Summarizing Quantitative Data: Measures of Spread

A histogram of a quantitative variable. The vertical dark blue line is the mean, and the vertical red lines represent -2, -1, +1, and +2 standard deviations from the mean.

Hi, in this video we're going to keep on using the EPA flight data set. This is, these are facility level emissions of greenhouse gas data, but now to exemplify how to calculate measures of spread. So we're going to use those data to find standard deviation and then the interquartile range in Python. So, let's get to it. So first off, to find the standard deviation, we can find this in a very similar way as to how we found the mean and median above. So I'll do this in a print function, as we did before. So standard deviation is equal to, and then we call our whatever variable it is we want to do this on, 
so total reported direct emissions in this case. And our function now is not mean or median, but STD for standard deviation. And this actually shares the same units as the underlying variable. So in this case, millions of metric tons of CO2 equivalent. Okay, so our measure of spread, basically our measurement of how much these direct emissions vary across all the facilities in this data set, comes out to be a little bit more than a million. Pretty large number! Okay. Well, let's compare this to the interquartile range now. So we're going to calculate the interquartile range using the describe function. And there's different ways to do this describe function. I want to do it this way because the describe function is actually quite handy for a number of things. So let's first find that. So we'll call our variable total reported direct emissions, and our function here is just describe, and here we have to put in parentheses at the very end. And then we'll view the output.

Here we go. So this is a series actually, and it's got some, a lot of nice values. So we see our standard deviation here. We see our mean value. It also tells us how many values are being used here; so 6,481, in this case. Note the use of scientific notation here. And then the minimum zero, max of 20 million or so, and then 25, 50, and 75 percent. This 50 percent is the median. So this agrees with exactly what we had in the median above because this 50 percent 
refers to the 50th percentile. And that is another name for the median. So those are synonymous. And then we also have the 25th percentile, and the 75th percentile. So 25 percent of the values are less than 31,000 and some change, here. And 75 percent of the data values are less than 186,000 and some change here.

So our interquartile range is the difference between the 75th percentile and the 25th percentile. So we can simply find this by referencing these values in the describe series above. So it's our 75th percentile minus our 25th percentile. And you can view that value. Turns out to be 154 000 or so. So quite a bit less than standard deviation. So these are two different measures of the spread. The standard deviation is you know the typical distance away from the mean. And if you recall, in the comparison of the mean and median we have a right-skewed distribution up here. And so, our mean was a lot larger than our median. And so, there's a lot more spread about that mean, if we're looking about it that way. Whereas, the median and the interquartile range are much more concentrated down in this lower end where we have more observations.

So in summary, that's how you can find the standard deviation and the interquartile range. Along the way we found this handy describe function which is useful for summarizing the data in one fell swoop. And again, we got to do all this with the fascinating flight data set on greenhouse gas emissions across the U.S. Okay, thank you.