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## Statistics and probability

### Course: Statistics and probability>Unit 4

Lesson 4: Density curves

# Worked example finding area under density curves

Worked example finding area under density curves.

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• tell now i dont really understand why the area always 1 , if we consider our set of data to be (1,3,4) and another set to be (321,4,21,4,51,3,321) how the two will confirm the same area of 1 . i understand the concept of the 100% that the curve represented the entire data of my set but why the area will be always 1 ?! • could you help me get out of that paradox please? i know that 1 represents 100% in general but if we have intervals of 10 (mean 60-70 and 70-80 and 80-90) for example then the area under the relative frequency graph will be 100% of that 10 means 10 not 1.
say:
10(20%)+10(15)+10(65%) this will sum up an area of 10 not 1 • If we were to make a density curve from this bar graph we would assume that the 20% that are in the 60-70 bucket are spread out evenly, so there are 2% between 60-61, 2% between 61-62, and so on...
The same process would be done to the 70-80 and 80-90 buckets, and we would end up with the sum
10(2%) + 10(1.5%) + 10(6.5%) = 1
• How do you deduce that height of A is 0.25 and of B is 0.50. It is simply not known.
Better way would have been to divide area into rectangle & triangle and calculate it. • how do you know it was 0.25 i dont understand that • How do you find Q1 and Q3 in this case? • is there a mathematical reason why the vertical lines are dotted instead of drawn in a line? (perhaps because a density 'curve' must be a function?)
(1 vote) • It indicates the boundary. The dotted lines imply '1' and '3' are not part of the probability density function.

As such it doesn`t matter in terms of finding the area as including '1' and '3' will make an infinitely small difference to the area.

In other word including 1 and 3 has no noticeable impact on the percentage for the probability density function.
• Right so in the first problem, when you find the area of the trapezoid, won't you be finding the area of the curve when x is more than or equal to two instead of when x is more than two?
(1 vote) • Suppose we have the data : 10,10.5,11,17,19 and we map it into two intervals 10-15 and 15-20. So, the interval 10-15 will contain 3/5 data points and the interval 15-20 will contain 2/5 data points.
Now if we calculate the area (as sum of areas of two rectangles), it comes out to be:
area=3/5*5 + 2/5*5 =5 but not 1.
Can anyone explain this to me?
Thanks.
(1 vote) • Does anyone know how you could find the area of different percentiles if the hight was not given?

I'm working on a problem where the density curve is a perfect triangle. The base is given (1.6) and I understand that the total area is equal to 1.0. Using A=1/2(b)(h) you can find H.

But now I'm stuck in finding the percentages of selected areas within the triangle such as: What is the percentage of values that are below 0.4?

I see that a new triangle is formed and if you can find that area you can subtract from one and find your answer, but in this case, we have two variables and I'm stuck. • In both examples we could work out the answer without knowing the numbers on the y-axis... It's obvious in the second example, and in the first one we could say the height at the point "3" is `x` so the height at the point "1" is `x/3` and the area under the entire density curve would be `2.(x+x/3)/2` which equals "1" and so we solve the equation for `x` and it would be `3/4 or 0.75` >>> and then we calculate the area under the smaller trapezoid 