Median used for non-parametric analysis. Likert (ordinal) scale usually violets the normality assumption. 

When there is more variation in the data then the mean is not suitable measure that's why we use median. Also when there is extreme values mean is not preferred.

When extreme values exist in data then we move towards median.

The mean became more popular than the median for things like t-test (e.g., y_i = B0 + B1 G_i + e_i) because it works out computationally much easier to minimize the squared residuals (min sum(e_i^2)). If you minimize the absolute value of the residuals (min sum |e_i|) you lessen the impact of large residuals, but this is computationally a harder problem so in the first half of last century it was seldom used. With the robust statistics revolution (say starting with Tukey, 1960) people have used lots of loss functions and generally compare these with least squares. People including Rand Wilcox has argued that we should use these robust methods much more (or always). So, the answer to part of your question, why are comparisons generally done with the mean (least squares) versus the median (least absolute value ... or M or ... estimators) is that traditionally we have used least squares for lots of things.

On using the median versus the mean for descriptive statistics, both are often used. For something like income, the choice depends on what is meaningful and purpose of the statistics. I see median lots for reporting with respect to social welfare in recent years in the US, because of the growth of income disparity (not so much variation, but skewness).

Salary data do not conform to a Normal Distribution or any symmetrical distribution. Hence the mean will be a biased statistic describing the typical salary in a population. Salaries tend to have a very long tail towards higher salaries and that skew drags the mean to a value that is less representative of the population at large. This statistic can be useful when compared to the median (the salary that sits right in the middle with half of the measures below and half above the median. If the mean is much larger than the median than salary inequity is very large.

hi,

The median is recommended over the mean when data is skewed. Othewise, for normally distributed data, we use the mean.

Regards

Patrice and Muayyad,

I would not go as far as you two on stating that the distribution implies which to use.

The distribution only comes into play when deciding between least squares statistics (like comparing means) and more robust loss functions (like least absolute values for comparing medians) because of the assumptions about residuals.  If two people have the same data set, and therefore the same distribution, your comments suggest that regardless of their purposes, they should use the same measure of central tendency. For some purposes I might want to get measure of the typical person's income (and arguably the median is a better measure of this than the mean), but for other purposes I might want a measure of say, how much tax is coming in (for which a weighted mean would be better).

This is basically the same thing Lord presents in his football numbers example to show that level of measurement is at least in part in the eye of the beholder.

I should also add, this thread has just been comparing the mean and median. There are lots of other measures, but these are two of the most popular and two of the most easily understood.

The Mean is one of the measures of central tendency which is very sensitive to extreme observations, as a result the Median is preferred since more than 50% of the data have to be outlying before it will be affected. Hence the median is robust and it gives better estimate in highly perturbed data sets.

However, the Median does not use all the information in the data set. The mean does use all the information and it is preferred when the data follows the classical assumptions. 

Regards.

The question was about the median and the mean not other measures.

And of course the choice between median and mean is largely based on the question being investigated. 

Least squares has nothing to do with the mean itself. It is only one of several methods for estimating the mean form the sample data. Maximum likelihood is another and a rather good one and much more robust. Both estimators, however, turn out to be, for the Normal Distribution, simply the sum of the observed values divided by the number of observation (arithmetic mean). The median is, in contrast, a ranked statistic being the value at the 50% quantile.

You are right (although I had already pointed that out) that the measure used depends on what your initial question is. If it is about salary (say) then the mean will be misleading (because of the skew) and the median will better describe the typical salary.

Perhaps if taxes are calculated as a linear function of salary, the mean would be useful for estimating mean taxes but of course tax rates are not linear with respect to income.

The remark about Distributions is absolutely relevant because the mean is a parametric estimate (dependent on a known Distribution that best describes the data) and the median in nonparametric (no need for a known Distribution)

the mean is preferred to median b/c it takes all data in its calculation but the median takes only one value in the middle of odd sorted values or the mean of the two middle points for even number of data but actually the median is preferred for skewed data b/c the  mean is more  affected by extreme values .

It may be relevant and of interest to point out that the distance between the mean and the median is no larger than the standard deviation as amply elaborated in                     D. Gilat & T.P. Hill, Quantile-locating functions and the distance between the mean and quantiles, Statistica Neerlandica 47:4, 279-283 (1993)   

Median used for non-parametric analysis. Likert (ordinal) scale usually violets the normality assumption.