Interquartile Range (IQR)

We need a backup for the standard deviation. Something to use when we’ve got skew or extreme scores. We have a backup for the mean, it’s the median. So what’s our backup for the standard deviation? The answer to that is the semi inter-quartile range. The semi inter-quartile range is by definition half the distance between the 25th percentile and the 75th percentile. To which I expect you to say, what? Let’s make this really concrete and clear. In fact, rather than giving a technical or text based definition, I think it’s easier to see visually. So let me work through this definition with images. Here is an example of a distribution of data, and I have divided the data into four equal sized chunks.

They might not look equal sized to you, but the area– the pink area– in each of those chunks is the same. So I’ve got four points. The first of those green lines is the point at which 25% of the scores are below me, the next point in the middle is the point at which half the points are below me, and the third line is the point in which 3/4 of the points are below me. So I’ve got four equal sized chunks or quartiles. Well, the distance between those outer lines is called the inter-quartile range. In this case, it’s 2.69 units. So it’s just the distance between the first quartile, that first line, and the third quartile, that third line.

It’s that whole range there. And it measures how spread out the scores are, same as a standard deviation would. It’s actually larger than a standard deviation, so to get it closer to a standard deviation we’re going to go ahead and cut that in half. So if we have that and divide it by 2 it’s about 1.34, which is fairly close to what the standard deviation would be. In fact, if you plot the standard deviation there in blue, you see it’s actually pretty close to what the semi interquartile range would be, plus or minus from the mean. So we’re going to use this as our backup for the standard deviation.

So in summary, if we can use the mean, we’ll find the mean and standard deviation. We’re going to always report those together. Tells me what the average score is and how far scores differ from that. But if we can’t use that because of skew or extreme scores, we’re going to report the median and the semi interquartile range. Going to give me the same information. Where a typical score is and how much scores differ from that. It’s just not influenced by skew and extreme scores as much.