Median – Measures of Central Tendency
Statistics is the study of gathering, presenting, evaluating, interpreting, and administering data. The collected data is analyzed and interpreted, and used by various interested parties. Since it is difficult for anyone to understand and remember a large group of data, measures of Central Tendency are used. It helps to find representative value for the given data.
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.
Formula for Median
In Statistics, the median is represented by M. The median value is calculated by arranging the data either in ascending or descending order.
Formula:
M = Size of (N + 1)^{th}/2 item
Calculation of Median
A median is a number that divides a data or population in half. Depending on the type of distribution, the median differs. By sorting the data from least to largest and identifying the midway value, the median may be simply estimated.
(i)Individual Series and Median:
Example: Let’s have the following observation 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.
Position of median = (N + 1)^{th}/2 item.
Median= (9 + 1)^{th}/2 item
= 10/2=5
So, 5th term is 17, which is the median.
(Note: Also, here the middle value is “17”, so half of the scores are greater than 17, and the other half are lower. If the data contains even numbers, there will be two observations in the center. In this example, the median is calculated as the arithmetic mean of the two middle numbers.)
Example: 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.
As we can see that there are two observations in the middle, 24 and 25. Taking the mean of these two observations gives the median.
Median = 24 + 25/2 = 24.5
Therefore Median is 24.5
- To determine the median, the position of the median, i.e., the item/items at which the median lies, must be known. The median position can be computed using the following formula:
Position of median = (N + 1)^{th}/2 item.
Here, N = number of items.
(ii)Discrete Series and Median:
In the case of discrete series, the cumulative frequency can be used to locate the median, i.e., the (N + 1)/2^{nd} item. The median value is the comparable value at this place.
Example: The frequency distribution of the frequency and their class intervals (income in Rs) are given below. Calculate the median income.
Class interval | 10 | 20 | 30 | 40 | 50 |
Frequency | 5 | 10 | 15 | 8 | 12 |
The frequency distribution shown below can be used to get the median income.
Calculation of median in discrete series |
||
Class interval | Frequency (f) | Cumulative frequency (cf) |
10 | 5 | 5 |
20 | 10 | 15 |
30 | 15 | 30 |
40 | 08 | 38 |
50 | 12 | 50 |
Here the median is located in the (N + 1)/2 = (50 + 1)/2 = 25.5th observation. This can easily be found using cumulative frequency. The 25.5th observation lies in the c.f. of 30. The income in the relevant class interval is Rs 30, hence the median income is Rs 30.
(iii)Frequency Distribution Series and the Median/Continuous series
In the continuous series, we have to locate the median class where N/2^{nd} item [not (N+1)/2^{nd} item] lies. The median can then be given as follows:
M = L + (N/2 – cf)/f × h
Where
L denotes the lower limit of the median class
c.f. represents the cumulative frequency of the preceding class before the median class,
f denoted the frequency of the median class,
The magnitude of the median class interval is given by h.
Example: The following table contains information about the marks scored by students in a class. Calculate the median of marks.
Marks obtained | Students |
10-20 | 10 |
20-40 | 5 |
40-60 | 20 |
60-80 | 15 |
80-100 | 5 |
As the given data is arranged in ascending order. The median class is the value of (N/2)^{nd} item (i.e., 55/2) = 27.5th item of the series, which lies in the 40–60 class interval.
Calculation of median in Continuous series | ||
Marks obtained | Students (f) | Cumulative frequency (cf) |
10-20 | 10 | 10 |
20-40 | 5 | 15 |
40-60 | 20 | 35 |
60-80 | 15 | 50 |
80-100 | 5 | 55 |
Applying the formula of the median as:
Median = L + (N/2 – cf)/f × h
Here L = 40, N/2= 27.5, c.f =15, f= 20 and h = 60-40
= 40 + (27.5 – 15)/20 × (60-40)
= 52.5
Thus, the median of marks is 52.5. This means that 50% of students are getting less than or equal to 52.5 marks, and 50% of the students are getting more than or equal to these marks.
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:
- Simple: It is one of the simplest measure of central tendency. It is quite simple to compute and understand. It can be easily calculated through inspection in case of simple statistical series.
- Unaffected by Extreme Values: The extreme values have little effect on the median. It is unaffected by the number of series.
- Graphical Representation: The Median can be presented graphically, which makes the data more presentable and understandable.
- Qualitative Data: When dealing with qualitative data, the optimum average to utilize is the median.
- 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.
Demerits of Median
The demerits of Median are as follows:
- 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.
- Lack of Representative Character: The median does not represent a measure of such a series in which the values are widely apart.
- Unpredictable: When the number of objects is minimal, the median is unpredictable, which does not depict the true picture of the data.
- Affected by Fluctuations: The sampling fluctuations have a significant impact on the median. So, the data cannot be completely accurate.
- Lack of Algebraic Treatment: Median cannot be further algebraically treated.
Sample Questions
Question 1: The following observation in a data set: 15, 17, 26, 10, 9, 20, 22, 14, and 23. Calculate the median of the set.
Solution:
Let’s arrange the given data set in ascending order: 9, 10, 14, 15, 19, 20, 22, 23, 26.
Here the middle value is “19”, so half of the scores are greater than 19, and the other half are lower.
Question 2: The frequency distribution of the frequency and their class intervals (income in Rs) are given below. Calculate the median income.
Class interval | 10 | 20 | 30 | 40 | 50 |
frequency | 20 | 19 | 25 | 21 | 15 |
Solution:
You can use the frequency distribution shown below to get the median income.
Calculation of median in discrete series Class interval frequency (f) Cumulative frequency (cf) 10 20 20 20 19 39 30 25 64 40 21 85 50 15 100 Here the median is located in the (N+1)/2 = (100+1)/2 = 50.5th observation. This can easily be found using cumulative frequency. The 50.5th observation lies in the c.f. of 64. The income in the relevant class interval is Rs 30. Hence the median income is Rs 30.
Question 3: The following table contains information about the marks scored by students in a class. Calculate the median of marks.
Marks obtained | Students |
10-20 | 10 |
20-40 | 12 |
40-60 | 36 |
60-80 | 28 |
80-100 | 14 |
Solution:
As the given data is arranged in ascending order. The median class is the value of (N/2)^{nd} item (i.e., 100/2) = 50^{th} item of the series, which lies in the 40–60 class interval.
Calculation of median in Continuous series Marks obtained Students (f) Cumulative frequency (cf) 10-20 10 10 20-40 12 22 40-60 36 58 60-80 28 86 80-100 14 100 Applying the formula of the median as,
Median = L+(N/2-cf)/f x h
Here L = 40, N/2 = 50, c.f = 22, f = 36 and h = 60-40
= 40 + (50 – 22)/36 × (60-40)
= 55.5
Thus, the median of marks is 55.5. This means that 50% of students are getting less than or equal to 55.5 marks and 50% of the students are getting more than or equal to this marks.