The standard normal distribution is an important member of the normal distributions family. Learn how to work with it.
- [Instructor] What a statistician does is estimate population parameters. One assumption that a statistician makes is, the attribute being measured is distributed in a particular way throughout a population. Most people have average amounts of the attribute, few people have extreme amounts, and that's an assumption that over time has worked out very well. A particular family of distributions fits this description. It's called the normal distribution family. To say the normal distribution is actually a misnomer, as this is really a family of distributions. Some general points. A normally distributed variable is continuous, its probability distribution is a probability density function, and it's generated by a complex mathematical rule. In that mathematical rule, you supply a value for x, for the mean, and the standard deviation, and the rule returns a value for f of x, the height of the distribution for x. The height is the probability density. This is that mathematical rule, and you could ignore the details as long as you understand the three points I just mentioned. Probability density is the thing that allows the area under the curve to be probability. So, this is a picture of what happens when you apply that mathematical rule for the normal distribution family. As you can see, the height of the curve, the probability density, is greatest at the mean. That is, when x is equal to mu, f of x is the highest. It gradually drops off toward the extremes. We can imagine many attributes being distributed like this, a lot of people around the average and progressively fewer toward the extremes, or tails, of the curve. Now, what exactly do we mean by probability density? Probability density allows the area under the curve to represent probability. X is a continuous random variable, so we can't find an exact value for probability of x. What we can find is the probability that x is between some lower bound and some upper bound. And so, the area bounded by the curve, the lower bound, the upper bound, and the x-axis, is that probability, and this shows how the percentage of area under a curve corresponds to probability. Let's look at some general characteristics of the family. As I pointed out earlier, the mean is at the maximum point, which is in the middle. Because it's the maximum point, that's where the mode is, too, and because it's symmetrical, the maximum point cuts the curve in half, that's where the median is as well. And so, the curve is symmetric around the mean. Note that the mean and standard deviation are independent of one another. So, here are two members of the normal distribution family, the distribution of the heights of women and the distribution of the heights of men. They have different means, and they have different standard deviations. Some additional family traits. Regardless of the mean and the standard deviation, notable proportions of the area are constant. On the x-axis, we can mark off mu, mu plus one standard deviation, and mu plus three standard deviations, and mu minus one standard deviation, mu minus two standard deviations, and mu minus three standard deviations. The proportion of area between the mean and each of these boundaries is always the same. Here's some notable proportions of area under the normal distribution curve. Between the mean and one standard deviation on either side is .34, so about 68% of the area is covered from one standard deviation below the mean to one standard deviation above the mean. That means that the probability of drawing someone from this normal distribution who is one standard deviation below the mean to one standard deviation above the mean is about .68, or 68%. Here's an example. IQ is normally distributed. For one version of the IQ test, the mean is 100 and the standard deviation is 16. On the x-axis, we can mark off the mean, the mean plus one standard deviation, which would take us to 116, two standard deviations, 132, and three standard deviations, 148. to minus one standard deviation, or 84, minus two standard deviations, 68, and three standard deviations, 52. And it would look like this. So, the probability of finding someone in the population who has an IQ between 100 and 116 is .3413, about 34%. And that's a look at the normal distribution family.
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