The sum of the deviations of the items for the  median ignoring signs is the least. For e.g. the median of 4,6,8,10,12, is 8. The deviation from 8 ignoring signs are 4, 2, 0, 2, 4  and the total is 12. This total is smaller than the one acquired, if the  deviations are taken from any other value thus if deviations are taken from 7, the values ignoring signs would be 3, 1,1,3,5 and the total is 13.

The Merits and limitations of the median are as shown below:

Merits:

1) It is especially useful in case of open-end classes as only the position and not the values of items must be known. The median is also suggested if the distribution has unequal classes as it is easier to compute than the mean.

2) The Extreme values do not affect the median as strongly as they do the mean, for example the median of 10,20,30,40 & 150 would be 30 whereas the mean is 50. And hence very often when extreme values are present in asset of the observations the median is a more satisfactory measure of the central tendency than the mean.

3) In markedly skewed distributions such as income distribution or price distribution where the arithmetic means would be distorted by extreme values the median is especially useful. Accordingly, the median income for some purposes be regarded as a more representative figure for half of the income. The earners must  receiving at least the median income one can say as many receive the median income and as many do not.

4) It is the most exact average in dealing with qualitative data where ranks are given or there are other kinds of items that are not counted or measured but are scored.

5) Perhaps the greatest merits of median is however the fact that the median actually does indicate what many people incorrectly believe the arithmetic mean indicates. The median points out the value of the middle item in the distribution. This is a clear meaning and made the median a measure that can be easily described.

Limitations:

1) For median calculation  it is necessary to arrange the data while other averages do not need an arrangement.

2) As it is a positional average its value is not determined by each and every observation.

3) It is not capable of algebraic treatment for example the median cannot be use of determining the combined median of two or more groups as is possible in case of mean. On the same way the median wage of a skewed distribution ties the number of workers will not give the total payroll  of this limit on the median is much less popular as compare to the arithmetic mean.

4) The value of medians are affected  by sampling the fluctuations than the value of the arithmetic mean.

5) The median income situation cannot be computed exactly as the mean. Whenever the number of items included in a series of data is even, then the median is determined approximately as the mid-point of the two middle items.

6) It is erratic if the number of items are  small.

7) The median is useful for the distributions containing, open. end intervals as these intervals do not enter its computation also since the median is affected by the number rather than the size of items it is frequently used instead of the mean as a measure of central tendency in case where such values are likely to distort the mean.

Related positional measure

Besides the median there are another measures which splits a series into equal parts. The main important amongst these are the quartiles, these are those values of the variety which split the total frequency into four equal parts. Deciles splits the total frequency into 10 equal parts and the percentiles divide the total frequency into 100 equal parts. Likewise one point splits a series into two parts, three points would split it into four parts, 9 points into 10 parts and 99 points into 100 parts. Accordingly  there are only 3 quartiles, 9 deciles and 99 percentiles for a series. The quartiles are represented  by symbol Q, deciles by D and percentiles by P. the subscripts 1,2,3, etc are beneath the Q,D etc. Q₁ would denote first quartile, Q₂ second quartile, D₁ first docile D₈ 8th docile P₁ first percentile and P₆₀ 60th percentile etc.

Graphically any set of these partition values divides the area of the frequency curve or histogram into equal parts if vertical lines are drawn as third quartiles for example the area of the histogram will be divided by theses lines to four equal parts. The 9 deciles splits the area of the histogram or frequency curve into 10 equal parts and the 9 percentiles dived the area into 100 equal parts.

In economics and business statistics quartiles are more widely used than deciles &  percentiles. The quartiles are the points on the X_ scale that divide the distribution into four equal parts. Obviously there are three quartiles the second coinciding with the median more precisely stead the lower quartile Q₁ is that point on the z-scale such that one fourth of the total frequency is less than Q₁ and one-fourth is above it.

The deciles and percentiles are important in psychological and education statistics concerning grades rates ranks etc. they are of use in economics and business statistics in personnel work productivity ratings another such situations.

It must be noted that quartiles, deciles etc. are not the averages they are measure of dispersion and as such shall be discussed in detail in the next chapter here only a passing reference is made the method of computing these portion values is the same as discussed for median.

Just as quartiles splits the series into 4 equal parts, pentiles dived into 5 equal parts reptiles into 7 equal parts and cortiles into 8 equal parts however these partition values are really used in practices.