{"id":260,"date":"2023-07-28T12:49:22","date_gmt":"2023-07-28T12:49:22","guid":{"rendered":"https:\/\/tactical-gear.biz\/how-to-predict-the-future-statistics-for-shooters-part-1\/"},"modified":"2023-07-28T12:49:22","modified_gmt":"2023-07-28T12:49:22","slug":"how-to-predict-the-future-statistics-for-shooters-part-1","status":"publish","type":"post","link":"https:\/\/tactical-gear.biz\/how-to-predict-the-future-statistics-for-shooters-part-1\/","title":{"rendered":"How To Predict The Future \u2013 Statistics For Shooters Part 1"},"content":{"rendered":"
\n\t\t\t\t\t <\/p>\n

I actually believe the average shooter might get more value from this Statistics for Shooters series of articles<\/a> than anything I\u2019ve published in a long time. I promise it\u2019s worth your time to read!<\/strong><\/p>\n

I realize most shooters aren\u2019t engineers or math nerds. Many people have an uncomfortable relationship with math and aren\u2019t impressed with fancy formulas. However, statistics and probability are insanely<\/em> applicable when it comes to long-range shooting in particular. Understanding just a few basics can help you gain actionable insight and put more rounds on target. Venturing beyond \u201caverage\u201d and \u201cextreme spread\u201d will lead to better decisions.<\/p>\n

I have spent an absurd amount of time arduously crafting this article with the math-averse shooter in mind. I pulled from a dozen books, white papers, magazine articles, and other sources on the subject (see works cited<\/a>) to deliver a comprehensive, but approachable, overview of the most relevant aspects for fellow shooters. I literally spent months trying to make this content simple and balanced because I firmly believe this can help a lot of shooters once they wrap their heads around a few basics.<\/p>\n

Predicting the Future<\/h2>\n
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\u201cEach time you pull the trigger, the bullet chooses a single outcome from infinite possibilities based on countless random factors. Without a time machine, you can never know exactly where the next bullet will go. However, you can predict the most likely outcome, and precisely describe the chances of it being high or low, left or right, fast or slow. Many people shy away from statistics because, well, math. It seems complicated and unnecessary. On the contrary. It is a way of thinking that hones your intuition and helps you make better decisions. \u2026 Just by understanding the relationship between a sample and a population, you can learn how to predict the future.\u201d \u2013 Adam MacDonald, Statistics for Shooters<\/a><\/p>\n<\/blockquote>\n

Often as shooters, we use stats to make some kind of comparison or decision. Here are two simple examples:<\/p>\n

    \n
  1. Fred has a big hunt coming up, so he bought 3 different kinds of factory ammo to see which groups best from his rifle.<\/li>\n
  2. Sam is a competitive long-range shooter and reloader, and he tried several different powder weights to find the one that produces the most consistent velocity.<\/li>\n<\/ol>\n

    In both of those examples, the shooter is trying to make an informed, data-driven decision. Fred and Sam will both fire a bunch of rounds during a practice session at the range and use the results to decide what they\u2019ll use in the future. But what Fred and Sam measure in the practice session isn\u2019t what they actually care about. What really matters to them is what happens in the big hunt or the upcoming competition. Whether they realize it or not, both are collecting a sample of data and using that to predict the future performance of their rifle\/ammo.<\/strong> Let\u2019s say Fred fires a 3-shot group from each box of ammo, and the extreme spreads of those groups measured 0.54, 0.57, and 0.94 inches. He should just go with the smallest, right? Does he need to fire more shots? How can he know he\u2019s making the right choice? Those are questions statistics can help answer.<\/p>\n

    The Plan for This \u201cStatistics For Shooters\u201d Series<\/h2>\n

    I plan to publish 3 articles focused on how stats can help us as shooters in a way that is practical and applicable. There will be one article focused on helping us make better decisions related to these common applications:<\/p>\n

      \n
    1. Quantifying muzzle velocity consistency<\/a>: Gaining insight to minimize our shot-to-shot variation in velocity<\/li>\n
    2. Quantifying group dispersion<\/a>: Making better decisions when it comes to precision and how small our groups are<\/li>\n<\/ol>\n

      This article will lay a foundation that we\u2019ll use in the others. So let\u2019s dive into some important basics.<\/p>\n

      Descriptive Statistics: The Good & The Bad<\/h2>\n

      When we talk about our average muzzle velocity or the extreme spread of a group, both of those are descriptive statistics. So is a baseball player\u2019s batting average or an NFL passer rating. Sports fans use descriptive statistics in everyday conversation. How good of a baseball player was Mickey Mantle? He was a career .298 hitter. To a baseball fan, that is a meaningful statement and is remarkable because that tiny statement encapsulates an 18-season career with over 8,000 at-bats.<\/p>\n

      Descriptive statistics are very good at summing up a jumble of data, like 18 seasons of baseball or a 10-shot group, and boiling it down to a single number. They give us a manageable and meaningful summary of some underlying phenomenon. The bad news is any simplification invites abuse. Descriptive statistics can be like online dating profiles: technically accurate and yet pretty darn misleading! Descriptive statistics exist to simplify, which always implies some loss of detail or nuance. So here is a very important point: An over-reliance on any descriptive statistic can lead to misleading conclusions.<\/strong><\/p>\n

      Even under the best circumstances, statistical analysis rarely unveils \u201cthe truth.\u201d Statistics can help us make more informed decisions, but I\u2019ll caution that some professional skepticism is appropriate when it comes to statistics. That is why smart and honest people will often disagree about what the data is trying to tell us. \u201cLies, damned lies, and statistics\u201d is a common phrase describing the persuasive power of numbers. The reality is you can<\/em> lie with statistics \u2013 or you can make inadvertent errors. In either case, the mathematical precision attached to statistical analysis can dress up some serious nonsense.<\/p>\n

