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

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

[Jeff Miller]

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]

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)

Answer 1

[Jeff Miller]

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.