Example: Describing a distribution

sometimes in life like say on an exam in particular like an AP exam you might be asked to describe or compare a distribution and so we're gonna get an example of doing that right over here sometimes in life say on an exam especially on something like an AP exam you're asked to describe or compare a distribution and what we're going to do in this video is do exactly that in fact this one we're going to describe and then in a future video we're going to compare distributions now before we even read about this distribution or look at this distribution if you're asked to describe distribution there's four things that you should be thinking about you should be thinking about the shape of the distribution and when we're talking about shape it's going to be there could be left skew there could be right skew and we'll see examples of these and we've talked about them in detail in other videos it could be symmetric these are the ones that we typically see although there might be other types of shapes you will have your center of distribution and there's multiple ways of thinking about the center of distribution we've talked about this before you have your mean you have your median these are the two most typical ones you have a notion of spread and for spread you could use range you could use inter quartile range you could use something like a mean absolute deviation you could use a you could use the standard deviation these are all measures of spread and then you probably should at least comment about outliers even if you don't see them it's a good idea to comment just to make sure that you are being relatively comprehensive so now given that let's do let's describe the distribution right over here it says in the state of Connecticut the Department of Motor Vehicles the DMV requires 16 and 17-year olds to take a 25 question knowledge test in order to obtain a learner's permit to pass prospective drivers most correctly answer at least 20 questions on one Monday 22 teenagers took the test the dot plot below shows their scores so why don't you pause this video and see if you can take a shot at describing the shape the center the spread and the outliers some of these you might be able to come with the actual numbers you might be able to calculate some of these but really just to get a sense of it why don't you take a shot at it all right now let's do this together so first on the shape so what we see is we have most of the distribution is in this part between 20 and 25 but that we have this fairly long tail to the left and so this tells us that it we have a left skew or it is a left skewed distribution right over here so we have done the shape it's a left skewed distribution because the tail goes to the left now what about the center of this distribution so there's a few ways to measure Center mean or median just for the sake of simplicity I'll think about the median here I can actually do I can eyeball that to some degree you could also calculate the mean it would take a little bit more time I would guess that it's some place not even calculating it I would guess that it's someplace in this range right over there but let me actually calculate it so the median there's 22 data points so the median is whatever number has 11 on to the right of it and 11 to the left half of 22 so let's see we have 1 2 3 4 5 6 7 8 9 10 11 so the median here is going to be let's see this is 23 because we have a bunch of 23 s 1 2 3 4 5 6 23 s and if we were to just order all of the data points 11 of the data points would be 23 or less and then 11 would be 23 or more so our median here so I could say our center is 23 if we use the median and actually let me write that down so our median is 23 that's the measure of center that I decided to use now what about spread well the simplest measure of spread is just the range which is the highest value minus the lowest value and so our range here would be 25 minus 4 25 minus 4 is equal to 21 so that is a measure of range you could have others but this one is very easy to calculate and then if we think about outliers well there are a few outliers I would consider and that's it's very subjective there's use people can debate you know if there's a dot right over there is that an outlier or not but I would say that these four right over here I would consider outliers so I would say approximately four outliers but once again this is subjective the main point of this exercise is to just get in the habit of thinking about these things and statistics is all about creating engineering one could say different measurements for center for spread and different ways to describe the shape but the point is is to just think about these various dimensions