The median of a set of data values is the middle value of the data set when it has been arranged in ascending order. That is, from the smallest value to the highest value.
Median of distribution is that value of the variate which divides it into two equal parts. In terms of frequency curve, the ordinate drawn at median divides the area under the curve into two equal parts. Median is a positional average because its value depends upon the position of an item and not on its magnitude.
Determination of Median
(a) When individual observations are given
The following steps are involved in the determination of median: (i) The given observations are arranged in either ascending or descending order of magnitude. (ii) Given that there are n observations, the median is given by:
The size of n+1/2 th observations, when n is odd. The mean of the sizes of n/2th and n+1/2 of observations, when n is even.
Example : Find median of the following observations:
20, 15, 25, 28, 18, 16, 30.
Solution: Writing the observations in ascending order, we get 15, 16, 18, 20, 25, 28, 30.
Since n = 7, i.e., odd, the median is the size of 7+1/2 th, i.e., 4th observation. Hence, median, denoted by Md = 20.
Note: The same value of Md will be obtained by arranging the observations in descending order of magnitude.
Example : Find median of the data : 245, 230, 265, 236, 220, 250.
Solution: Arranging these observations in ascending order of magnitude, we get 220, 230, 236, 245, 250, 265. Here n = 6, i.e., even. Median will be arithmetic mean of the size of 6/2th, i.e., 3rd and (6/2 +1)th, i.e., 4th observations. Hence Md =236+245/2=240.5
Remarks: Consider the observations: 13, 16, 16, 17, 17, 18, 19, 21, 23. On the basis of the method given above, their median is 17.
According to the above definition of median, "half (i.e., 50%) of the observations should be below 17 and half of the observations should be above 17". Here we may note that only 3 observations are below 17 and 4 observations are above it and hence, the definition of median given above is some what ambiguous. In order to avoid this ambiguity, the median of a distribution may also be defined in the following way:
Median of a distribution is that value of the variate such that at least half of the observations are less than or equal to it and at least half of the observations are greater than or equal to it.
Based on this definition, we find that there are 5 observations which are less than or equal to 17 and there are 6 observations which are greater than or equal to 17. Since n = 9, the numbers 5 and 6 are both more than half, i.e., 4.5. Thus, median of the distribution is 17.
Further, if the number of observations is even and the two middle most observations are not equal, e.g., if the observations are 2, 2, 5, 6, 7, 8, then there are 3 observations (n/2=3) which are less than or equal to 5 and there are 4 (i.e., more than half) observations which are greater than or equal to 5.
Further, there are 4 observations which are less than or equal to 6 and there are 3 observations which are greater than or equal to 6. Hence, both 5 and 6 satisfy the conditions of the new definition of median. In such a case, any value lying in the closed interval [5, 6] can be taken as median. By convention we take the middle value of the interval as median. Thus, median is 5+6/2= 5.5
(b) When ungrouped frequency distribution is given
In this case, the data are already arranged in the order of magnitude. Here, cumulative frequency is computed and the median is determined in a manner similar to that of individual observations.
Example : Locate median of the following frequency distribution: 
Solution:
Here N = 95, which is odd. Thus, median is size of (95+1/2)th i.e.,48th observation. From the table 48th observation is 12, Therefore Md = 12.
Alternative Method:
N/2 = 95/2 = 47.5 Looking at the frequency distribution we note that there are 48 observations which are less than or equal to 12 and there are 72 (i.e., 95 - 23) observations which are greater than or equal to 12. Hence, median is 12.
Example : Locate median of the following frequency distribution :
Solution:
Here N = 252, i.e., even.
Now N/2= 252/2 = 126 and N/2+1 = 127
Median is the mean of the size of 126th and 127th observation. From the table we note that 126th observation is 4 and 127th observation is 5.
Md = 4+5/2= 4.5
Alternative Method: Looking at the frequency distribution we note that there are 126 observations which are less than or equal to 4 and there are 252 - 75 = 177 observations which are greater than or equal to 4. Similarly, observation 5 also satisfies this criterion. Therefore, median = 4+5/2 = 4.5.
(c) When grouped frequency distribution is given (Interpolation formula) The determination of median, in this case, will be explained with the help of the following example.
Example : Suppose we wish to find the median of the following frequency distribution.
Solution: The median of a distribution is that value of the variate which divides the distribution into two equal parts. In case of a grouped frequency distribution, this implies that the ordinate drawn at the median divides the area under the histogram into two equal parts. Writing the given data in a tabular form, we have:
For the location of median, we make a histogram with heights of different rectangles equal to frequency density of the corresponding class. Such a histogram is shown below:
Since the ordinate at median divides the total area under the histogram into two equal parts, therefore we have to find a point (like Md as shown in the figure) on X - axis such that an ordinate (AMd) drawn at it divides the total area under the histogram into two equal parts.
We may note here that area under each rectangle is equal to the frequency of the corresponding class. Since area = length ´ breadth = frequency density× width of class = f/h× h = f.
Thus, the total area under the histogram is equal to total frequency N. In the given example N = 70, therefore N/2= 35. We note that area of first three rectangles is 5 + 12 + 14 = 31 and the area of first four rectangles is 5 + 12 + 14 + 18 = 49. Thus, median lies in the fourth class interval which is also termed as median class. Let the point, in median class, at which median lies be denoted by Md.
