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.
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 .
The relationship between the values and quantiles of is described by:
- For a uniform distribution with minimum and maximum :
- For a normal distribution with mean and standard deviation , denoting as the quantile function of the standard normal distribution:
- For a pareto distribution with minimum and tail index :
- For a logistic distribution with mean and scale :
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.