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docs/visualizing_uncertainty.md

@@ -10,9 +10,9 @@ Two commonly used approaches to indicate uncertainty are error bars and confiden
 
 Before we can discuss how to visualize uncertainty, we need to define what it actually is. We can intuitively grasp the concept of uncertainty most easily in the context of future events. If I am going to flip a coin I don't know ahead of time what the outcome will be. The eventual outcome is uncertain. I can also be uncertain about events in the past, however. If yesterday I looked out of my kitchen window exactly twice, once at 8am and once at 4pm, and I saw a red car parked across the street at 8am but not at 4pm, then I can conclude the car left at some point during the eight-hour window but I don't know exactly when. It could have been 8:01am, 9:30am, 2pm, or any other time during those eight hours.
 
-Mathematically, we deal with uncertainty by employing the concept of probability. A precise definition of probability is complicated and far beyond the scope of this book. Yet we can successfully reason about probabilities without understanding all the mathematical intricacies. For many problems of practical relevance it is sufficient to think about relative frequencies. Assume you perform some sort of random trial, such as a coin flip or rolling a die, and look for a particular outcome (e.g., heads or rolling a six). You can call this outcome *success,* and any other outcome *failure.* Then, the probability of success is approximately given by the fraction of times you'd see that outcome if you repeated the random trial over and over again. For instance, if a particular outcome occurs with a probability of 10%, then we expect that among many repeated trials that outcome will be seen in approximately one out of ten cases.
+Mathematically, we deal with uncertainty by employing the concept of probability. A precise definition of probability is complicated and far beyond the scope of this book. Yet we can successfully reason about probabilities without understanding all the mathematical intricacies. For many problems of practical relevance it is sufficient to think about relative frequencies. Assume you perform some sort of random trial, such as a coin flip or rolling a dice, and look for a particular outcome (e.g., heads or rolling a six). You can call this outcome *success,* and any other outcome *failure.* Then, the probability of success is approximately given by the fraction of times you'd see that outcome if you repeated the random trial over and over again. For instance, if a particular outcome occurs with a probability of 10%, then we expect that among many repeated trials that outcome will be seen in approximately one out of ten cases.
 
-Visualizing a single probability is difficult. How would you visualize the chance of winning in the lottery, or the chance of rolling a six with a fair die? In both cases, the probability is a single number. We could treat that number as an amount and display it using any of the techniques discussed in Chapter \@ref(visualizing-amounts), such as a bar graph or a dot plot, but the result would not be very useful. Most people lack an intuitive understanding of how a probability value translates into experienced reality. Showing the probability value as a bar or as a dot placed on a line does not help with this problem.
+Visualizing a single probability is difficult. How would you visualize the chance of winning in the lottery, or the chance of rolling a six with a fair dice? In both cases, the probability is a single number. We could treat that number as an amount and display it using any of the techniques discussed in Chapter \@ref(visualizing-amounts), such as a bar graph or a dot plot, but the result would not be very useful. Most people lack an intuitive understanding of how a probability value translates into experienced reality. Showing the probability value as a bar or as a dot placed on a line does not help with this problem.
 
 We can make the concept of probability tangible by creating a graph that emphasizes both the frequency aspect and the unpredictability of a random trial, for example by drawing squares of different colors in a random arrangement. In Figure \@ref(fig:probability-waffle), I use this technique to visualize three different probabilities, a 1% chance of success, a 10% chance of success, and a 40% chance of success. To read this figure, imagine you are given the task of picking a dark square by choosing a square before you can see which of the squares will be dark and which ones will be light. (If you will, you can think of picking a square with your eyes closed.) Intuitively, you will probably understand that it would be unlikely to select the one dark square in the 1%-chance case. Similarly, it would still be fairly unlikely to select a dark square in the 10%-chance case. However, in the 40%-chance case the odds don't look so bad. This style of visualization, where we show specific potential outcomes, is called a *discrete outcome visualization,* and the act of visualizing a probability as a frequency is called *frequency framing.* We are framing the probabilistic nature of a result in terms of easily understood frequencies of outcomes.
 
@@ -52,7 +52,7 @@ In Figure \@ref(fig:election-prediction), I showed a "best estimate" and a "marg
 
 We are normally interested in specific quantities that summarize important properties of the population. In the election example, these could be the mean vote outcome across districts or the standard deviation among district outcomes. Quantities that describe the population are called *parameters,* and they are generally not knowable. However, we can use a sample to make a guess about the true parameter values, and statisticians refer to such guesses as *estimates.* The sample mean (or average) is an estimate for the population mean, which is a parameter. The estimates of individual parameter values are also called *point estimates,* since each can be represented by a point on a line.
 
-Figure \@ref(fig:sampling-schematic) shows how these key concepts are related to each other. The variable of interest (e.g., vote outcome in each district) has some distribution in the population, with a population mean and a population standard deviation. A sample will consist of a set of specific observations. The number of the individual observations in the sample is called the *sample size.* From the sample we can calculate a sample mean and a sample standard deviation, and these will generally differ from the population mean and standard deviation. Finally, we can define a *sampling distribution,* which is the distribution of estimates we would obtain if we repeated the sampling process many times. The width of the sampling distribution is called the *standard error,* and it tells us how precise our estimates are. In other words, the standard error provides a measure of the uncertainty associated with our parameter estimate. As a generaly rule, the larger the sample size, the smaller the standard error and thus the less uncertain the estimate.
+Figure \@ref(fig:sampling-schematic) shows how these key concepts are related to each other. The variable of interest (e.g., vote outcome in each district) has some distribution in the population, with a population mean and a population standard deviation. A sample will consist of a set of specific observations. The number of the individual observations in the sample is called the *sample size.* From the sample we can calculate a sample mean and a sample standard deviation, and these will generally differ from the population mean and standard deviation. Finally, we can define a *sampling distribution,* which is the distribution of estimates we would obtain if we repeated the sampling process many times. The width of the sampling distribution is called the *standard error,* and it tells us how precise our estimates are. In other words, the standard error provides a measure of the uncertainty associated with our parameter estimate. As a general rule, the larger the sample size, the smaller the standard error and thus the less uncertain the estimate.
 
 (ref:sampling-schematic) Key concepts of statistical sampling. The variable of interest that we are studying has some true distribution in the population, with a true population mean and standard deviation. Any finite sample of that variable will have a sample mean and standard deviation that differ from the population parameters. If we sampled repeatedly and calculated a mean each time, then the resulting means would be distributed according to the sampling distribution of the mean. The standard error provides information about the width of the sampling distribution, which informs us about how precisely we are estimating the parameter of interest (here, the population mean).