Median

Median

When the scores are arranged in ascending or descending order of magnitude, the measure or value of the central item in the series is called Median. Median is a value that divides the distribution into two equal parts, ie., half of the values lie above the Median and half below. That is median is the middle most item in the distribution, when the measures are arranged in the order of magnitude.

 Characteristics of Median

 It is the value that occupies the middle point of the distribution, such that half the items fall above it and half below it.

 Median is a positional average.

 It is not based on all observations in a given distribution.

 A distribution has only one median.

 Median cannot be used for further algebraic treatment.

When there is missing data, if their position is known, median can be calculated.

Median can be calculated in open end distributions.

Median is not affected by extreme values

It is not stable as mean.

Calculation of Median

In the case of raw data median is the n+1/2 th value when the scores are arranged in ascending or descending order when n is odd. If n is even, median is the average of n/2th and (n/2+1) th item.

For example suppose 43, 34, 26, 45, 13 are the marks obtained by five students in an examination. When arranged in the descending order, 45, 43, 34, 26, 13 are the scores. Since n=5 is odd, (n+1)/2= 3, that is the third item ,34 is the median.

If one more item 42 is added to the data, the ordered data will be 45,43,42,34, 26,13.

 n/2th score, that is 6/2th score = 3^rd score=42,

(n/2+1) th item = 4^th score = 34

Then median = (42+34)/2 = 76/2 = 38.

Suppose instead of 42, the mark was 1, then the arranged data will be 45,43, 34, 26, 13, 1, then the median will be 30 [(34+26)/2=30]. If the score added is 25 instead of 1, then also the median is 30 as the scores in the central position are the same as the previous case. This is a major drawback of median.

When the data is in the form of a discrete frequency table

When the data is in the form of a discrete frequency table, first calculate the cumulative frequency which can be calculated by adding the frequencies of all classes upto that class, either in the upward direction or downward. (Either less than or greater than cumulative frequency). As a second step locate the score that contains N/2 in the cumulative frequency for the first time. That score is the median of the given data.

 The table gives marks in Science of 50 students. Compute the median.

Median is the N/2 th item and it is 50/2= 25 th item. In the cumulative frequency, 25 is included in the cf 35, therefore median is 21, the score corresponding to the cf that include N/2 for the first time.

In the case of continuous frequency table

where Md = Median, l = Lower limit of the median class,  i = class interval, N = Total frequency, F = Cumulative frequency upto the median class, f = frequency of the median class, Median class is the class with cumulative frequencyequal to or greater than N/2 for the first time. 

Example:

           = 734.79