I assume that the y
variable 1.81827
is a typo that should be 1818.27
.
The common practice in fittin a log-normal distribution is to fit a normal distribution to a set of logarithmic data:
data = Transpose@{Log10@x, y};
I use the NonlinearModelFit
:
nlm = NonlinearModelFit[data, a + b Exp[-(μ - z)^2/(2 σ^2)], {a, b, μ, σ}, z];
Normal@nlm
731.951 + 5607.06 E^(-2055.8 (2.71362 - z)^2)
To plot the data and the fit:
pts = ListPlot[data, Frame -> True];
plot = Plot[Normal[nlm], {z, Min@Log10@x, Max@Log10@x}, Frame -> True];
out = Show[pts, plot, Frame -> True, FrameLabel -> {"log10x", Rotate["y", 270 Degree]}, Frame -> True]

The parameters and uncertainties of the fit can be obtained via
table1 = nlm["ParameterTable"]

or with another useful command:
param = nlm["BestFitParameters"]
{a -> 731.951, b -> 5607.06, [Mu] -> 2.71362, [Sigma] -> 0.0155953}
which allow, e.g., to get the location of the mean in the initial units:
10^(μ /. param)
517.156
The mean, standard deviation and variance of the log-normal distribution may be obtained with the formulae displayed, e.g., on wikipedia.
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button above the edit window. The edit window help button?
is also useful for learning how to format your questions and answers. You may also find this meta Q&A helpful -- The idea is to make it so that others can copy-paste it into the software Mathematica. It makes it convenient for them and more likely you will get someone to help you. You will need to put the data in proper Mathematica syntax. $\endgroup$ – Michael E2 Sep 11 '16 at 17:26