The position of this point should be such that the ordinate AMd (in the above histogram) divides the area of median rectangle so that there are only 35 - 31 = 4 observations to its left. From the histogram, we can also say that the position of Md should be such that
Md-30/40-30= 4/18
Thus, Md=40/18 + 30 =32.2
Writing the above equation in general notations, we have
Where, Lm is lower limit, h is the width and fm is frequency of the median class and C is the cumulative frequency of classes preceding median class. Equation (2) gives the required formula for the computation of median.
Remarks:
1. Since the variable, in a grouped frequency distribution, is assumed to be continuous we always take exact value of N/2, including figures after decimals, when N is odd. 2. The above formula is also applicable when classes are of unequal width. 3. Median can be computed even if there are open end classes because here we need to know only the frequencies of classes preceding or following the median class.
Determination of Median When 'greater than' type cumulative frequencies are given
By looking at the histogram, we note that one has to find a point denoted by Md such that area to the right of the ordinate at Md is 35. The area of the last two rectangles is 13 + 8 = 21. Therefore, we have to get 35 - 21 = 14 units of area from the median rectangle towards right of the ordinate. Let Um be the upper limit of the median class. Then the formula for median in this case can be written as
Note that C denotes the 'greater than type' cumulative frequency of classes following the median class. Applying this formula to the above example, we get
Md=40-(35-21)/18 x 10 = 32.2
Example: Calculate median of the following data :
Since N/2=100/2= 50, the median class is 7- 8. Further, Lm = 7, h = 1, fm = 22 and C = 38.
Thus, Md = 7 +50-38/22x1 = 7.55 inches
Example : The following table gives the distribution of marks by 500 students in an examination. Obtain median of the given data.
Solution: Since the class intervals are inclusive, therefore, it is necessary to convert them into class boundaries.
Since N/2 = 250, the median class is 49.5 - 59.5 and, therefore, Lm = 49.5, h = 10, fm= 162, C=192
Thus Md= 49.5 + 250-192/162 x10 = 53.08 marks
Example: The weekly wages of 1,000 workers of a factory are shown in the following table. Calculate median.
Solution: The above is a 'less than' type frequency distribution. This will first be converted into class intervals
Since N/2= 500, the median class is 625 - 675. On substituting various values in the formula for median, we get
Example : Find the median of the following data:
Solution: Note that it is 'greater than' type frequency distribution
Since N/2=230/2= 115, the median class is 40 - 50.
Example : The following table gives the daily profits (in Rs) of 195 shops of a town. Calculate mean and median.
Solution:
Example : Find median of the following distribution:
Solution: Since the mid-values are equally spaced, the difference between their two successive values will be the width of each class interval. This width is 1,000. On subtracting and adding half of this, i.e., 500 to each of the mid-values, we get the lower and the upper limits of the respective class intervals. After this, the calculation of median can be done in the usual way.
Determination of Missing Frequencies
If the frequencies of some classes are missing, however, the median of the distribution is known, and then these frequencies can be determined by the use of median formula.
Example : The following table gives the distribution of daily wages of 900 workers. However, the frequencies of the classes 40 - 50 and 60 - 70 are missing. If the median of the distribution is Rs 59.25, find the missing frequencies.
Solution: Let f1 and f2 be the frequencies of the classes 40 - 50 and 60 - 70 respectively.
Since median is given as 59.25, the median class is 50 - 60. Therefore, we can write
Graphical location of Median
So far we have calculated median by the use of a formula. Alternatively, it can be determined graphically, as illustrated in the following example.
Example : The following table shows the daily sales of 230 footpath sellers of Chandni Chowk:
Locate the median of the above data using (i) only the less than type ogive, and (ii) both, the less than and the greater than type ogives.
Solution: To draw ogives, we need to have a cumulative frequency distribution.
The value N/2= 115 is marked on the vertical axis and a horizontal line is drawn from this point to meet the ogive at point S. Drop a perpendicular from S. The point at which this meets X- axis is the median.
(ii) Using both types of ogives
A perpendicular is dropped from the point of intersection of the two ogives. The point at which it intersects the X-axis gives median. It is obvious from figures that median = 2080.
Properties of Median 1. It is a positional average. 2. It can be shown that the sum of absolute deviations is minimum when taken from median. This property implies that median is centrally located.
Merits and Demerits of Median (a) Merits
It is very easy to calculate and is readily understood. Median is not affected by the extreme values. It is independent of the range of series. Median can be measured graphically. Median serves as the most appropriate average to deal with qualitative data. Median value is always certain and specific value in the series. Median is often used to convey the typical observation. It is primarily affected by the number of observations rather than their size.
(b) Demerits
Median does not represent the measure of such series of which different values are wide apart from each other. Median is erratic if the number of items is small. Median is incapable of further algebraic treatment. Median is very much affected by the sampling fluctuations. It is affected much more by fluctuations of sampling than A.M. Median cannot be used for further algebraic treatment. Unlike mean we can neither find total of terms as in case of A.M. nor median of some groups when combined. In a continuous series it has to be interpolated. We can find its true-value only if the frequencies are uniformly spread over the whole class interval in which median lies. If the number of series is even, we can only make its estimate; as the A.M. of two middle terms is taken as Median.
Uses
It is an appropriate measure of central tendency when the characteristics are not measurable but different items are capable of being ranked. Median is used to convey the idea of a typical observation of the given data. Median is the most suitable measure of central tendency when the frequency distribution is skewed. For example, income distribution of the people is generally positively skewed and median is the most suitable measure of average in this case. Median is often computed when quick estimates of average are desired. When the given data has class intervals with open ends, median is preferred as a measure of central tendency since it is not possible to calculate mean in this case.
No comments:
Post a Comment