Median(Measures of Central Tendency): Meaning, Formula, Merits, Demerits, and Examples
What is Median?
The median is a centrally located value that splits the distribution into two equal portions, one including all values more than or equal to the median and the other containing all values less than or equal to it. The median is the ” middle ” element when the data set is organized in order of magnitude. As the median is established by the position of several values, it is unaffected if the size of the greatest value increases. The data or observations might be arranged in either ascending or descending order. In Statistics, Median is denoted by M.
According to Yule and Kendall, “The median may be defined as the middle most value of the variable when items are arranged in order of magnitude or as the value such that greater and smaller values occur with equal frequency.”
Calculation of Median
A median is a number that divides a data or population in half. Depending on the type of distribution (individual, discrete, and continuous), the median differs. By sorting the data from least to largest and identifying the middle value, the median may be simply estimated.
1. Individual Series:
The steps required to determine median of an individual series are as follows:
Step 1: Firstly, arrange the given data in ascending or descending order.
Step 2: Apply the following formula of Median:
Where, N is the Number of Items
Median in case of Odd and Even Number of Items
In case of odd number of items, Median is the Middle term of the observation. However, in case of even number of items, Median is the average of two middle terms and is determined by using the following formula:
Example (Odd Numbers):
Determine Median for the following observations in a data set: 15, 17, 26, 11, 8, 20, 22, 14 and 23.
Let’s arrange the given data set in ascending order: 8, 11, 14, 15, 17, 20, 22, 23, 26.
N = 9
Median = 17
Here the middle value is 17, so half of the observations are greater than or equal to 17, and the other half are less than or equal to.
The following data has marks for 10 students: 25, 27, 36, 21, 18, 30, 22, 24, 13 and 28. Calculate the Median marks.
Let’s arrange the given data set in ascending order: 13, 18, 21, 22, 24, 25, 27, 28, 30, 36.
N = 10
Median = 24.5
Alternatively, we can also determine median with the formula used in case of odd number of items.
Now, to get median, the sum of 5th and 6th item is divided by 2.
Median = 24.5
2. Discrete Series or Frequency Array:
The steps required to determine median of a discrete series are as follows:
Step 1: Arrange the given distribution in either ascending or descending order.
Step 2: Denote the variables as X and frequency as f.
Step 3: Determine the cumulative frequency; i.e., cf.
Step 4: Calculate the median item using the following formula:
Where, N = Total of Frequency
Step 5: Find out the value of We can find it by firstly locating the cumulative frequency which is equal to higher than and then find the value corresponding to this cf. This value will be the Median value of the series.
Calculate the median from the given data.
25.5th item falls under the cumulative frequency 30 and the size of the distribution against this cf value is 30.
Therefore, Median = 30
3. Frequency Distribution Series or Continuous Series:
It is not possible to straight-forwardly locate median in a continuous series as the value of median lies between the lower and upper limit of a class-interval. To determine the exact median value, it is essential to estimate or interpolate median with the help of a formula. However, it should be kept in mind that when the median class of a series is first class, then the c.f. in the formula will be taken as zero. The steps required to determine median of a continuous series are as follows:
Step 1: Arrange the given data in either descending or ascending order.
Step 2: Determine the cumulative frequency; i.e., cf.
Step 3: Calculate the median item using the following formula:
Where, N = Total of Frequency
Step 4: Now inspect the cumulative frequencies and find out the cf which is either equal to or just greater than the value determined in the previous step.
Step 5: Now, find the class corresponding to the cumulative frequency equal to or just greater than the value determined in third step. This class is known as the median class.
Step 6: Now, apply the following formula of median:
l1 = lower limit of the median class
c.f. = cumulative frequency of the class preceding the median class
f = simple frequency of the median class
i = class size of the median group or class
Note: While calculating median of a given distribution, we have to assume that every class of the distribution is uniformly distributed in the class interval.
The following table contains information about the marks scored by students in a class. Calculate the median of marks.
Therefore, median lies in the class 40-60.
l1 = 40, c.f. = 15, f = 20, i = 20
= 40 + 12.5
Median = 52.5
Properties of Median
1. The sum of deviations of the items from median (ignoring signs) is the minimum. For example, the median of series, 2, 4, 6, 9, 10 is 6. Now, the deviations from 6 after ignoring signs are 4, 2, 0, 3, 4. The total of these deviations is 13. This total is minimum from the total obtained from deviations taken from any other number. If the deviations are taken from 4, the deviations after ignoring the signs will be 2, 0, 2, 5, 6 and the total of these deviations is 15. Therefore, it can be said that median is centrally located.
2. Median is not influenced by the extreme values as it is a positional average.
Merits of Median
It is one of the easiest and widely used measure of Central Tendency. Some of the merits of Median are as follows:
1. Simple: It is one of the simplest measure of central tendency. It is quite easy to compute and simple to understand. It can be easily calculated through inspection in case of some statistical series.
2. Unaffected by Extreme Values: The extreme values do not effect the median of a series. For example, median of 20, 25, 30, 35, and 120 is 30; however, its mean is 46. With this, we can see that the median is not affected by the extreme value; i.e., 120, and is thus a better average.
3. Graphical Representation: The Median can be presented graphically through ogive curves, which makes the data more presentable and understandable.
4. Appropriate for Qualitative Data: When dealing with qualitative data, the optimum average to utilize is the median. For example, measuring honesty quantitatively is not possible. However, one can easily locate an individual with average honesty through array of a group of persons arranged in ascending or descending order of honesty.
5. Helpful in case of Incomplete or Typical Data: Median is even used in case of incomplete data as median can be calculated even if one knows the number of items and middle item/items of a series. The median is frequently used to express a typical observation. It is influenced mostly by the number of observations rather than their magnitude.
6. Ideal Average: Median has definite and certain value, which means that it is defined rigidly.
Demerits of Median
The demerits of Median are as follows:
1. Arrangement of Data: The given data must be arranged in ascending or descending order in order to compute the Median, which can be a time-consuming task, if the number of items are large.
2. Lack of Representative Character: The median does not represent a measure of such a series in which the values are widely apart.
3. Unpredictable: When the number of items is minimal, the median is unpredictable, which does not depict the true picture of the data.
4. Affected by Fluctuations: The sampling fluctuations have a significant impact on the median. It means that if the class intervals of a series are not uniform, then the median value becomes inappropriate.
5. Lack of further Algebraic Treatment: Median cannot be further algebraically treated. For example, just like in case of mean, it is not possible to determine the combined median of two or more groups.
6. Not based on all Observations: As median is a positional average, it is not based on each and every item of a distribution. For example, median of 10, 20, 30, 40, and 50 is 30. Now, if we replace the values 10 and 20 with 80 and 90, and replace the values 40 and 50 with 15 and 22 respectively, then it will not have any impact on the median value of the series.
7. Unrealistic assumptions in case of Grouped Distribution: The assumption while determining median in case of grouped distribution is that the median class of the series are uniformly distributed, which in reality rarely happens.
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