I have a sample of data that follows a lognormal distribution. I would like to represent the distribution as a "Gaussian" histogram and overlayed fit (along a logarithmic x-axis) instead of a lognormal representation. For simplicity, I'll call the average and sigma of the lognormal data
sigma_log, respectively. It is my (possibly incorrect) understanding that the average of the normal representation should then be
mu_norm = exp(mu_log), and sigma of the normal represenation should be
sigma_norm = exp(sigma_log). In order to make the histogram follow a Gaussian shape, I can take the log of every value in my data; I'll call it
data_log, which becomes
data_norm when normalized to one.
Q1) I am performing a Chi Square analysis to find the optimized mu and sigma that produce the best fitting curve to my histogram. When counting observed values per bin and computing expectation values per bin, do I use the original lognormal data or data_norm? Does it necessarily matter? For the expectation values, should I integrate the formula given for a normal distribution or a lognormal distribution?
Q2) Will performing the Chi Square analysis produce the mu and sigma for a histogram fit of my lognormal data or a histogram fit of data_norm (or could it be either, depending on Q1)?
Q3) Once I have the parameters mu and sigma that give a Gaussian shape to my histogram and data overlay, I need the normalization constant (let's call it c_norm). In the case of a plain Gaussian,
c_norm = 1 / ((2 * pi)^0.5 * sigma). But in the case of a lognormal distribution,
c_norm = 1 / (x * (2 * pi)^0.5 * sigma). I am guessing that I use sigma and c_norm from the normal distribution to find the normalization constant.
PS: I am asking because I have tried repeatedly and failed. As seen in this picture, I was able to fit a curve to a normal distribution (left), but my Gaussian fit for a lognormal distribution (right) does not look correct. I can post/message my python code for that plot, but it is a bit lengthy.