How can I fit a flexible distribution to a set of percentiles?

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I want to estimate a distribution for test scores however the only data available is in percentile scores {10,25,50,75,90}. I can measure skew and kurtosis from these percentiles but how can I fit the percentile scores into a continuous distribution while preserving the shape. here is some example data:
Test_Score_Data = [352, 410, 478, 543, 597]
Percentiles = [10, 25, 50, 75, 90]
Thanks
  2 Comments
Jeff Miller
Jeff Miller on 27 Oct 2020
Do you want to fit particular continuous distribution shapes (e.g., normal, gamma), or do you want to fit the percentiles into some generic, unspecified continuous distribution?
Matthew Jane
Matthew Jane on 28 Oct 2020
It does not need to be a particular distribution. Ideally I want to be able to capture skew, kurtosis, and the heaviness of the tails. Whatever distribution that can capture the precise shape of the distribution. (if I can calculate a leptokurtic shape or a slight positive skew in the percentiles I want the continuous distribution to reflect that)

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Accepted Answer

Jeff Miller
Jeff Miller on 28 Oct 2020
A normal distribution fits very well: see attached for a figure and the matlab code to generate it.
There is no obvious skew (543-478 is nearly equal to 478-410), and I think you would need wider percentiles to get much information about kurtosis.

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