Density curve worked example | Modeling data distributions | AP Statistics | Khan Academy

Consider the density curve below. It's depicted right over here; it's a little unusual looking. It looks more like a triangle than our standard density curves, but it's valid.

Which of the following statements are true? Choose all answers that apply:

The mean of the density curve is less than the median. Pause this video and see if you can figure out whether that's true. Well, we don't know exactly where the mean and median are just by looking at this. But remember, the median is going to be the value for which the area to the right and the left are going to be equal. So, I would guess the median is going to be some place like that. So that's my guess, my approximation.

That is the median, and because our distribution goes off further to the left than it does to the right, you could view this as something of a tail. It's reasonable to say that this is left skewed. Left skewed, and generally speaking, if a distribution is left skewed, the mean is to the left of the median.

So, because it is left skewed, the mean might be some place like right over there. Another way to even think about the mean is that the mean would be the balance point where you would place a fulcrum.

If this were a mass, and you might say, why doesn't that happen at the median? Well, remember, even when you're balancing something, a smaller weight that is far away from the fulcrum can balance out a heavier weight that is closer to the fulcrum.

So, in terms of this first one, the mean of the density curve is less than the median in this case, or you could say to the left of the median. We can consider this to be true.

Now, what about the median of the density curve is three? Well, I already approximated where the median might be, saying, hey, this area looks roughly comparable to this area. The median, definitely, I might not be right there, but the median is definitely not going to be three.

This area right over here is for sure smaller than this area right over here, so we can rule that out. The area underneath the density curve is one. Pause this video; is that true? Yes, this is true. The area underneath any density curve is going to be one. If we look at the total area under the curve, it's always going to be one.

So, we answered this question. I'll leave you with one extra question that we can actually figure out from the information they've given us. What is the height of this point of this density curve right over here? What is this value? What is this height going to be? See if you can pause this video and figure it out, and I'll give you a hint. The hint is this third statement: the area under the density curve is one.

All right, now let's try to work through it together. If we call this height H, we know how to find the area of a triangle. It's 1/2 base times height. Area is equal to 1/2 base times height. We know that the area is one. This is a density curve, so one is going to be equal to what's the length of the base? We go from one to six, so from one to six, the length of this base is five.

1 = 1/2 * 5 * height. Or we could say 1 is equal to 5 times height. Multiply both sides by two-fifths to solve for the height. And what are we going to get? We're going to get the height is equal to 2/5.

So, if you have a very clean triangular density curve like this, you can actually figure out the height, even if it was not directly specified.