Measures of Central Tendency - Median

A median is a positional number that determines the position of the middle set of data. It divides the set of data into two parts. In which, one part includes all the greater values or which is equal to a median value and the other set includes all lesser values or equal to the median. In simple words, the median is the middle value when a data set is organized according to the magnitude. The value of the median remains unchanged if the size of the largest value increases because it is defined by the position of various value.

To evaluate the median the value must be arranged in the sequence of numbers, and the numbers should be arranged in the value order starting from lowest to highest. For instance, while evaluating the medium if there is any sort of odd amount of number in the list, the median will be the middle number, with a similar number presented below or above. However, if the amount is an even number than the middle pair must be evaluated, combined together, and divided by two to find the median value.

Meaning, Merits and Demerits of Median

Q.1 MEDIAN
  • “The median is that value of the variable which divides the group into two equal parts, one part comprising all values greater and the other values less than the median.”….L.R. Connor
  • Median is the middle value of the series when items are arranged either in ascending or descending order.

It divides the series into two equal parts. One part comprises all values greater than the median and the other part comprises all values smaller than the median.

Q.2-BRIEFLY EXPLAIN THE MERITS AND DEMERITS OF MEDIAN.
ANSWER:
(A) FOLLOWING ARE SOME OF THE MERITS OF MEDIAN:
(1) EASY TO CALCULATE AND SIMPLE TO UNDERSTAND
  • It is easy to calculate and simple to understand.
  • In many situations median can be located simply by inspection.
(2) NOT AFFECTED BY EXTREME VALUES
  • It is not affected by the extreme values i.e. the largest and smallest values. Because it is a positional average and not dependent on magnitude.
(3) RIGIDLY DEFINED
  • It has a definite and certain value because it is rigidly defined.
(4) BEST AVERAGE IN CASE OF QUALITATIVE DATA
  • Median is the best measure of central tendency when we deal with qualitative data, where ranking is preferred instead of measurement or counting.
(5) USEFUL IN CASE OF OPEN ENDED DISTRIBUTION
  • It can be calculated even if the value of the extremes is not known. But the number of items should be known.
(6) REPRESENTED GRAPHICALLY
  • Its value can be determined or represented graphically with the help of Ogive curves. Whereas it is not possible in case of Arithmetic Mean.
(B) FOLLOWING ARE SOME OF THE DEMERITS OF MEDIAN:
(1) ARRANGEMENT OF DATA IS NECESSARY
  • Since the median is an average of position, therefore arranging the data in ascending or descending order of magnitude is time-consuming in the case of a large number of observations.
(2) NOT BASED ON ALL THE OBSERVATIONS
  • It is a positional average and doesn’t consider the magnitude of the items.
  • It neglects the extreme values.
(3) NOT A REPRESENTIVE OF
THE UNIVERSE
  • It is not dependent on all the observations so, it cannot be considered as their good representative.
  • In case there is a big variation between the data, it will not be able to represent the data.
(4) AFFECTED BY FLUCTUATIONS IN SAMPLING
  • It is affected by the fluctuations of sampling and this effect is more than in case of Arithmetic Mean.
(5) LACK OF FURTHER ALGEBRAIC TREATMENT
  • It is a positional average so further algebraic treatment is not possible. Like, we cannot compute the combined median of two groups of data.
Q.3-LIST THE PROPERTIES OF MEDIAN.
ANSWER:
PROPERTIES OF MEDIAN
  1. The sum of deviations of items from median, ignoring the signs, is minimum.
  2. Median is a positional average and hence it is not influenced by the extreme values.
Q.4-DEFINE PARTITIONAL VALUES. WHAT IS MEANT BY FIRST, SECOND AND THIRD QUARTILE?
ANSWER:
(A) PARTITIONAL VALUES Partition values are the values which are obtained by dividing a series into more than two parts.
(B) QUARTILES:
TYPES WITH THEIR MEANING
  • Quartile divides a series into four equal parts.
  • For any series, there will be three quartiles.
  • First Quartile also known as Lower Quartile (Q1):

It divides the distribution in such a way that the one-fourth (25%) of total items fall below it and three-fourth (75%) are above it.

