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### Course: Statistics and probability>Unit 4

Lesson 4: Density curves

# Density curve worked example

Analyzing skew, median, mean and height of a density curve.

## Want to join the conversation?

• At you placed the fulcrum in the middle of the hypotenuse. If it were to be placed there, the triangle would have fallen on the right. Therefore, the fulcrum should be placed a bit more to the right. But then it will be so close to the median that the distinction will be very hard to make. Our eyes might be playing us. How should we work it out?
• Nice catch! you're right the fulcrum should be a bit more to the right, and maybe the mean will be so close to the median but it will be to the left of it, and that's because the curve is a very little bit skewed to the left... If it were more skewed to the left the mean would be further left from the median
• the density curve in the video is typically a perfect triangle and calculation of h(height)is possible. But what if the curve is actually curvy as happens in reality? Can we apply the same formula ? area=1/2 b*h
• No, A = 1/2 b*h is specifically used for the area of a triangle. If the density curve was actually curvy, we would probably have to use calculus to find the height (if it was even possible without more information).
Hope this helps!
• For the last problem of the exercise, wouldn't the area be less than one though? You're using the area for everything left of the vertical line to calculate for height.
• No, the area of the whole triangle is the base times the height that is perpendicular to the base, which equals 1 in this case.
(1 vote)
• Density curve worked example () Isn't the base of the triangle 6 not 5? Thanks.
• No, the base starts at 𝑥 = 1 and ends at 𝑥 = 6,
so its length is 6 − 1 = 5
• we can calculate mean and median by given values, both are equal to 3.5, so why can't mean and median are equal in this example?
• At Sal said "the area underneath any density curve is going to be 1". I was confused at first. But this statement is true if the density curve is relative density curve. The area under the curve corresponds to the cumulative relative frequencies, which should sum up to 100% or 1.
Therefore, the last answer is a common sense. Looking for the `h` of the triangle is not required to select the last answer.
What do you think?
• If the values would spread from 0.1 to 0.6 the calculation based on the area would give us h = 4.

What is the purpose of calculating this height? (other than solving easy geometry problem) ?
• Calculating the height of the density curve, as demonstrated in the example, serves the purpose of understanding the scale or relative magnitude of the density represented by the curve. While in this particular case it may seem like an easy geometry problem, in more complex scenarios or when dealing with different types of distributions, understanding the height of the density curve can provide insights into the relative concentration or probability density of the data at different points along the curve.
(1 vote)
• How could determine a base for density curves like unicorn or else and what method we use to decide the base factor in general also?
(1 vote)
• Determining the base for density curves depends on the context and the nature of the data being represented. In general, the base of a density curve is determined based on the range of values observed in the data set and the desired level of granularity for the visualization. There isn't a fixed method for deciding the base factor; it often involves considerations of data distribution, visualization clarity, and statistical analysis goals.
(1 vote)
• mean is the balancing point 50% area left or right
so median is the middle 50% of data point
how they different and simalarility in density curve
Can someone explain me
(1 vote)
• In a density curve:

The mean represents the balancing point of the distribution, where the total area under the curve is evenly distributed on both sides. It is influenced by the magnitude of each data point and their distances from the center.
The median represents the middle value of the distribution, where 50% of the data lies above and 50% lies below. It is less influenced by extreme values (outliers) compared to the mean. In symmetric distributions, the mean and median are often equal, but in skewed distributions, they can differ.
(1 vote)
• This fulcrum estimate always feels really wrong to me.
(1 vote)

## Video transcript

- [Instructor] Consider the density curve below. It's depicted right over here. It's a little unusual looking. It's 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 someplace 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 can 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 someplace like right over there. Another way, to even think about the mean is that the mean would be the balance point, where you'd 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 in 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. Alright, 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 this base, the length of this base is five. 1/2 times five times height. Or we could say one is equal to 5/2 times height or multiply both sides by 2/5 to solve for the height and what are gonna get? We're gonna 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 with even if it was not directly specified.