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# Density curve worked example

AP.STATS:
UNC‑1 (EU)
,
UNC‑1.M (LO)
,
UNC‑1.M.2 (EK)

## Video transcript

consider the density curve below its 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 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 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 someplace like right over there another way to even think about the mean is that the mean would be the balance point where you 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 into 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 1 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 1 this is a density curve so 1 is going to be equal to what's the length of the base we will go from 1 to 6 so from 1 to 6 this base the length of this base is 5 1/2 times 5 times height or we could say 1 is equal to 5 halves times height or multiply both sides by 2/5 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 width even if it was not directly specified
AP® is a registered trademark of the College Board, which has not reviewed this resource.