In this video I explain the difference between calculating an exclusive interquartile range and an inclusive interquartile range. While we generally use an exclusive interquartile range with an even set of scores, when we have an odd-numbered set of scores we can choose which method we’d like to use. The inclusive method will tend to give us a slightly smaller interquartile range by increasing the 25th percentile and decreasing the 75th percentile.

Video Transcript

Hi, I’m Michael Corayer and this is Psych Exam Review. In the video explaining the interquartile range I didn’t go into detail on the fact that there’s actually two different calculations for finding the interquartile range. These are an inclusive method and an exclusive method, and this refers to whether we’re including the median as being part of our upper and lower halves of the data or if we’re excluding the median.

Now if we’re excluding the median what we’re really saying is that the lower half of the data is less than the median and the upper half is greater than the median. But if we’re using an inclusive method what we’re saying is the lower half of our data is less than or equal to the median and the upper half of the data is greater than or equal to the median. This will slightly change the boundaries of where our 25th and 75th percentiles lie and this means that it will slightly change the interquartile range. If we have an even number of scores then we naturally tend to exclude the median because it falls between two of our values. It’s the mean of the two middle scores; it’s not actually one of the values of our data, so it makes sense to exclude it from being part of the upper and lower halves of our data.

So let’s take a look at a set of 10 scores and see what I mean. Since we have 10 scores our median is going to be the mean of the 5th and 6th positions: 9 and 11. So this gives us a median of 10. And we can see that this line for 10 neatly divides our data into two halves and now we take the median of each of those two halves to find the 25th and 75th percentiles. So we have a 25th percentile of 6 and a 75th percentile of 15. This gives us an interquartile range of 15 minus 6 equals 9. Now in this case we’re excluding the median because it’s not actually one of our values. So it makes sense to say that well 10 is not really part of the lower half of the data and it’s not part of the upper half of the data.

But in the case of an odd number of scores we can see that our median is going to be one of our values and that means we might want to include it as being part of the lower and upper halves of the data. So let’s take a look at a group of 11 scores. In this case our median is going to be the 6th position, which gives us a median of 11. Now if we want to find an exclusive interquartile range we’re going to ignore this 11. We’re going to define the lower half as being below this and the upper half as being above it. So now we find the medians of those two halves and we get a 25th percentile of 4 and a 75th percentile of 18. So to find our exclusive interquartile range we take 18 minus 4 equals 14.

But we could decide to do an inclusive interquartile range here. This means we’d say that the lower half of our data extends up to and including 11 and the upper half of our data starts at 11 and moves up. This means that the 25th percentile will be the mean of 4 and 5, which would be 4.5 and the 75th percentile would be the mean of 16 and 18 which would be 17. So now we’d have an inclusive interquartile range of 17 minus 4.5 equals 12.5.

As we can now see, if we use this inclusive method we’ll tend to get a slightly smaller interquartile range. This is because by including the median as part of the lower and upper halves we’re moving the 25th percentile up and we’re pulling the 75th percentile down, so we’re looking at a narrower range of our data now.

Depending on our data, this distinction between an inclusive and exclusive interquartile range might be trivial or we might find if we have repeated scores around those points that it actually doesn’t change the interquartile range at all. This distinction isn’t always fully explained in introductory statistics textbooks but I think it’s important to understand that there’s two calculations and to understand the difference between them. It can also help you understand why the same data set in different software programs might lead to slightly different interquartile ranges if one is using an inclusive method and the other is using an exclusive method.

I hope you found this helpful. If so, let me know in the comments, like and subscribe, and make sure to check out the hundreds of other psychology tutorials that I have on the channel. Thanks for watching!