      What Is the \u201cMiddle\u201d?<\/h2>\n

      We use averages all the time, right? Average is one of the most common descriptive statistics, which is easy to understand and helpful \u2013 but sometimes average can be deceptive. Here is a great story that illustrates the point, which is from Naked Statistics by Charles Wheelan<\/a>:<\/p>\n

      \n


      \n\t\t\t10 guys are sitting in a middle-class bar in Seattle, and each of them earns $35,000 a year. That means the average annual income for the group is $35,000. Bill Gates then walks into the bar, and let\u2019s say his annual income is $1 billion. When Bill sits down on the 11th stool, the average income rises to around $91 million. Obviously, the original 10 drinkers aren\u2019t any richer. If we described the patrons of this bar as having an average annual income of $91 million, that statement would be both statistically correct and grossly misleading. This isn\u2019t a bar where multimillionaires hang out; it\u2019s a bar where a bunch of guys with relatively low incomes happen to be sitting next to Bill Gates.\n\t\t\t<\/p>\n<\/div>\n

      We often think about the average as being the \u201cmiddle\u201d of a set of numbers \u2013 but it turns out that the average is prone to distortion by outliers<\/strong>. That is why there is another statistic that is often used to signal the \u201cmiddle\u201d, albeit differently: the median. Okay, don\u2019t check out or let your eyes glaze over! I promise this is applicable and easy to understand.<\/em> The median is simply the point that divides a set of numbers in half, meaning half of the data points are above the median and half are below it.<\/strong><\/p>\n

      If we return to the barstool example, the median annual income for the 10 guys originally sitting at the bar is $35,000. When Bill Gates walks in and perches on a stool, the median income for the group is still $35,000<\/em>. Think about lining up all 11 of them on stools in order of their incomes, as shown in the diagram below. The income of the guy sitting on the 6th stool (bright yellow shirt) represents the median income for the group because half of the values are above him and half are below him. In fact, if Warren Buffet came in and sat next to Bill the median would still be $35,000!<\/p>\n

      \n
      <\/figure>\n<\/div>\n

      If you had to bet $100 on what the income was of the very next guy who walked in the door, would $35,000 or $91 million be more likely? When we\u2019re talking about what is most likely to happen in the future, the median can often be a better choice than the average.<\/p>\n

      When there aren\u2019t major outliers, average and median will be similar \u2013 so it doesn\u2019t matter which we use. It\u2019s when there are major outliers that it does matter. Neither is \u201cwrong!\u201d The key is determining which measure of the \u201cmiddle\u201d is more accurate for a particular situation: median (less sensitive to outliers) or average (more affected by outliers)?<\/strong><\/p>\n

      How Spread Out Is the Data?<\/h2>\n

      Often as shooters, we want to understand how spread out a group of shots are, so we measure the extreme spread (ES). That is easy to measure by hand and it is useful. However, like any descriptive statistic, we are simplifying multiple data points into a single number \u2013 so we lose some level of detail. There is another statistic we can use to describe how spread out data points are. To understand it, let\u2019s go to another example from Naked Statistics by Charles Wheelan<\/a>:<\/p>\n

      \n

      \n\t\t\tLet\u2019s say we collected the weights for two sets of people:<\/p>\n
        \n
      1. 250 people who qualified for the Boston Marathon<\/li>\n
      2. 250 people on an airplane flying to Boston<\/li>\n<\/ol>\n
        \n
        \"Standard<\/a><\/figure>\n<\/div>\n

        Let\u2019s assume the average weight for both groups was 155 pounds. If you\u2019ve ever been squeezed into the middle seat on a flight, you know many American adults are larger than 155 pounds. However, if you\u2019ve flown much you also know there are crying babies and poorly behaved children on flights, all of whom have huge lung capacity but not much weight. When it comes to calculating the average weight, the 320-pound football players on either side of your middle seat is likely to offset the six-year kicking the back of your seat from the row behind.<\/p>\n

        In terms of average and median weights, the airline passengers and marathon runners are nearly identical. But they\u2019re not! Yes, the weights have roughly the same \u201cmiddle,\u201d but the airline passengers have far more dispersion, meaning their weights are spread farther from the midpoint.<\/strong> The marathon runners may all appear like they all weigh the same amount, while the airline passengers have some tiny people and some bizarrely large people.\n\t\t\t<\/p>\n<\/div>\n<\/div>\n

        Standard deviation (SD) is the descriptive statistic that allows us to communicate how spread out values are from the average with a single number.<\/strong> How to calculate SD isn\u2019t straight-forward, but virtually nobody does it by hand \u2013 so I won\u2019t bore you with the formula. Typically, a chronograph or app calculates it for us, or we can use a formula in a spreadsheet.<\/p>\n

        Some sets of numbers are more spread out than others, and that is what SD will provide insight into. The SD of the weights for our 250 airline passengers will be much higher than the SD of the weights for our 250 marathon runners because the weights of the marathon runners are much less spread out.<\/p>\n

        The Normal Distribution<\/h2>\n
        \n
        \"\"<\/figure>\n<\/div>\n

        Not only does SD describe how spread out data is, but it also helps introduce one of the most important, helpful, and common distributions in statistics: the normal distribution. Data that is distributed \u201cnormally\u201d is symmetric and forms a bell shape that will look familiar.<\/strong><\/p>\n

        A normal distribution can be used to describe so many natural phenomena. Wheelen points out a few practical examples:<\/p>\n