![]() You know what the standardĭeviation means in general but this is the standard deviation We'll play with that a littleīit in this chart and see what that means. Sigma right here - that is just the standard deviation Have all these Greek letters there, but this is just - this Were to do a search for a normal distribution - let meĪctually get my pen tool going - this is what you would see. Wikipedia and if you were to type in normal distribution or Then download/normalintro.xlsĪnd you'll get this spreadsheet right here. You know, is downloadable at and if you just type that part in you'll see everything Say, oh I know what that is, this is a formula and I Someone says, we're assuming a normal distribution you can Video, in this spreadsheet, is to essentially give you as deepĪn understanding of the normal distribution as possible. Some degree, based on the normal distribution. Everything we do or almostĮverything we do in inferential statistics which isĮssentially, making inferences based on data points, is to Sorry for the length of this, but it was an interesting question that is hard to describe in a shorter post.Īrguably, the most important concept in statistics. ![]() Again, this is so small that we'd just round it off to zero, and say that it is impossible to have more than 60 inches of rain in this area. ![]() There's not really an upper bound on rainfall, so 60 inches isn't a physically impossible value like -1, but the probability is 0.000000000001. Similarly, we could ask about the probability of more than 60 inches of rain. However, it is so small as to be practically zero, so this isn't going to affect much of anything, and it is much easier than the alternative of using a different distribution to model the rainfall. It's clearly an impossible event, but the probability is not equal to zero. Let's try it out, say we want to calculate the probability of less than -1 inches of rain. Why? Because the probabilities down near zero will be so small that they'll round off to zero, so it makes no difference really. For example, let's say we're measuring the rainfall in a certain city, and that the mean is 25 inches and the standard deviation is 5 inches.Ĭlearly the amount of rainfall cannot be negative, but we can still put a normal distribution on this. However, in spite of whether or not something is theoretically possible, on a practical level it is impossible. At some point, we just get sick of the whole process and round it off to 0.Īre those values actually impossible to obtain? Possibly, but not necessarily. What about 0.0000001 ? Well, that's even more unlikely. If something has probability 0.0001, okay, that's pretty unlikely. What you are thinking about as 0% probability, is actually just rounded off. So all of those values are possible values, they just have extremely small probability. You are right that on a theoretical level, it goes out to infinity in either direction. We're simply only interested in the non-zero probability, so we don't write all the items for which the probability is zero. How can we describe the probability distribution? Well, we'd start with the obvious: Say I place 3 red marbles, 3 yellow marbles. What probability does is assign a value (the probability) to a particular outcome. The rest of this answer is a somewhat lengthy explanation, but I couldn't think of a way to shorten it without sacrificing a point. In fact, that is how we define impossible. The shortest answer would be: having a probability of zero is equivalent with being impossible. This is a good question about probability. For more information on the nature of the normal distribution, take a look at At last, the exponential gives the function its asymptotic behavior. The variable squared gives this function is parabolic look, while the negative sign makes its concavity look downward. So the core of the normal distribution is exp(-x²/2). f(x/2) is tighter, while f(x/0.5) is wider than the original f(x). Finally, if you try out exp(-((x-mu)/sigma)²/2) you'll then find out that you have the same shape shifted by the mean and elongated or shrunk by the standard deviation. That is a basic characteristic of the normal distribution. If you then graph exp(-(x-mu)²/2), you'll see the same function shifted by its mean - the mean must correspond to the function's maximum. If you try to graph that, you'll see it looks already like the bell shape of the normal function. Actually, the normal distribution is based on the function exp(-x²/2). So it must be normalized (integral of negative to positive infinity must be equal to 1 in order to define a probability density distribution). The integral of the rest of the function is square root of 2xpi.
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