Sign in to continue learning
Probability & Statistics for Machine Learning & Data Science
33 hours 26 mins
In the last video, you learned about the expected value, or mean, of a random variable. The mean is not the only way to measure the center of a distribution, however, and that is important because there are other ways to measure what are the central or typical values for a probability distribution. In this video, you will learn about two more measures of central tendency and why they're important. So you've seen the mean before, but let me show you an example where the mean may not work so well. In the 1980s, the starting salary for a geography graduate at the University of North Carolina was actually $250,000. Sounds like a very, very big salary, especially considering that for the rest of the United States, the starting salary for a geography graduate was $22,000. So why do you think this happened? Well, it could have been that the program was excellent, although I'm sure there were other excellent programs too, so that may not be it. It could have been that the university had great connections, but you know, I'm sure other universities also had great connections, so maybe that's not it. Or perhaps it was that one student made a lot of money, and that was indeed the case. Michael Jordan graduated from the geography program at the University of North Carolina in the 80s, and he went out and made a lot of money, and he brought the average up for everybody. So it's not really that the average student was making a lot of money, it's that he was making so much money that it looked like they were all making money. But the average here is a little deceiving, and here's a plot that shows us why the average is deceiving. So imagine that this is the histogram of the salaries, so salary is in the horizontal axis, and the number of students with that salary in the vertical axis, and if you were to balance it, it goes over here, around $22,000. However, Michael Jordan is all the way here, and then what happens? He tips the scale, and in order to balance it, we have to move it up. So he brings the average up for everybody, and the average is no longer measuring what the average student makes, it got skewed by that one point on the right. So how can we fix that? Well, let's consider a different metric. Let's actually put them all in order by salary. So now it's just the position that matters, and once we do that, we can pick the number in the very middle. So the salary in the very middle is the medium, and notice that that is no longer so high, because now Michael Jordan doesn't contribute a lot of money, he just contributes one person to this list. That is the median. Now, if the median were to be in the middle, because you have an even number of points, then you simply take the average of the two points in the middle. So the moral here is, when the average is deceiving, go for the median, and you may get a better idea of your data. There's one more way to describe the center of a distribution, apart from the mean and the median, and it's called the mode. Remember our call center example, where the probability that the operator picks up the phone in a given time looks something like this? Mode is simply the value at which the distribution has the highest probability, which makes it the most frequent one. For example, in this case, it's this number over here. You can see that it's the outcome with the highest probability. And the same thing happens here, in the more granular discrete distribution, it's the point where the tower is the highest. And for a continuous distribution, the mode is simply the value that has the highest height. Now, the mode may not be unique, you may be in a distribution like this, where the maximum value gets repeated many times. And that's called a multimodal distribution. And in places like the uniform distribution, for example, everything is a mode. So in summary, you have the mean, the median and the mode. And it's always good to look at the three of them in order to really see where the center of your data is. Now let's look at a real example, the binomial distribution. Here's an example of five trials with the probability of one half like flipping five fair coins. What does this mean? Well, remember that we use the scale to balance distribution in the scale distribution, it's pretty simple to find the balance and it's perfectly symmetric, it's over here, the mean is at a value of 2.5. Now what about the median? Well, same thing is the average of two values in the middle, because we have an even number of values. So it's also 2.5. And what about the modes? Well, the highest points in distribution are two and three, so the modes are two and three. Now let's look at an asymmetric normal distribution. So flip a coin five times and this time the probability of heads is 0.3. So you're going to balance in the scale and find out that it's here at 1.5. So that is the mean. As the blue line shows, median is where half of the data is to the right and half the data to the left. So that one is over here. And mode is again the outcome of the highest probability, so it's going to be one. Now let's look at a continuous example, the normal distribution. What's the mean? Well, the mean is the center, because that's where you would balance the distribution as it's perfectly symmetrical. What about the median? Well, the median is also that point in the middle, and the mode is also that point in the middle, which is the highest one. So for a normal distribution, they all match.