  • Second Quartile (Q2) or Median: It divides the distribution in two equal halves.
  • Third Quartile also known as Upper Quartile (Q3):

It divides the distribution in such a way that three-fourth (75%) of total items fall below it and one-fourth (25%) are above it.

Practice Questions:

INDIVIDUAL SERIES

Q.1 FOLLOWING IS THE DATA REGARDING HEIGHT OF SEVEN STUDENTS. YOU ARE REQUIRED TO COMPUTE THEIR MEDIAN HEIGHT:
Height (in Cm.) 162 122 161 165 160 169 198
Q.2 IN A HOSPITAL A PATIENT WAS OPERATED UPON AND POST OPERATION DATA WAS COLLECTED REGARDING HIS BODY TEMPERATURE AFTER EVERY HOUR FOR THE FIRST EIGHT HOURS TO TAKE DECISION ABOUT MEDICINES. FIND MEDIAN BODY TEMPERATURE OF THE PATIENT.
Time 1pm 2pm 3pm 4pm 5pm 6pm 7pm 8pm
Temperature (°f) 98.5 99 100 100 101 101 101 102

DISCRETE SERIES

Q.1 CALCULATE MEDIAN SIZE OF SHOES OF STUDENTS OF CLASS XI FROM THE FOLLOWING DATA:
Size of Shoes 5 6 7 8 9 10
No. of Students 2 10 15 11 1 1
Q.2 FIND MEDIAN WEIGHT OF PINEAPPLE FROM THE FOLLOWING DATA:
Weight of Pineapple (in Gms.) 800 950 1100 1150 1250 1300 1400 1500 1850 2,000
Quantity (Pcs.) 3 11 8 10 18 7 6 7 3 2
Q.2 SUBHASH A FRUIT VENDOR BOUGHT A SAC OF PINEAPPLE HAVING 75 PIECES FROM THE WHOLESALE FRUIT MARKET. HE HAS ARRANGED ALL THE PIECES AND KEPT THEM IN THE ASCENDING ORDER OF ESTIMATED WEIGHT. HE WANT TO KNOW THE ESTIMATED AVERAGE WEIGHT HELP HIM TO FIND AVERAGE WEIGHT HE WANT TO KNOW THE AVERAGE FROM THE FOLLOWING, FIND MEDIAN WEIGHT OF PINEAPPLES:

CONTINUOUS SERIES

(A) MEDIAN FROM EXCLUSIVE CONTINUOUS SERIES
Q.1 USING MEDIAN, FIND ESTIMATED CASH WITHDRAWN BY ONE CUSTOMER FROM THE FOLLOWING:
CASH

WITHDRAWN (RS.)

0–500 500–1,000 1,000–1,500 1,500–2,000 2,000–2,500 2,500-3,000
NO. OF CUSTOMERS 12 20 25 23 7 13
(B) MEDIAN FROM INCLUSIVE CONTINUOUS SERIES
Q.2 FROM THE FOLLOWING, FIND MEDIAN SIZE.
SIZE (Units) 10 – 19 20 – 29 30 – 39 40 – 49 50 – 59 60 – 69
FREQUENCY 5 9 10 14 8 4
(C) MEDIAN FROM MID VALUE SERIES
Q.3 FIND MEDIAN FROM THE FOLLOWING:
Mid Values 8 16 24 32 40 48
Frequency 3 8 16 15 10 8
(D) MEDIAN FROM INVERTED/DESCENDING EXCLUSIVE CONTINUOUS SERIES
Q.4 USING MEDIAN COMPUTE AVERAGE AGE FROM THE FOLLOWING.
Age (in Years) 50 – 60 40 – 50 30 – 40 20 – 30 10 – 20 0 – 10
Number of Persons 9 12 15 7 5 2
(E) MEDIAN FROM CONTINUOUS SERIES WITH UNEQUAL INTERVALS
Q.5 FIND MEDIAN FROM THE FOLLOWING.
Class 0 – 2 2 – 10 10 – 30 30 – 50 50 – 80 80 – 100
Frequency 20 5 2 3 12 6
(F) MEDIAN FROM LESS THAN CUMULATIVE FREQUENCY DISTRIBUTION
Q.6 FIND AVERAGE AGE USING MEDIAN FROM THE FOLLOWING DATA.
Age (in Years) Less than 10 Less than 20 Less than 30 Less than 40 Less than 50 Less than 60
No. of Persons 5 20 45 78 90 100
(G) MEDIAN FROM MORE THAN SERIES/CUMULATIVE FREQUENCY DISTRIBUTION
Q.7 FIND MEDIAN MARKS FROM THE FOLLOWING DATA:
Marks More than 10 More than 20 More than 30 More than 40 More than 50 More than 60
No. of Students 100 85 75 45 5 2
(H) MEDIAN FROM OPEN END SERIES
Q.8 CALCULATE MEDIAN WEIGHT OF CLASS XII STUDENTS FROM THE FOLLOWING DATA:
WEIGHT (In Kg) Below 40 40 – 45 50 – 60 60 – 75 75 – 85 85 and Above
Number of Students 3 5 14 10 7 1

