Median in statistics :
Median is one of the measures of central tendency which can be defined as follows.
Median is the middle-most value when the observations are arranged either in an ascending order or a descending order of magnitude.
For example, if the marks of the 7 students are 72, 85,56,80,65,52 and 68, then in order to find the median mark, we arrange these observations in the following ascending order of magnitude:
52, 56, 65, 68, 72, 80, 85.
Since the 4th term i.e. 68 in this new arrangement is the middle most value, the median mark is68
i.e. Median (Me) = 68.
As a second example, if the wages of 8 workers, expressed in dollars are 56, 82, 96, 120, 110, 82, 106, 100 then arranging the wages as before, in an ascending order of magnitude, we get Rs.56, Rs.82, Rs.82, Rs.96, Rs.100, Rs.106, Rs.110, Rs.120.
Since there are two middle-most values, namely, Rs.96, and Rs.100 any value between Rs.96 and Rs.100 may be, theoretically, regarded as median wage.
However, to bring uniqueness, we take the arithmetic mean of the two middle-most values, whenever the number of the observations is an even number.
Thus, the median wage in this example, would be
M = (96 + 100) / 2 = 98
In case of a grouped frequency distribution, we find median from the cumulative frequency distribution of the variable under consideration.
We may consider the following formula, which can be derived from the basic definition of median.
l₁ = lower class boundary of the median class i.e. the class containing median.
N = total frequency.
Nl = less than cumulative frequency corresponding to l₁. (Pre median class)
Nu = less than cumulative frequency corresponding to l₂. (Post median class)
l₂ being the upper class boundary of the median class.
C = l₂ – l₁ = length of the median class.
Compute the median for the distribution
Computation of median
For the given data, the formula to find median is given by
We find, from the table N/2 = 308/2 = 154 lies between the two cumulative frequencies 119 and 201
i.e. 119 < 154 < 201
Thus, we have
Nl = 119
Nu = 201
l₁ = 409.50
l₂ = 429.50
C = 429.50 – 409.50 = 20
Plugging these values in the above formula, we get
M = 409.50 + [ (154-119) / (201-119) ] x 20
M = 409.50 + 8.54
M = 418.04
1) If x and y are two variables, to be related by y=a+bx for any two constants a and b, then the median of y is given by
Mᵧ = a + bMₓ
For example, if the relationship between x and y is given by
2x – 5y = 10
and if the median of x is known to be 16.
Then 2x – 5y = 10 -------> y = -2 + 0.40x
Now, we have
Mᵧ = -2 + 0.40Mₓ
Mᵧ = -2 + 0.40(16)
Mᵧ = 4.4
2) For a set of observations, the sum of absolute deviations is minimum when the deviations are taken from the median. This property states that ∑|xᵢ–A| is minimum if we choose A as the median.
3) We cannot treat median mathematically, the way we can do with arithmetic mean.
4) It is rigidly defined.
5) It is easy to comprehend.
6) It is not based on all the observations.
7) It is not much affected by sampling fluctuations.
8) It is the most appropriate measure of central tendency for an open-end classification.
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