best way to measure my groups? What is the most effective and accurate way to compare groups between two loads? I spent more time writing this article than any other single post I’ve ever written! It challenges some common beliefs held by some in the shooting community, so I tried to carefully present the reasoning behind that in a way that was approachable to shooters who aren’t math nerds. It also contains a lot of practical tips. I firmly believe these concepts will help a TON of people in the long-range community.