this is how they did it: got the percentile of log transformed data, then applied exponential to it. what they call geomean is simply a mean of long transformed data.

• your transformed variable's 97.5 percentile is $-0.047+1.96*0.125\approx 0.198$
• exponentiate it to get the answer $10^{0.198}\approx 1.578$

this is how they did it: got the percentile of log transformed data, then applied exponential to it. what they call geomean is simply a mean of long transformed data.

• your transformed variable's 97.5 percentile is $-0.047+1.96*0.125\approx 0.198$
• exponentiate it to get the answer $10^{0.198}\approx 1.578$

logarithm is a monotonic function so if you obtain percentiles analytically then the order of operations (percentile vs log) doesn't matter. when you deal with empirical estimation, then the results depend on the order of operations, as you saw. in this case your source seems to think that the data comes from lognormal distribution. if that's the case, then what they did is appropriate. they are using parametric approach to percentiles. it's fairly common.

Percentile function uses some sort of an interpolation at some point, e.g. see NIST handbook. this will lead to a different answer depending on whether you took the logarithm before or after applying percentiles.

If you were obtaining the percentiles analytically then the order of operations wouldn't matter because Log is a monotonic function. you could get the percentiles on original variable, then get the log of the percentiles. Alternatively, you could first transform the variable, then get the percentiles. both would render the same answer analytically, but when you do it from the empirical dataset the results are different.

this is how they did it: got the percentile of log transformed data, then applied exponential to it. what they call geomean is simply a mean of long transformed data.

• your transformed variable's 97.5 percentile is $-0.047+1.96*0.125\approx 0.198$
• exponentiate it to get the answer $10^{0.198}\approx 1.578$