# How to determine distribution parameters from quantiles

May 30 20222 min read 6

# 23

TLDR: Feel free to download or make a copy of this Sheets to calculate the parameters of uniform, normal, loguniform, lognormal, pareto and logistic distributions (including the mean and median), based on the values of 2 quantiles.

# Scenario

Given a random variable  with cumulative distribution function , and two values  and  respecting the quantiles  and , what are the parameters of the underlying distribution? Answering this question is relevant to determine the expected value (mean) of .

As a concrete example,  could represent the cost-effectiveness distribution of an intervention whose 10th and 90th percentiles are 5 and 15. For this case, the inputs would be:

•  and .
•  and .

# Distribution parameters

The relationship between the values and quantiles of  is described by:

The parameters which define the distributions could be determined solving a system of 2 equations for each of the above relationships:

• For a uniform distribution:
• For a normal distribution:
• For a loguniform or lognormal distribution:
•  follows a uniform or normal distribution.
• This means the parameters referring to the logarithm of  could be calculated replacing  and  by  and .
• For a pareto distribution:
• .
• .
• For a logistic distribution:
• .
• .

Feel free to download or make a copy of this Sheets to calculate the parameters of uniform, normal, loguniform, lognormal, pareto and logistic distributions, based on the above formulas. I have also included formulas for the mean and median for all these distributions.

1. ^

The standard normal distribution has mean 0, and standard deviation 1. The quantile function is the inverse of the cumulative distribution function. The quantile function of the normal distribution could be calculated via NORMINV in Sheets, and scipy.stats.norm.ppf in Python.

# Reactions

Sorted by Click to highlight new comments since:

Useful stuff! I was working on something similar months ago and ended up eyeballing things.

Sorry, and thanks! They are working now.

The hyperlink on the word "this" (in both instances) is broken. I don't see how to get to the calculator.

Sorry, and thanks! They are working now.

Thank you for this! I had been trying to solve this exact problem recently, and I wasn't sure if I was doing it right. And this spreadsheet is much more convenient than the way I was doing it.

Great to know that you found it useful!

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