How do I find the p50 value in a lognormal distribution given p10 and p90 using excel/@RISK ??
Good point BGM. But I still like squareandrare's answer because it can shed some light on the process of how to figure it out yourself.
I actually got a lot further in this problem than I needed to.
I took the cdf for the lognormal from wikipedia:
I set up this system of equations:
I solved for and :
I started to plug and into the cdf, setting it equal to .5. Then the answer suddenly hit me...
The logarithm function is monotonic. Therefore, the median of the normal distribution with parameters and will be the logarithm of the median of the lognormal distribution with parameters and (remember that normal distributions are symmetric, so the mean and median are both equal to ).
Thus, the median for the lognormal distribution is:
Then the logic of the formula made sense. The 10th and 90th percentiles are symmetric about the mean of any normal distribution, so the average of the logarithms of the 10th and 90th percentiles of the lognormal distribution with parameters and must be equal to the mean (and median, by symmetry) of the normal distribution with parameters and . To find the median of the lognormal distribution with parameters and , you just need to take the antilog of .
It didn't occur to me that can be simplified to . Good catch, BGM.
Last edited by squareandrare; 01-15-2011 at 02:57 AM.
Yes I totally agree that using the normal distribution is the most natural way
to tackle the problem, because we are much more familiar with the properties
of the normal distribution
The geometric mean is just a little point to add that there is some analogy
from the lognormal distribution.
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