We have learned that shape will dictate center. We have learned that shape can either be symmetric or skewed. When your shape is symmetric, you use mean. When your shape is skewed, you use median. But we learned in section 3.1, we learned in Chapter 2 that center is not enough. Knowing the typical value is not enough. You also need to know how is my data spread out. Now, when we discuss symmetric graphs and mean, we said its corresponding spread is standard deviation. And so, the last thing we're going to talk about then is what is the spread for data that is skewed. And here we go, guys, here we go. It's called IQR. It's called IQR, alright, guys, let's talk about IQR. IQR is the abbreviation for interquartile range. Where the definition of interquartile range is: it's representing what is the range in which the middle 50% of my data will live. Now, this is kind of an ambiguous definition, so I find the best way to understand IQR is to actually talk about how you find it. So how do you find IQR? Step one is you need to find the median. Step one is you need to find the median. We already talked about how to find median before this. We already said median is going to be some value where 50% of my data lives below it, 50% of my data lives above it. So that's why with this red line, we have 25 + 25 or 50 below it, 25 plus 25 or 50 blue data points above it. Why? Because what we've done now is come created a lower 50% of data and an upper 50% of data. And with each half data set, lower and upper 50%, we will then find the quartiles Q1 and Q3. Where quartile one is going to be taking that lower 50% and finding the median of that data set, meaning half of the blue dots will be above, half of the blue dots will be below that Q1 that are part of that lower 50% of the data. That will then give me my Q1. In the same way, we will take that upper 50% of data and slice that in half so that 25% of the data is above and below that value of Q3. Why do we want to find Q1 and Q3? Well, remember the definition of IQR is Middle 50%. So, I want you to see we've now sliced up my data into four regions and that the two regions in the middle are then between Q1 and Q3. And those two regions in the middle, 25% + 25% make 50%. That, guys, is what IQR represents. It's that distance from Q1 to Q3. That distance from Q1 to Q3 is then calculated by taking Q3 minus Q1 and that will give you the IQR. So let's make the rubber meet the road. Let's do a practical example of this. So let's go back to those six quiz scores again. Let's go back to those six quiz scores again and put them in order: 19, then 20, and 24, and 27, 28, and 30. Now, Step One is finding the median, which we actually already did. Did we already find the median was 25.5? But what I want you to see is that the median has now sliced my data into two subgroups. We have the lower 50% of data and now the upper 50% of data. And the idea of Q1 is that Q1 is going to be the median of this lower set of data. What is Q1? What is the median? What is the middle of this lower 50% of data? What will be Q1 here? Yeah, it's going to be this number: 20. And Q3 is then going to be the median of this upper set of data. So can you guys give me a hand? What is then the median, the middle data value of the upper 50%? Yeah, it's 28. And here's the cool thing, guys. Yes, your calculator will once again do its job. Once again, that one bar stat going to give us the Q1 and Q3. If you guys go back to your graphing calculator where you haven't even exited that screen yet, notice above the median and below the median is Q1 and Q3. And so, I want to emphasize to you that it's already done for you on the calculator. The only thing the calculator won't do for some reason is compute the IQR. But never fear, we have the IQR formula at our fingertips. The IQR formula is Q3 minus Q1. So literally just take that Q3 value of 28 and subtract it by that Q1 value of 20. Just do simple, simple, simple subtraction to give us eight. So then, of course, comes the all-important interpretation. How do we then interpret IQR? It will always be the range of the middle 50%. So, again, this is just another great template that you can put on your note sheet. The range of the middle 50% of quiz scores is going to be eight points. Yes, median and IQR still use the same unit value. Why don't you guys try example four? We're in example four. It's the same CO2 emissions problem. I want you to note I gave you Q1 and Q3. Alright, I calculated it for you. So, I just want you to work on finding IQR and doing the interpretation for IQR. I'll give you a few moments to work on that. And IQR is subtracting the bigger value minus the smaller value. I need to emphasize IQR represents a distance. There's no such thing as negative distance. So therefore, when you find IQR, it should always be a positive number: bigger minus smaller number. And that the interpretation of IQR is the range of the middle 50%. And in this case of what CO2 emissions per capita was 1.07 tons per person.