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Now we can talk about the normal distribution. In order to talk about the normal distribution, first we’ll talk about histograms. The histogram is a graphical way to display the data in a set. Suppose a set of 30 elements. If we wanted to know where the numbers fell relatively to each other, scanning the list of numbers would take some time.
In other words, if we look at a page of numbers, it can be hard to determine, and that’s only with 30 numbers. And you can imagine if you had 100 numbers or 1000 numbers, it would be very hard to scan that list. So instead, a histogram allows us to see the whole distribution at a glance. So the way we read this is, so for example, between 95 and 100 there was on person with a score in that range.
Between 90 and 95, there were two people in that range. So there were three scores between 90 and 100. Between 80 and 90 there were 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 scores, and so forth. So each score is representative of the box, we don’t have the exact score. In other words, we know that those 7 scores are between 80 and 85.
We don’t know the exact score, but that’s okay. This gives us the overall sense of the distribution. For example, here we can see that the person who had the low score, between, somewhere between 55 and 60, they were quite far away from everyone else. So it was rare to get that low and it was rare to get in the 90s on this particular test.
The horizontal axis of a histogram is always the quantitative variable in question, the number variable that we’re exploring. The vertical axis is always something we call the frequency. That is, how many people or items fall in that particular category. So for example with the test scores the verti, the horizontal axis was the actual scores on the exam, and the vertical axis, that told us how many people fell into each bucket.
And we call that the frequency. How frequent was each bucket inhabited by a, a particular score? We could do a histogram for 100 individuals, or 1000, or 10,000. And of course, the higher the number, the much easier the histogram is than to deal with the list of numbers. If we had 10,000 numbers, that would be a short book.
Rather than flipping through all those numbers, we could just look at a single histogram and right away, tell the high points and the low points. And of course, when we make the numbers higher, we have to make the bars skinnier. Well, you could imagine where this is going, instead of 100,000, suppose we had millions or hundreds of millions, suppose we’re dealing with a population the size of an entire country.
Well then, what’s going to happen essentially, the shape become, the bars get so skinny that it becomes continuous, and the shape becomes a smooth curve called the distribution of a population. And so, in statistics, there are many possible distributions, but for the test, you just need, you need to know just one. The most common and most famous distribution in the whole world, the normal distribution, sometimes it’s called the Gaussian, that’s a more mathematical term, the common term is just the Bell curve.
I’ll be calling it the normal distribution. And so if we imagine really, really skinny bars, what we have here is we have a lot of people right in the center. And then it drops off sharply and then we just have some rarities out toward the edges. Now why is this a famous curve?
Well in measurements of bodily features or innate abilities. So for example suppose you went and you surveyed everyone on Earth. What is the length if you have them bend their arm and you measure from the tip of their middle finger to their elbow? Or you measure the distance between their s, eyeballs. Or you measure the height of their ear, or any measurement on the body you could possibly make.
Or you could measure some kind of psychological ability, their reaction time. An image pops on the screen and they have to click on something, and how fast should they react to that. You measure that for every, everybody on earth, those abilities would always fall into a Bell curve.
you could do that for humans, animals, plants, any living thing, or biological systems. Anything you could possibly measure about a biological system over a whole population, it follows a Bell curve. Similarly, it’s also common in measurements of populations of mass-produced items.
So for example, a machine that’s making screws, or, making any kind of part. Well, for if each part is not gonna be identical, there’s gonna be some small variance, but it’s very typical that the overall variation in all the screws produced by that single machine, that they would vary according to a normal distribution. So you’ll see the normal distribution pop up in many word problems, and I just want to assure you, it’s not at all that the test makers making up.
In fact, many things in the world do follow the normal distribution. Now, it’s very important to realize, all normal distributions have a number of properties in common. These properties become clear, if the horizontal axis, instead of measure in the ordinary units, you know, pounds, or kilograms, or inches, or something like that, instead if we measure it in standard deviations.
In fact, when we measure in standard deviations, all normal distributions look the same. So in the following graphs, I’m just going to use the abbreviation, M for the mean, and S for the standard deviation. And this is what, essentially, every normal distribution in the world looks like.
So one property is that first standard deviation, from the mean to one standard deviation above the mean, 34% of the population falls there. And of course, the Bell curve is completely symmetrical, so it means that in this corresponding region here, there’s another 34%. So, within one standard deviation, one, from one side to the other side, 68, that’s 34 times 2, 68% of the population is within one standard deviation of the Bell curve.
And so here we just have it stated between the mean and the mean plus 1 standard deviation is 34% of the bell curve, also between the mean and one mean minus 1 standard deviation that’s above it 34% of the bell curve. Also that region up there, here is 13.5%. So this is the region between one standard deviation and two standard deviations, and of course, the corresponding region here would also be 13.5, and from these pieces you realize you can build all kinds of things.
For example we can ask for the mean to two standard deviations above the mean, how much would that be. Well, that would be 34% plus 13.5, and so that would be 47.5% of the population. All you have to do is memorize those two numbers, 34% and 13.5% and you can calculate everything else.
First, the mean equals the median. So 50% are above the mean and 50% are below the mean, that’s always true for a normal distribution. How many points are further than two standard deviations above the mean? Well, think about this. In other words, we’d have to take the 50% that’s above the mean, subtract the 34% that’s between 1 and 2 stand, between 0 and 1 standard deviation, and then subtract the 13.5% that is between 1 standard deviation and 2 standard deviations.
And that will just leave the little bit up in the tail. So 50 minus 34 minus 13.5, and that gives us 2.5. So 2.5% of the population is above two standard deviations above the mean. And incidentally, 3 standard deviations above the mean, then you’re getting up into people that are 1 out of say 1 out of a 1000. You get up to four standard deviations at something close to 1 out of 10,000.
So those are the truly exceptional people, the people that are experts or professionals. And just world renowned in their abilities in one particular field, music, athletics, anything like that. Here we have incidentally, the little tail region, 2.5% in the tail, that is just two standard deviations above the mean, from there all the way up to infinity.
Here’s a practice problem. Pause the video and then we’ll talk about this. So this is approximately true, for adult males, heights are normally distributed with a mean of 175 centimeters and a standard deviation of 10 centimeters. What percent of adult males have a height of less than 185 centimeters?
Well, notice that 185, is 175 plus 10 in other words the mean plus 1 standard deviation. So we want to know how many people are below that point that is one standard deviation above the mean. So of course what this is going to be is, the, first of all the folks between the mean and one standard deviation above the mean, as well as everyone below the mean.
So that’s gonna be the 50% that’s below the mean, and the 34% that is between the mean and one standard deviation above the mean, and that’s going to equal 84%. Then we can see it graphically here. Everyone below one standard deviation above the mean. In summary, histograms visually display data, the heights of the bars represent the frequency, that is how many people fall into each slot.
For population-size sets, histograms become smooth distributions, you can just imagine those bars getting skinnier and skinnier until just the overall shape remains. You need to know the normal distribution, the bell curve. You need to know the basic properties of that, because it’s very common in any natural population, or in any kind of population of machined items.
And on any normal distribution, 34% of the population is between the mean and one standard deviation above the mean, and 13.5 is bringing one standard deviation above the mean and two standard deviations above the mean. And if you know those two numbers, 34% and 13.5%, you can ask just about anything the test is going to ask about this.
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