# Representation of population data mean/median (similar to BMI/age curve)

I want to represent my data (n=150 samples with a curve of value over time, similar to BMI representations in figure).

Possibilities are either mean/std and median/percentiles.

The question are:

• Can we chose the representation we want independently of the data distributions?
• How can we justify the use of either (besides one being prettier than the other)?
• Would it be better to use confidence intervals for either ?

Thank you for your help.

Can we chose the representation we want independently of the data distributions?

Yes, in principle, both methods are not restricted to any distribution. (For some distributions e.g. mean is not defined, but that is more the exception.)

How can we justify the use of either (besides one being prettier than the other)?

The representation with just mean and std is providing less information than the one with median and percentiles, presuming you provide sufficient percentiles. Think of a presentation that provides percentiles for all integer percent values. This gives you a very accurate description of the distribution, probably much more detailed than necessary. So the answer is: it depends on how detailed you want your data to be described.

Would it be better to use confidence intervals for either ?

Roughly speaking, the information provided by confidence intervals can also be gleaned from (appropriate) percentile presentations. Confidence intervals would provide a rather small amount of information, but maybe exactly what you need, so it really depends on what your requirements are.

Note that you cannot obtain confidence intervals from mean/std representations, but you also cannot obtain mean/std representations from a provided confidence interval.

In my experience, people without much statistical education can have a hard time correctly interpreting mean and std, while their interpretation of confidence interval and percentiles is more likely to be correct.

TL;DR For your case, use median/percentiles.

I would decide between those two options based on the following criteria: Is your distribution generally (at any point in time) symmetric? If yes, do mean and standard deviations. If not, do median and percentiles.

Perhaps someone more knowledgeable can give a more rigorous explanation (as mine will be a bit hand-wavy), but my reasoning here is simply that standard deviation is a "symmetric" measure. It is used to describe symmetric distributions, because those are the only ones it can describe accurately. Since it is a single number expressing the "spread" of the data away from the mean, std. dev. has no way of differentiating the spread on one side of the mean from the other. So for example, the classic "~95% of the data is contained within two standard deviations of the mean" is only true if you have a normal distribution because a normal distribution is symmetric, so one std. dev. to the right of the mean has the same number of points as one std. to the left.

However, if your distribution is skewed, this breaks down:

You can see that the std. dev. left of the mean has way more points than the right one, since the distribution is skewed and non-symmetric. However, in this same image, you can see that percentiles adjust to this (by definition, as they are designed to).

So, if your example graphic is a reasonably accurate representation of your actual data, I would use the median/percentiles because that distribution is clearly skewed (the spread of points "above" the mean at any x value is much greater than the spread of points "below" the mean). Confidence intervals are based on mean/SD, so they also can't capture the asymmetric nature of your distribution (so I wouldn't use it).

I hope this explanation managed to answer all 3 of your questions!