@Noetsi

I'm not sure I follow your argument here. At the end of the day, the median is just a summary statistic. In other words we are trying to use one (or a few) numbers to describe a data set. When there are a lot of tied values, however, the median, as it is traditionally calculated, falls short of this.

The example I was struggling with in my previous post was this. Students rated a video and a practical demonstration from 1 to 5 on a questionnaire. One set of results was the following:

Group 1 shown video

3 3 4 2 4 4 3 4 2 3 4 3 4 4 2 3 4 3 4 4 4 3 4 4 3

Group 2 given presentation

4 4 3 5 4 4 4 4 4 5 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 4 4 4 4 5 5 5 5 5 4

Now, anyone looking at this dataset would immediately say the presentation did better (got higher scores). And if you do Mann Whitney Test you get p<0.0001 which would validate this.

But the median of both these datasets is 4. Therefore, in this particular example, the median (as it is normally calculated) does a poor job of summarising the data. It fails because of the large number of tied values in the middle. It just isn't a good summary statistic for this data.

So what do you do? You could take the mean of the data since the mean is not affected by tied values in the middle but this is technically 'wrong' in this case as Likert data is ordinal and the mean shouldn't be used.

So, for my purposes, bootstrapping worked well because it told me what the mean of many medians from the samples would be. Comparing the mean of medians (or, alternatively put, median + bias) was much better for my data as it gave me the following:

Group 1 3.58

Group 2 4.03

This, to me, summarises the data much better.

I guess the confusion arises when we say 'is there a better way to calculate the median'. In a sense, I don't have the median anymore, I have a new statistic (median + bias for want of a better term). But for me, median + bias does a better job of summarising the data than does the median (or mean).

I guess maybe the OP just worded it badly when he said 'The median is 3.7'. Maybe if it was given a different name, such as the 'interpolated median', it might offend a little less.