Median
and mode
Median Definition
Median is that value of distribution which
divides the distribution in two equal parts.
There are equal no. of observations below the
median and above the median.
Median is the middle most observation.
Ungrouped data
First arrange the values in ascending or
descending order.
Count number of observations N
If N is Odd then Median = (N + 1)/2th observation
If N is Even then Median = average of N/2th and
(N/2 + 1)thobservation
Example
In a survey of 10 households number of children
per family were found to be
4 , 1, 5, 4, 3, 7, 2, 3, 4, 1
Arranging the data in ascending order
1, 1, 2, 3, 3, 4, 4, 4, 5, 7
Example countiued
Number of observations in the above problem
are 10 i.e. even
Therefore N/2 = 5th observation is 3
And N/2 + 1 = 6th observation is 4
Average of the two is (3 + 4)/2 = 3.5
Example
Fifteen students were asked their shoe sizes.
The results are given below.
8, 6, 7, 6, 5, 4½, 7½ , 6½ , 8½ , 10, 7, 5, 5, 8, 9
Arranging in ascending order
4½, 5, 5, 5, 6, 6, 6½, 7, 7, 7½, 8, 8, 8½, 9, 10
Here N = 15 i.e. odd
Therefore Median = (N + ½)thobservation
Which is (15+1)/2 = 16/2
= 8th observation
The 8th observation is 7
therefore 7 is Median of the given data
Discrete frequency distribution
• Step 1: find the cumulative frequencies of
each observation
• Step 2: find N/2 where N = ∑ f
• Step 3: find the cumulative frequency (c.f.) just
greater then N/2
• Step 4: corresponding value of observation is
Median
Example
Obtain the median for the following frequency
distribution:
X 1 2 3 4 5 6 7 8 9
F 8 10 11 16 20 25 15 9 6
Example continued
X F C.F.
1 8 8
2 10 18
3 11 29
4 16 45
5 20 65
6 25 90
7 15 105
8 9 114
9 6 120
N = 120; N/2 = 60
65 is c.f just greater then 60. X corresponding to c.f. 65 is 5 hence 5 is Median
Median = 5
Continuous frequency distribution
Example
Find the median wage of the following
distribution:
Wages (in Rs.) No. of Labourers
20 - 30 3
30 – 40 5
40 – 50 20
50 – 60 10
60 - 70 5
Solution
Wages (in Rs.) No. of Labourers CF
20 - 30 3 3
30 – 40 5 8
40 – 50 20 28
50 – 60 10 38
60 - 70 5 43
N/2 = 43/2 = 21.5, Median Class = 40 – 50
L = 40, n/2 = 21.5, C.f of proceeding class = 8, h = 10, f = 20
Median = 40 + (21.5 – 8) 10/20 = 46.75
Exclusive and inclusive class intervals
Inclusive class intervals Exclusive Class Intervals
Class interval Frequency Class interval Frequency
0–9 68 10 – 20 8
10 – 19 165 20 – 30 17
20 – 29 110 30 – 40 25
30 – 39 29 40 – 50 30
Merits of Median
• It is rigidly defined.
• It is easily understood and is easy to
calculate. In some cases it can be located
merely by inspection.
• It is not at all affected by extreme values.
• It can be calculated for distributions with
open-end classes.
Demerits of Median
• In case of even number of observations
median cannot be determined exactly. We
estimate it by taking the mean of two middle
terms.
• It is not based on all the observations
• It is not amenable to algebraic treatment.
Mode
Definition: the observation that occurs
maximum no. of times
It means the observation around which the
other items cluster densely.
Ungrouped data
In a survey of 10 households number of children
per family were found to be
4 , 1, 5, 4, 3, 7, 2, 3, 4, 1
Arranging the data in ascending order
1, 1, 2, 3, 3, 4, 4, 4, 5, 7
4 children occurs three times therefore 4 is modal
value.
Example
Fifteen students were asked their shoe sizes.
The results are given below.
8, 6, 7, 6, 5, 4½, 7½ , 6½ , 8½ , 10, 7, 5, 5, 8, 9
Arranging in ascending order
4½, 5, 5, 5, 6, 6, 6½, 7, 7, 7½, 8, 8, 8½, 9, 10
Here 5 occurs maximum times therefore 5 is
modal shoe size.
Discrete frequency distribution
Obtain the mode for the following frequency
distribution:
X F
1 8
2 10
3 11
4 16
5 20
6 25
7 15
8 9
9 6
Continuous frequency distribution
Example
Find the median wage of the following
distribution:
Wages (in Rs.) No. of Labourers
20 - 30 3
30 – 40 5
40 – 50 20
50 – 60 10
60 - 70 5
20 is highest frequency therefore 40 – 50 is modal class
L1 = 40, f0 = 5, f1 = 20, f2 = 10, i = 10
Mode = 40 + (20 – 5)/ (40 – 5 – 10) * 10 = 46
Hence Mode is 46
Merits of Mode
• Mode is readily comprehensible and easy to
calculate. Like median, mode can be located in
some cases merely by inspection.
• Mode is not at all affected by extreme values.
• Mode can be conveniently located even if the
frequency distribution has class-intervals of
unequal magnitude provided the modal class and
the classes preceding and succeeding it are of the
same magnitude.
• Open ended classes also do not pose any problem
in the location of mode.
Demerits of Mode
1.Mode is ill-defined. It is not always possible to find
a clearly defined mode.
2.In some cases, we may come across distributions
with two modes. Such distributions are called bi-
modal. If a distribution has more than two modes
it is said to be multimodal.
3.It is not based upon all the observations.
4.It is not capable of further mathematical
treatment.
