Describing Distributions in Statistics

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“I am. Mr. Tarrou now that we have talked nabout. Some of the quantitative graphs graphs you might want to read about something about the ndot plot.

We have covered plots and histograms. We want to discuss or talk about how we describe ndistributions. The four things you want to remember when describing a distribution is nshape center spread and outliers now. It is very very important to be able to describe ndistributions well you need to use number use your data make sure you mention all four nof these variables the first free response question of every.

Ap stats test. I have seen nso far they release the questions and you can look them up on. Apcentralcollegeboardcom. Nthe very first question is always about.

Datadescribing graphical situations like this shape center. Nspread and outliers so shape is it symmetric or left skewed or right skewed. Is there na big gap in the. Middle so it is bi modal or is there only one peakis it unimodal ncenter.

Well you have three choices mean median and mode. Mean is when you add up nall the numbers and divide by the number of numbers. If you have 100 numbers you add nthem up and divide by 100 the median is when you take your numbers and you list them from nlow to high and you just gotick tick. Tickuntil you get to the middle one value nor the middle 2 values if you have the middle 2 values average those out or if you are ngoing to describe a distribution density curve .

The easiest measure of center to identify nis the mode. We will be doing that in a minute so for measures of center you have three choices nmean median and mode. If you just want to look at a graph and describe what you see nthe mode is the easiest measure of center to talk about spread well there is something nrange. Which is probably the most common measure of spread you just take the max value that nwould be to my right the max value.

And the min value and subtract them remember that nthe range is just one single value so here. We have a graph displaying a min of 20 and na max of around a 100 that would give you a range of 80. You don t ever want to say nthe range is between 20. And a 100 you can say the min is 20 and the max is 100.

But nyour range is that 1 value that one difference. Which is 80 at least. It will be in this example. Nthe iqr.

Just write this down there is a lot to talk about in the beginning of statistics ni am not sure if i am going sure if i am going to get two q1 and q3 type stuff in this little nvideo. But there is another measure of spread and that is the spread of the middle fifty npercent of the data you might want to just write that down because you don t really understand nhow to find the quartiles yet and then finally your outliers. So here we have a histogram nthat displays again some grades. We are just going back to that old example that i keep ndoing in these videos.

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So far. And we have one or two kids doing poorly a lot of kids ngetting in the b range. And then 90 up to a 100 90 up to a 100 would be your a s why nis this a histogram to begin with why is this not a bar graph because i am letting nmy numerical values represent a continuous random variable. I am not using the scores nto group them into letter grades like a b.

C. D and. F i am keeping the grades as the nraw scoreso histogram. I have my values in the middle of my classes here again if nyou have not watched.

My other video you can also have this displayed as 20 30 40 50 nnow. I common mistake. I see my students make especially if they are pulling numbers out nof their textbook. That is in a table.

Like 20 30. 1. Kid. 30 40.

0. Kids. 40. 50.

4. Kids nwhere. They make. The x axis.

Look. Like this 20 to 30 30 to 40. They are attempting to nlabel the x axis. But really what they are using is the table labelling to put into their ngraph and this is not correct.

If you are going to put numbers on each of these tic nmarks. Just do it. Once 20. 30.

40. 50. 60. 70.

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80. 90. 100. And for the one kid.

That nscored. A perfect 100. I have to go to 110 this is because again this last. Interval represents nx s that are greater than or equal to 100.

And less than 110. So this interval is for nx s that are greater than or equal to 90 and less than 100. So. If some kid scored a perfect n100 you have to make a new class interval for them and put them over here ok so how nwould you describe this distribution.

That is skewed to the left. See i have not even ntalk about that with you guys yet so. If you have a peak on the right side of your graph nand it tapers down to the left that is called a left skewed distribution. I am going to nbreak these shapes down for you and talk about these three measures of center would show na left skewed distribution.

And if it is right skewed their relationship will be the opposite ni will highlight that before this 15 minute period is up. Now. I don t have time to write nthis. But please pause and reply.

The video and write it down or read your textbook for nexamples about how to describe a distribution. This is a left skewed. Distribution the pointy nside is to the left this is a left skewed distributionshape. It has a min of 20 and na max in the 100 s giving my a range of approximately 80.

