1) There might be some situations where median is preferred to Geometric Means. One reason why  median could be preferred over geometric mean is when you have negative values in your data-set or when you have some zero (0) observations. This is because geometric mean involves product term.

However, for a data which follows log-normal distribution, geometric mean should be same as median.

2) Now, to get to your pressing question:

Geometric mean have some nice statistical properties. 

Let X be a random variable which follows lognormal distribution with parameters mu and sigma^2.

X~ LN (mu, sigma^2)

Then your sample geometric mean is the maximum likelihood estimator (m.l.e) of e^mu. And MLEs have a lot of nice properties. 

Please refer to the paper below for some nice comparisions between GM and median.

http://www.tandfonline.com/doi/pdf/10.1080/00949658708811013#.VCsaXPldUcc

Hope this helps.

Central Tendency can be measured by using the mean, median, or mode. 

"The median is determined by sorting the data set from lowest to highest values and taking the data point in the middle of the sequence. There is an equal number of points above and below the median...The median can be determined for ordinal data as well as interval and ratio data. Unlike the mean, the median is not influenced by outliers at the extremes of the data set. For this reason, the median often is used when there are a few extreme values that could greatly influence the mean and distort what might be considered typical. This often is the case with home prices and with income data for a group of people, which often is very skewed. For such data, the median often is reported instead of the mean".

For more details you may look at the following link:

http://www.quickmba.com/stats/centralten/

Thank you, Prof. Mahfuz. I agree with you, but this depends on certain criteria such as the type of variables and the type of data distribution. In my question above, being  exponential, data will need to be summarized using the geometric mean  instead of the arithmetic one as its distribution will be asymmetric with extreme values. It remains unclear to me the reason for not using the median which suits such types of distribution.

Kind regards,

Rashad 

You can use whatever suits your aim (given the application is done in a technically correct way).

Median and GM are just different statistics. The median is an order statistic, saying something about the data. The GM is a model parameter of a log-normal model, so it tells you something about a model.

Now the question is if you want to argue about the data or about the model.

Dear Jochen, 

Thank you for your contribution. In fact, I am still a little bit confused. Geometric modeling could possibly  be apart from the statistical applications of geometric mean in biostatistics. We often use it to summarize certain characteristics such as growth rates. I feel that its application in routine statistical procedures for the sake of comparing some study characteristics could be a different situation.  I do not know whether my impression was to the point. 

Kind regards,

Rashad

This wiki maybe will be useful:

http://en.wikipedia.org/wiki/Geometric_mean

Thank you, Demetris. I have read the contents of the link before asking my question. It remains unclear whether these two measures of central tendency could be interchangeably used to describe or compare two sets of data, at least, in certain situations . 

Kind regards,

Rashad

I think that if you add on your study of observations also the quartile, you can better describe the shape of your data, and then you can decide.

Kind regards, Bye mariano.

Dear Mariano,

Thank you for contribution. As Dr. Jochen stated, geometric mean is used with log-normal distribution (i.e., the logarithms of variable data are normally distributed even if the distribution of these data is not so). It is known that for symmetric distributions both mean and median are almost equal. The pressing question that comes to my mind is why median is not used.  Is there a possibility to use these two measures interchangeably? If not, what is the reason?

Kind regards,

Rashad

1) There might be some situations where median is preferred to Geometric Means. One reason why  median could be preferred over geometric mean is when you have negative values in your data-set or when you have some zero (0) observations. This is because geometric mean involves product term.

However, for a data which follows log-normal distribution, geometric mean should be same as median.

2) Now, to get to your pressing question:

Geometric mean have some nice statistical properties. 

Let X be a random variable which follows lognormal distribution with parameters mu and sigma^2.

X~ LN (mu, sigma^2)

Then your sample geometric mean is the maximum likelihood estimator (m.l.e) of e^mu. And MLEs have a lot of nice properties. 

Please refer to the paper below for some nice comparisions between GM and median.

http://www.tandfonline.com/doi/pdf/10.1080/00949658708811013#.VCsaXPldUcc

Hope this helps.

Dear Prabin,

Many thanks for the detailed answer. I got it now.  It became clearer that both can be used for log-normal distributions and that it is preferred to use the geometric mean. Thank you for the paper as well.

Kind regards,

Rashad 

I found this explanation extremely helpful, particularly not being a mathematician:

https://medium.com/@JLMC/understanding-three-simple-statistics-for-data-visualizations-2619dbb3677a