I have some empirical data from a research article (attached) that shows the climatic range of a particular group of plants. For each month (x axis), the mean, extreme min and max, and 0.05 and 0.95 quantiles are plotted.

The article says that the data is basically a frequency distribution. Would it therefore be correct to say that the mean is the 'optimum' conditions for this plant, and that away from this optimum in each direction, the probability that a plant will be present decreases, until there is basically no potential or probability above and below the extreme limits?

Can I infer from this that for any month, the relationship between temperature and plant distribution is a normal distribution (because the mean is not skewed one way or the other)?

Secondly if this is the case, I would like to take a given temperature (say -10 degrees) and be able to state

(a) the percentile of the distribution (i.e. 0.4) that it represents within the plant climatic range at a particular month.

(b) I would like to rescale it so that 0 represents no probability and 1 represents the mean, with an index in between.

If this makes sense, which equations would I need? Is it the Gaussian pdf, or the inverse Gaussian pdf, or the cumulative gaussian, or something else altogether? Does it make sense to return the percentile at one particular point (not the percentile above or below?)

Thank you for any help.