 Statistics and Excel. Normal distribution, bell-shaped curve or Gaussian curve. Got data? Let's get stuck into it with statistics and Excel. After this presentation, we will be able to understand the significance properties and applications of the Gaussian or normal distribution, otherwise known as the bell curve. In prior presentations, we've been thinking about how we can represent different data sets using both mathematical calculations like the average or mean, median, mode, quartile, and with pictorial representations like the box and whiskers and the histogram. The histogram being the primary tool we use to envision to visualize the spread or distribution of the data using descriptive terms to describe the spread or distribution of data on a histogram, such as it's skewed to the left, the data is skewed to the right. We then thought about different types of curves and lines that can be represented with formulas that may approximate different data sets in real life. If we can approximate a data set with a curve that has a formula related to it, that gives us more predictive power over whatever the data set is about. Some of the common types of formulas and curves that are representative of real life type situations are a uniform distribution that we looked at, a binomial distribution, the Poisson distribution, the exponential distribution. Now, of course, we're moving to the most famous of them all, the bell shaped distribution. Now, because the bell shaped distribution is so famous, oftentimes people have an idea that the bell shaped distribution is going to be applicable basically to all data sets, but that's not the case. It's just like what we saw in prior sections in that if we have a data set, we want to see whether or not that data set will conform to some type of curve that we can apply a formula to. If it does, such as the bell shaped curve, then again, that would be great. And the bell shaped curve has some unique characteristics related to it, which are also quite useful that we'll take a look at. So historical concept, it's named after Carl Frederick Gauss, a mathematical prodigy and is foundational to the realm of statistical studies. Obviously, the bell shaped curve because of its specific and special unique characteristics, very important to the field of statistics. Carl Frederick Gauss, so the genius behind the discovery, of course, historical application used the method of least squares to predict the position of the asteroid series. So the error distribution originated as the distribution of errors in measurements. So in other words, one of the early contexts of the use of the normal distribution was for errors. In other words, when you make predictions, such as making predictions about where something is going to show up in the sky, if you have multiple people that are good at making predictions, they're all going to get close, but none of them are going to be exact in the exact location, because it's basically impossible to be exact on something that's so precise. And therefore, if you were to plot those points, you would think you might get something that would be errors around the actual result, which would be the mean. Therefore, the error distributions often have this bell shaped type of curve. And that's one of the early places where we saw this normal distribution being applied often. Characteristics and properties of the normal distribution, the shape, so bell shape and symmetrical about its mean. So what's the shape of the bell shape curve? It's in a bell. It looks like a bell. It's an upside down bell. It's symmetrical. So if we drew a line down the middle, which is an even line because I'm drawing it here by hand, but it would be symmetrical the same on the left to the right, which is a great characteristic because that symmetry will give us some mathematical concepts that will, of course, be useful. If we're looking at a histogram, then clearly it's not always going to be symmetrical on the right and the left of the middle point. That middle point being the mean also quite useful. Remember that when we were looking at a histogram of data, we could say that the mean was kind of like the focal point of the data. So if you thought of it as a teeter-totter, that would be the point where it would be balanced on both sides. But here we have the mean right in the middle. And then if you folded the page in half, it would be folding right on top of each other because it's symmetrical around that middle point. So inflection points at one standard deviation away from the mean. We'll talk a little bit more about the standard deviation concept a bit as it relates to the bell shaped curve. But the point here is that you could see when you're down here, the concavity of the curve is up like this. So it's increasing at an increasing rate. And then somewhere up here, it starts, it's still going up, but no longer is concaved up, it's concaved down. And that happens, the inflection point where that happens is at one standard deviation away from the mean. Parameters. So the mean, which we can represent with the Greek letter mu, represents the center of the distribution. So that's always going to be the center point where you have the highest peak of the distribution. Standard deviation, which can be represented with the Greek letter sigma, indicates the spread or dispersion. Just like we've seen in prior section, the standard deviation represents the spread of the data. The great thing about a normal distribution is the spread of the data is going to be very uniform, because note that when we define the bell curve, the normal distribution, we're saying that it's got the middle point. So the middle point is the mean. If I change the mean, and I keep everything else the same, then it'll simply shift on the x axis to the left or the right, the left or the right. If I was to change the other characteristic of it is the standard deviation, which measures the spread. So if I increase the standard deviation, what you would see is the shape of the curve getting squat and wider. Whereas if you lessen the standard deviation, it's going to be taller and more compact around that middle point. So notice we're basically defining the curve with simply those characteristics. You're saying this is the mean.