Measures a variation so the problem tells us to use the table bellow and answer the following questions. So here’s a table that talks about the calories and serving of juice, okay. The questions we need to answer are one to find the range of the calories, two to find the medium and the upper and lower quartiles, three to find the inter-quartile range and last number four to find any out-wires in the data.
Before we move in, let’s talk about measures of variations. So Measures of variations are used to describe the distribution of data. So Measures of variations are used to describe the distribution of data that indicate house spread out. The data is, let’s take a look at our first question number one. So find the range of the calories, so when you're finding the range, your range is the difference between the least and the greatest.
So it’s the difference between your least and your greatest numbers in your data side. So for looking at our data set here, our greatest value is 180 which is the prune juice and our least number of calories is the tomato juice which is 35. Now we’re going to take the difference, so our range is going to equal between this numbers. So I'm going to take 180-35 which will give me 145 calories, so the range of calories is 145 calories
Let’s move on and take look at find the Median and the upper and lower quartiles. So when you're talking about quartiles, quartiles they divide a data set into, they divide the data into four equal parts. So when you're looking at the median, your median separates the data into two equal parts. So the median separates your data into two equal parts, so the data in this problem, I'm just going to give myself a little bit room here.
The data in this problem is followed, so we have a 120 from the apple juice, we have 80 for the carrot, we have 170, 100, 120, 110, 180 and 35. Now to find the median, we first need to order our data from least to greatest. So I'm going to go and reorder this from least to greatest, I'm going to start with 35 then I'm going to go to 80, 100, 110, 120 I have another 120, because I actually forgot to add the apple juice in there. So I did, I just didn’t see it at first, okay, 170 and then 180.
So now, your median, I'm just going to get a little bit more room here. so my median is going to be my two middle numbers, well if I look, my two middle numbers are 110 and 120. So this is my median, so if I have two middle numbers I need to find the average or the median of it. So to find the average I'll take 110+120 and I divide that by two to find your average and that would give me 115, okay. So the median is 115, now I'm going to take, now I have my, I'm going to give myself a little bit room here.
So now I have my median, so I'm going to take my numbers which are still in order and I'm just going to rewrite it, so I have 35, 80, 100, 110, 120, 120, 170, 180. So now I know my median is between 110 and 120 right at 115, okay. Now I'm going to look and find my upper and lower quartiles. So to find the lower quartile, this will be the lower half of your median. So this will be my lower quartile and this side will be my upper quartile, okay. So now I have my lower and my upper and to find it, I'm going to take the median of the lower. So my lower quartile is actually going to be the median, the middle values of my lower quartile. So my actual lower quartile is going to be the median, but since I have the two values I have to take the median of this again. I'm going to take 80+100/2 and that will give me 90. So my lower quartile is going to be 90 and to find my upper quartile in my upper quartile range to find the upper quartile I need to find the median, well again I have even numbers. So my median is going to be the mean or the average of this two middle numbers, I'm going to take 120+70/2 and that will give me 145.
So now I have my lower quartile, okay is 90 and my upper quartile is 145. All right let’s go ahead and move on to find the inter-quartile range. So the inter-quartile range is the range of the middle half of the data. So we are looking at the middle half of the data. So it is the difference between the upper quartile and the lower quartile, so you’re looking for the difference between your upper quartile and your lower quartile. So we know that our, so basically what we’re doing is we’re taking our upper quartile and subtracting it by our lower quartile. So our upper quartile is 145, so we’re going to take 145- and our lower quartile was 90 so we’re going to take 90 and if subtract this, I get 55. This will be our inter-quartile range, so our inter-quartile range is 55. So please note though that a small inter-quartile range means that the data in the middle of the set are close in values.
Our large inter-quartile range means that the data in the middle are spread out, so that’s something to keep in mind. Let’s take a look at our last one to find out-layers in our data. So data that are more than 1.5*a value of the inter-quartile range beyond the quartiles are called out layers. So if it is 1.5* the value of your inter-quartile range beyond the quartiles are considered, are your out-layers.
To find the out-layers in this data then, the first thing we need to do is multiply our inter-quartile range. So we’re going to take our inter-quartile range and we’re going to multiply it by 1.5, so that’s our first step. So our inter-quartile range for this data set was 55. So here we’re going to take 55 and we’re going to multiply it by 1.5 and that will give us 82.5. So now our second step is to find the limits for the out-layer, so now we need to find the limits for the out-layer.
So here step one we found finding our out-layer and number two is we’re finding the limits, okay. I'm sorry number one is we’re setting up to find our out-layer and number two is we’re finding the limits for our out-layer. So now to find the limits for the out-layer we’re going to subtract 8.82.5 from the lower quartile range, so our lower quartile was 90. So we’re going to take 90-82.5 and that’s going to give us 7.5. Now to find the other limits, what we’re going to do then is find, is going to add 82.5 to the upper inter-quartile range, so here we added, sorry here we subtracted from the lower inter-quartile now we’re going to add to our upper quartile, so our upper quartile was 145 and we’re adding the 82.5 and that gives us 227.5. So the limits for this out-layer, so our limits for the out-layer are 7.5 and 227.5, no value in the data set beyond this limits. So if we look at our data set, there is no value, there is nothing lower than 7.5 and there is nothing greater than 227.5 in our data set.
So that tells us that there are now out-layer in this data set, all right. So let’s do a quick review of what we just went over, okay. So remember the measures of variations, measures of variations are used to describe the distribution of the data. They indicate how spread out that data is. Some of the measures are, we talked about range, we talked about quartiles and we talked inter-quartiles. So this are some of the measures on that we used to measures of variation. So let’s look at the range for a second, remember the range is the difference between your least and the greatest value in your data set. The median is your middle value, incase of add numbers of value. So if we have an add number of values, your median is the middle value but in the case of even values which we had in this data set. The median then is the average or the mean of those two middle values and then you have your lower quartile which is the median of the lower half of the data, you have your upper quartile which is the median of the upper half of your data.
The inter-quartile range remember is the range of the middle half of the data and it is the difference between your upper quartile and your lower quartile, and then remember, data that are more than 1.5* the value of the inter-quartile range beyond the quartiles are called your out-layers.
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