I just talked about spread so shape and nspread are done most scores were in the the 80 s i just talked about the center that nwas mode. There appears to be no outliers. There is a gap in this class interval. No none scored between 30 and 40.

But the grades slowly taper down to the left. So i would not ncall. This data an outlier. So this is a left skewed distribution with a min in the 20 s nand.

A max in the 100 s giving a range of approximately 80. The most common score is in the 80 s and nthere is no apparent outliers within those very short simple sentences. I talked about nthe four things you need to discuss when you are describing a distribution shape center nspread and outliers here. We have a left skewed distribution.

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Now i have not talked nabout these concepts yet but you can hear them and we will discuss them later as we ncontinue on through the notes when you have a left skewed distribution and we are talking nabout these three measures of spread when you have a skewed distribution. The mean is nalways closest to the tail. Now i am not really sure where the mean is in this data. But i ndo know it is going to be toward the left side because that is where the tail tapers noff.

Probably not this far so don t think i am saying the mean if 50. I really don t nknow what it is when you have a left skewed distribution a little bit larger than the nmean is the median ok. I just eye balled that i have know idea exactly where the median nis. We will maybe actually stop this and recalculate where the median is in a minute and the mode nis always where you have the peak of your graph.

So the mode is in the 80 s when you nhave a left skewed distribution. The mean is smaller than the median and the median nis smaller than the mode in numerical value. If it is a right skewed distribution let nme put this here. If it is a right skewed distribution.

Then the mode is the smallest nmeasure of center. The median is a little bit larger and the mean is always closest nto. The tail is going to be. Larger so when you have a right skewed.

Distribution this nis right. Skewedi am running out of room to write you have your three measures of ncenter when it is a right skewed distribution the mode is the smallest measure of the center nthen comes the median and then always closest to the tail is the mean i jump around a lot nwhen. I talk about ap stats in the beginning. Because there is so little that you know nso.

I jump around a bit. But you will see this in notes later as well so if you don t have nthe. Written. Now you will have it written later as you keep watching my videos as we ngot through the year.

If you have a symmetrical distribution. Whether it is a bell curve like nthis or whether it is was bimodal. Such as this some of this might not make sense. Unless.

Nyou also read your textbook. But if you have a symmetric distribution your mean and your nmedian will be roughly equal so. If your mean and median are approximately equal that is nyour sign for symmetry. If your mean median and mode are equal this will get repeated nlater because i am jumping ahead a little bit.

If your mean median and mode are all nthe same that is a sign that you have a unimodal symmetric distribution. Let s see here let s ntry and find where the median actually is in this histogram. I have 1 number 2. Numbers ni have got 2 people in.

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This category that is 1 2. And. 2 make 4another 2 people make n6another 3. People.

Makes 9let s call that. 4 so a total of 13now 15 and 16. This ngraph represents the way it is made as a frequency. Histogram represents.

16 pieces nof data. So if finding the median goes from low to high and the median is the middle nnumber. So if there is 16 pieces of data that means that there is 8 over here in the nlow end and 8 over here in the upper end. I want to find where is the 8th or 9th number nto estimate the median from this histogram that i have now scribbled all over the eighth nnumber rightlow to highcounting the tops.

I have 16 pieces of data. So i want nthe eight ninth number 1 2 4 ok so i guess i was wrong. 6. Because i am not to the eighth nnumber yet actually i wanted to.

Draw this correctly then this comes out. And. 1246. Nand another 3 makes 9.

The median is actually if you want to go back to your notes and change nthat somewhere between 70 and 80 so left skewed distribution. Mean is the smallest nmeasure of center. Then median and then mode. If you have a right skewed distribution.

Then nit is just the opposite by the way this will come up in your textbook. As well all nmoney questions are right skewed. You always have a lot of people doing the grunt work nand then a few people at the top of the company making the big. Decisions and the ceo s of nthe.

Companythe stars of the football team or whatever you always have that handful nof people making a ton of money all money questions in our textbook are going to be nright skewed distributions that is a quick introduction on how you describe the shape nof a. Distribution don t forgetshape center spread and outliers don t forget to read nyour book read the examples and do your homework. Because you are going to want to have those nclass discussions and get help from your teacher. ” .

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