When the observations are arranged in ascending or descending order of magnitude, then the middle value is known as median of these observations.
Let x1, x2 ….xn be n observations arranged in the ascending order of magnitudes. Median is defined as the ‘middle most’ term, that is, the value of x at the position n+1/2, i.e, we can write.
Median can be calculated from the following formula:
In case of individual series, the following method is used:
Steps of Calculate:
1. First of all, arrange the data in order whether it is ascending or descending.
2. Put the value of N to find out the value of ‘M’e.
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Example 1:
Find the median of the following:
391, 384, 591, 407, 672, 522, 777, 753, 1490.
Solution:
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X= 384, 391, 407, 522, 591, 672, 753, 111, 1490,
Median N+1th /2 item N= 9+1/2=5th item.
∴ Median = 591.
If even number, like number of items are 10
Then,
N+1/2 10+1/ item =5th + 6th/2
In case of Discrete Series:
In discrete series, following formula is useful:
Steps to calculate:
1. Arrange the data in ascending or descending order accordingly.
2. Then, make the cumulative frequency.
3. Put the value of N in the formula (N+1/2).
4. Look just greater value which find in step 3 in cumulative frequency table, the value of corresponding variable is median.
Example 2:
Find the median from the following data:
Median = N+1th/2item = 61 + 1/2= 31th
Just greater than 31 is 33, and corresponding variable is 150
Therefore, Median = 150
Continuous Series:
The following formula is used:
Where,
M= median
L = lower limit of the median class
cf = cumulative frequency of the class preceding the median class
f= frequency of median class
N = sum of the frequencies
i = size of class interval of median class.
Steps to calculate:
1. First of all, calculate the cumulative frequencies.
2. Find out the value of median by formula,
M= N/2, find out median class where the value of median lies.
3. Put the value, that is, N, L, cf, f and i in the formula to find the value of median
Example 3:
Find the value of median of following data:
Median = N/2 = 20/2 = 10 lies 20-30
Median = 20 + 10-8/2× 10
= 20 + 10 = 30.
Graphical Presentation of Median:
Median is also possible to find out graphically with the help of ogive curves.
The following steps are involved in this method:
1. First of all, draw an ogive curve, it may be more than ogive or less than ogive.
2. Calculate N/2 and place item value on Y-axis as median value
3. Then, draw a horizontal line from this point Y-axis as to meet at ogive curve.
4. Then, draw a perpendicular from the intersection point in X axis.
5. Where the point of intersection lies on X-axis. That will be value of median.
Example 4:
Calculate the median from the following data:
Solution:
Since cumulative frequencies are given, first find the frequencies.
Median = Size of N/2th item = Size of 125/2 = 62.5th item
Hence, median lies in the class 30-40
Missing Frequency:
If one number is missing then the value of median is given but total numbers are not given. If two frequencies are missing, and total numbers are given. The value of average, which is given for the purpose of missing value then that average lies directly in X variable, then calculate the missing value.
Example 5:
Median = 25, N = 20, find missing value
Solution:
The given median is 25, it lies between 20 and 30
Now,
Merits and Demerits of Median Merits:
1. It is very simple to understand.
2. Its calculation is very easy and simple.
3. It is not effected by the extreme items.
4. It can be represented graphically very well.
5. It is most suitable average for open-ended class intervals.
6. It deals with quality more than quantity because it is a very suitable measure of average to find the qualitative facts that is intelligence, beauty and so on.
Demerits:
1. It needs extra labour to make the ascending and descending order of data than other average measures.
2. It does not involve all the observations at the time of calculation which affect its reliability.
3. It cannot be calculated exactly in the series of even number of items.
4. It is very difficult to calculate at the time of presence of very small or large number of items in the series.
5. It has no further, mathematical applicability like other methods of average.