# Real (not rounded) Lower and Upper limits of groups of frequency distribution and Median for infinitely continuous data

From here on page 5: https://www.pbarrett.net/techpapers/percentiles.pdf

So how can both be correct – yet seem to be more appropriate under different conditions? The clue is spread throughout the various texts quoted above. The test score, although in many cases an integer value, is in fact deemed a point‐estimate of a hypothetical interval of continuous real‐value number scores. So, a test score of 4 is actually considered to be a point‐estimate of scores that can range from 3.5 through to 4.49999999999999999999999999999999999999. Therefore, when computing the median of 2, 4, 5, 9 as (4+5)/2 = 4.5, we are in fact computing an average of 4.499999999999999999999999999999999999999 + 4.5 = 4.5 (rounded). The first number is the upper bound of the point‐estimate 4.0. The second number is the lower bound of the point‐estimate 5.0.

Now take the example 2, 4, 5, 8, 9. The median is 5. But, the upper bound of this number is 5.4999999999999999999999999999999. It is a verbal “shorthand” that states that 5 is the median – in fact the upper bound of the median is 5.49999 etc (note it could also be as low as 4.5 given the definition of a point‐estimate number).

Below, I describe how I understand it in theory. Let's assume that our data is infinitely continues (can have infinite decimal numbers).

From the information from link I understand it as:

## Median

real (without any rounding) median of any infinitely continuous data cannot be a single number but should be represented as an interval with an upper and lower limit?

If my statement above correct then, if we define Q2 as the second quartile, the lower limit of median interval should be calculated as:

Q2 - 0.5


and the upper limit of median interval as:

Q2+0.49999...


## Group limits of frequency distribution

Some logic should apply to groups limits of frequency distribution. If we have groups in frequency distribution like:

... 14-16 17-19 20-22 ...

The lower limit of group 17-19 is:

17 - 0.5


and for upper group limit:

19 + 0.49999...


## Question: Are my thoughts above correct?

Also, additional information regarding this subject will be interesting.