I am using someone else's research and only have part of the data. I want to find the percentile of a value, such as 350.2, based upon the data given.
The following is known:
mean = 308.2
The following percentiles are known:
5%: -27.4
25%: 3.2
50%: 56.3
75%: 260.4
90%: 736.0
95%: 1364.8
I assume you would want to calculate, the std deviation, and then from there you can calculate the given percentile for a value is that correct?
This data looks a little weird to me, as you would assume the difference between the 50th and 25th to be the same as the difference between the 75th and 50th.
Your final observation about the $25$th, $50$th, and $75$th percentiles just says the distribution is not symmetric. Do you have a reason to believe that it should be? Using the standard deviation assumes the distribution is Gaussian, which the evidence contradicts. All you really know is that $350.2$ is somewhere between the $75$th and $90$th percentile. If the events between $260.4$ and $736.0$ are equally distributed, you would have $350.2$ at the $\frac {350.2-260.4}{736.0-260.4}\cdot 15+75\approx 78$th percentile. It is probably rather higher than that-if the data is mostly near the middle it will be. It is hard to give a nice justifiable answer.