If I remember correctly you can calculate the standard deviation - sigma - of your datapoints using the FWHM when you have a gaussian (bell shaped curve).

FWHM = 2*sqrt(2*ln2)*sigma

Your data points in b) seem okay to be described by a gaussian. In general you might consider using a combination of a gaussian and lorentzian curve. In this case, as long as you have a maximum consisting of only these few datapoints, I think a simple gaussian is justified.

FWHM is related to the standard deviation by FWHM=2.354*sx

Dear Robin Duncan Wylie,

The above answers seem relevant. Still, I make some remarks, of which some may seem irrelevant to you but others  may be helpful.

1. A material is some matter applied for some technical purpose. So I got the impression that the subject of your study is a chemical or physical transformation.

2. Transformations develop in time, so kinetics is essential, given the thermodynamical conditions satisfied.

3. What is plotted on the Y- and X-axis of your graph a?

4. The accuracy of the transition will have values along the X- and the Y-axis.

5. What is the interpretation you would give to the transition point on the X-axis:

a. transformation completed for 50 % or b. maximum of transformation rate.

6. Supposing the transformation studied is thermally activated, would you like to use the transformation point for an activation energy analysis? 

As far as I can tell, your presentation is comparable to what is done with the Avrami equation to model kinetic transitions. It seems you are after the time to have 50% transition, t50. Note, you can also define other transition times such as t25, t75, ... 

You have two questions. First, how do I determine a transition time tX? Second, how do I establish the confidence in my value?

For the first question, you have two approaches. First, as you suggest, take the derivative of the raw data and proceed from there. Here are the problems with this approach.

* Since it has no basis in relevant theory and is just a "guess" to a pseudo-model, it _might_ give you what you are after (or it might not).

* It certainly and unnecessarily compounds the overall uncertainty in your final result.

The better method (that fixes both problems) is to use a justifiable kinetic model (e.g. Avrami equation) and fit your raw transition versus time data with a non-linear regression method.

I have done this level of analysis on examples for what are called TtT diagrams in austenite steel. I found the figures in materials textbooks, and I pulled the data out using a digitizing tool (a ruler and a pencil!). It is an interesting exercise for a graduate-course in materials science. I think that I have some home-built tools to do the analysis on-hand the raw data. You are welcomed to contact me off-line for further details.

Dear Robin Duncan Wylie,

Some remarks/questions:

1. The basic point in the above discussion is of course what about the physics/chemistry of the transformation studied to assume a Gaussian curve depicting the derivative of graph a.

2. The accuracy of independent variable plotted along the X-axis should be known in advance and can be compared to FWHM?

3. The concept of standard deviation belongs to the world of statistics; the above graphs reflect a number of measurements? 

4. Your plot suggests a thermally activated transformation, so I would suggest to analyse your data correspondingly.

I saw the chart and reminded me of the Logistic Equation.  I agree with the comments that point you to focus on the Physics (First Principle) of your system and try to match your data to that system.  So check the Logistic equation or the Avrami equation (maybe are related, but I do not know much about this one), and understand what best describe your physical system.  Fit your data to that Equation.  Excel can do this easily using "Solver".  If you decide that the Logistic Equation makes sense, here the equation I recommenced:  

y = FinalValue +(InitialValue - FinalValue) / (1+EXP(ROC*(x - MiddlePoint)))

ROC = Rate-of-Change

If you fix "FinalValue" and "InitialValue" then you only have to fit two variables (ROC and MiddlePoint) to your data.  Otherwise it will be four unknowns that could be resolved by iterations (minimization of the error² between the equation and the data)

Your resulting "MiddlePoint" will correspond to your Average "y". 

Regarding your uncertainty... Hard to recommend any approach with only one set of data.  Several Sets of data, and several "MiddlePoint" fittings could be used to determine an uncertainty value.

@R D Wylie: With all the supportive answers here, I have to ask something now. Can you do us the return favor to add some further background to the motivation for your question? Truth be told, the statement "a professor of mine" as a driving force seems to leave me with some questions about the professionalism and end goal of your posting.