MISSING FIGURE

Q.1 FIND THE MISSING FREQUENCY OF CLASS 30-40, IF MEDIAN OF THE SERIES IS 28.
CLASS 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60
FREQUENCY 12 18 20 19 6

LOCATING MEDIAN GRAPHICALLY

MEDIAN FROM LESS THAN OGIVE

Q.1 FROM THE FOLLOWING DRAW LESS THAN CUMULATIVE FREQUENCY CURVE i.e. “LESS THAN OGIVE” AND LOCATE MEDIAN:
MARKS 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60
NUMBER OF STUDENTS 6 15 25 20 10 4

MEDIAN FROM MORE THAN OGIVE

Q.2 DRAW MORE THAN OGIVE AND LOCATE MEDIAN.
MARKS 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60
NUMBER OF STUDENTS 6 15 25 20 10 4

MEDIAN FROM LESS THAN AND MORE THAN OGIVE

Q.3 DRAW LESS THAN AND MORE THAN OGIVE AND LOCATE MEDIAN. ALSO VERIFY YOUR ANSWER.
MARKS 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60
NUMBER OF STUDENTS 6 15 25 20 10 4

QUARTILE DEVIATION

INDIVIDUAL SERIES

Q.1 FOLLOWINGIS THE DATA OF MARKS OF 11 STUDENTS. YOU ARE REQUIRED TO LOCATE VALUE OF LOWER QUARTILE AND UPPER QUARTILE.
MARKS 45 39 38 67 90 86 76 40 84 53 70
Q.2 FOLLOWING IS THE DATA MARKS OF TWELVE STUDENTS. YOU ARE REQUIRED TO LOCATE VALUE OF LOWER QUARTILE AND UPPER QUARTILE:
MONTHLY INCOME (`) 3,500 5,000 2,800 10,000 50,000 4,500 1,00,000 3,000 8,000

DISCRETE SERIES

Q.2 CALCULATE LOWER QUARTILE AND UPPER QUARTILE FROM THE FOLLOWING DATA:
Scores 30 40 50 60 70 80
No. of Players 4 11 17 12 10 6

CONTINUOUS SERIES

Q.1 COMPUTE LOWER QUARTILE AND UPPER QUARTILE FROM THE FOLLOWING DATA:
ONLINE TRANSACTION (RS.) Below

1,500

1,500 – 3,000 3,000 – 4,500 4,500 – 6,000 6,000 – 7,500 7,500 & Above
NO. OF CUSTOMERS 21 10 13 7 10 14

Multiple Choice Questions:

Q.1 The ________ is that value of the variable which divides the group into two equal parts.
a. Mean
b. Mode
c. Median
d. Both (a) and (c)
Q.2 Which of the following is merits of Median value of data?
a. Easy To Calculate And Simple To Understand
b. Rigidly Defined
c. Not affected by extreme values
d. All of the above
Q.3 Which of the following is demerits of Median value of data?
a. Arrangement of data is mandatory
b. Affected by fluctuation in Sampling
c. Lack of further algebraic treatment
d. All of the above
Q.4 Median _______ extreme values.
a. includes
b. does not includes
c. rejects
d. None of the above
Q.5 Median is not dependent upon which of the following criteria?
a. All observations
b. Extreme values
c. Least values
d. All of the above
Answer Key
1-a, 2-d, 3-d, 4-b, 5-d

The above mentioned is the concept, that is elucidated in detail about the ‘Measures of Central Tendency – Median ’ for the class 11 Commerce students. To know more, stay tuned to BYJU’S.