Density Curves and their Properties (5.1)

in this video we will be learning about density curves and the properties of density curves but before we talk about this we have to do a bit of review let's say we have a room of 20 people and we record each of their weights then we take this information and create a histogram recall with any histogram we can transform a regular frequency distribution into a relative frequency distribution the only difference between these two is that a regular frequency distribution tells us the number of people within a given interval whereas a relative frequency distribution tells us the proportion or percentage of data values within that same interval for example there are six people that weigh between a hundred and a hundred ten pounds but we can also say that thirty percent of the people in our data set weigh between a hundred and 110 pounds to convert a frequency distribution into a relative frequency distribution we first needed to know the total number of individuals we were working with in this example we are working with a total of 20 people next we needed to determine the proportion of individuals in each interval we did that by dividing the number of people in each interval by the total amount of people this gave us our relative frequency values now you know that you have done this entire conversion process correctly if you add up the relative frequencies of each interval and they add up to one the total area of any frequency distribution is always equal to one or a hundred percent so what is a density curve anyways and where does it come from a density curve is just the curve that helps us visualize the overall shape of a distribution if we take the histogram we just work with and if we draw a curve around its distribution we have essentially made a density curve this can be done with any type of histogram with any shape and with any form and like a relative frequency distribution density curves always have an area that is equal to one or a hundred percent now you might be wondering what's the point of a density curve anyways why don't we just stick to using histograms well density curves have a few advantages over histograms first of all density curves give us an idealized picture of a population or dataset without considering irregularities and outliers because of this it really gives us a great overall picture of the actual distribution and its tendencies secondly the picture of a histogram really depends on how many intervals you have the more intervals you have the better you can see the distribution of the data but with a density curve you are not limited by the number of intervals that are are and you can actually have an infinite amount of intervals and third a smooth curve is generally easier to work with than a histogram especially when you are working with very large populations now the use of density curves it becomes more practical the larger your population is if we made a histogram from data we collected from 50 people and drew a density curve over it we have a lot of missing gaps and because of that our density curve can be inaccurate now imagine if we had a population of a hundred thousand people instead of 50 because their population is so big we can continuously reduce the length of each interval we have until we end up with so many intervals that we essentially end up with a histogram that can be accurately represented by a density curve this is why density curves are so valuable they aren't limited by these intervals and they can be very useful when working with very large populations we can also use density curves to make approximations for example if we have a density curve that represented the test scores of 1 million people by looking at the curve we can say that half of the people scored over 60 on this test we can also say that a large majority of students scored between 50 and 70 because there's a lot of area contained within this region we can also say that only a few students did very well on this test because there's only a small amount of area contained within the upper tail region of the curve you'll know how to calculate these exact areas in an upcoming video but now that you know what a density curve is we have to go over some very important rules for having a valid density curve in other words we'll have to talk about the properties of a density curve the first rule is that a density curve must lie on or above the horizontal axis density curves that are drawn along the y axis or ones that dip below the x-axis are invalid a density curve has to sit on the x-axis in order to be valid the second rule is that the total area under the curve is always equal to one now if I said that the total area was equal to 512 70 or any number other than 1 then I do not have a valid density curve however if I said that the total area was equal to one or a hundred percent then I do have a valid density curve this is a very important fact and it's one worth remembering density curves come in many different shapes and sizes some are well known mathematically and others aren't each type of density curve has its own name a common density curve you might encounter is the uniform distribution it is called this because each interval has the same frequency of data values and is uniform throughout the entire data set we also have the triangular distribution and it's called this because well it looks like a triangle and most importantly we have the normal distribution also known as the bell curve in statistics this is the most important density curve that you should familiarize yourself with we'll be talking a lot about this curve in the upcoming videos but for now it's very important that you understand the concept of density curves so let's do some practice questions feel free to pause the video at any point so you can try these questions for yourself question number one for the density curve below approximately what percentage of people weigh exactly 150 pounds a common mistake that students make in these types of questions is they see 150 on the graph they draw a line and then they see that it lines up with 0.20 so they will say that the answer is 20 percent now this is incorrect remember that the total area of a density curve is always equal to 100% this line definitely does not have an area of 20% in fact the area of this line is equal to zero because a line has no width as a result the answer is equal to zero logically this makes sense because realistically no one will ever weigh exactly 150.000 pounds usually you'll have some measurements very close to it like 150 point five one hundred fifty point seventy one hundred fifty point zero five and so on and so forth however if I asked you what percentage of people weigh between 150 and 152 pounds then we have this entire area to account for we can actually get a rough estimate of this area we know that the area of a rectangle is equal to the length times the width we have a length of 0.15 and a width of two multiplying these together gives us an answer of zero point thirty however we still have this top portion to account for to estimate this area we see that it is almost equal to half of a square the area of a square is equal to its length times its width which is equal to 0.05 times 2 which gives us zero point one and half of 0.1 is equal to 0.05 therefore the total area is roughly equal to 0.05 plus zero point thirty which is equal to 0.35 as a result the percentage of people that weigh between 150 and 150 two pounds is roughly equal to 35 percent as I mentioned before you'll know how to calculate these exact areas in an upcoming video using a different method question number two for the uniform distribution below what must be its width in order for it to be a valid density curve we know that the area of a rectangle is equal to the length times the width algebraically solving for the width tells us that it is equal to the area divided by the length we know that the area of any valid density curve is equal to one so the area is equal to one we have a length of eight so one divided by eight gives us an answer of 0.125 question number three for the uniform distribution below what proportion of values are located between twelve point three and eighteen point six on the graph 12.3 and 18.6 is located somewhere around here this is our area of interest now we know that the area of a rectangle is equal to its length times its width the length is equal to 18 point six minus twelve point three which gives us an answer of six point three and the width is equal to 0.05 multiplying these together gives us an answer of zero point three one five this means that the proportion of values contained within this interval is equal to zero point three one five or 31.5% if you found this video helpful consider supporting us on patreon to help us make more videos you can also visit our website at simple learning per com to get access to many study guides and practice questions feel 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Channel: Simple Learning Pro

Views: 118,843

Keywords: Statistics (Field Of Study), Statistics, stats, ap stats, ap statistics, Lesson, Tutorial, math, mathematics, Simple Learning Pro, Density Curves, Density Curve, Normal Distribution, Normal Curve, Bell Curve, Area, Distribution, Uniform, Triangular, Histogram, Practice questions, explained

Id: Txlm4ORI4Gs

Length: 9min 36sec (576 seconds)

Published: Thu Jun 28 2018