Median

Sometimes we can’t use the mean, either because we have too much skew, or we have extreme scores. In those situations we’re going to use the median. The median is the point in which half of the data are below you and half of the data are above you. This is functionally going to tell you where the middle of your data is, it’s going to tell you what a typical score would be, but it’s less biassed by skew, and therefore it forms a good backup for the mean. Now our symbol for this is mdn, which essentially just stands for median. So when you see this you know that somebody using the median instead of the mean.

Let’s take a look at this data, for example. In this case, we have age distributions, and in this case, you see that you have some pretty strong positive skew. Most scores are down on the lower end of this distribution, and so therefore you have a lot of positive skew here. Rather than using the mean in this example, the median would be the best measure of the average. There’s so much skew here that is going to pull that mean up toward that long tail, those extreme scores will. The median will give you a better sense as to what the middle of this data are doing, because it doesn’t care what those values are.

I could have somebody who was 100 years old in this data set. I could have a tonne of people who are up there, it doesn’t matter whether they’re up there, it doesn’t matter whether they’re in the 50s or 60s, it doesn’t matter what those values are. This is the beauty of the median. It just doesn’t care what the extreme scores are, so it can’t bias your test statistic. So how do we do the median? How we calculate it? Well, we’re going to organise our scores in order. We’re going to sort them from smallest to largest. Then we’ll just pick the middle score, or if you’ve got two middle scores, the average of those.

So for example, say I’ve got the numbers 2, 4, and 90. Doesn’t matter that that 90 is an extremely large score. The middle score is 4. So here we see the median in all its glory being unbiased by that score of 90. Or if we’ve got a couple of middle scores, 2, 4, 5, 90, we’re just going to have a median a 4.5, because we’re just going to average those two middle scores. So there we go. It’s a beautiful back up for the mean. It doesn’t care what the extreme scores are, and it is therefore a useful substitute when we’ve got too much extreme skew or extreme scores to use the mean.