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 their mean, median and mode, based on the values of 2 quantiles.

Scenario

Given a random variable $X$ with cumulative distribution function $F$, and two values $x_1$ and $x_2$ respecting the quantiles $F\left(x_1\right)$ and $F\left(x_2\right)$, what are the parameters of the underlying distribution? Answering this question is relevant to determine the expected value (mean) of $X$.

As a concrete example, $X$ 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:

• $x_1=5$ and $x_2=15$.
• $F\left(x_1\right)=0.1$ and $F\left(x_2\right)=0.9$.

Distribution parameters

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

• For a uniform distribution with minimum $a$ and maximum $b$:
  • $F\left(x\right)=(x-a)/(b-a);a\le x\le b.$
• For a normal distribution with mean $\mu$ and standard deviation $\sigma$, denoting as $Q$ the quantile function of the standard normal distribution^[1]:
  • $Q\left(F\right(x\left)\right)=(x-\mu )/\sigma .$
• For a pareto distribution with minimum $x_m$ and tail index $\alpha$:
  • $F\left(x\right)=1-{\left(\frac{x_m}{x}\right)}^{\alpha };x\ge x_m.$
• For a logistic distribution with mean $\mu$ and scale $s$:
  • $F\left(x\right)={(1+e^{-\frac{x-\mu }{s}})}^{-1}$.

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:
  • $a=\frac{F\left(x_2\right)x_1-F\left(x_1\right)x_2}{F\left(x_2\right)-F\left(x_1\right)}.$
  • $b=\frac{(1-F(x_1\left)\right)x_2-(1-F(x_2\left)\right)x_1}{F\left(x_2\right)-F\left(x_1\right)}.$
• For a normal distribution:
  • $\mu =\frac{Q\left(F\right(x_2\left)\right)x_1-Q\left(F\right(x_1\left)\right)x_2}{Q\left(F\right(x_2\left)\right)-Q\left(F\right(x_1\left)\right)}.$
  • $\sigma =\frac{x_2-x_1}{Q\left(F\right(x_2\left)\right)-Q\left(F\right(x_1\left)\right)}.$
• For a loguniform or lognormal distribution:
  • $\ln X$ follows a uniform or normal distribution.
  • This means the parameters referring to the logarithm of $X$ could be calculated replacing $x_1$ and $x_2$ by $\ln x_1$ and $\ln x_2$.
• For a pareto distribution:
  • $\alpha =\frac{\ln (1-F(x_1\left)\right)-\ln (1-F(x_2\left)\right)}{\ln x_2-\ln x_1}$.
  • $x_m=x_1{(1-F(x_1\left)\right)}^{\frac{1}{\alpha }}$.
• For a logistic distribution:
  • $\mu =\frac{\ln (\frac{1}{F\left(x_1\right)}-1)x_2-\ln (\frac{1}{F\left(x_2\right)}-1)x_1}{\ln (\frac{1}{F\left(x_1\right)}-1)-\ln (\frac{1}{F\left(x_2\right)}-1)}$.
  • $s=\frac{x_2-x_1}{\ln (\frac{1}{F\left(x_1\right)}-1)-\ln (\frac{1}{F\left(x_2\right)}-1)}$.

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, median and mode of all these distributions.