Quartile Definition, formulas, First Quartile, Third Quartile, How to calculate the Quartile.
Quartiles :
Quartile Definition :
" All data divided four equal parts is known as Quartile. "
It is clear that the items which divided the data into four equal parts lie at the quarter, half and three quarters, of the way down the arrangement.
Quartile denoted by Q1 means first quartile, Q2 means second quartile, Q3 means third quartile respectively.
Clearly the second quartile is the Median item.
Quartile Formulas :
For Individual items and Discrete Distribution :
For Individual items and Discrete Distribution :
1) First Quartile = Q1 = Size of [ ( N + 1 ) / 4 ]th item.
2) Third Quartile = Q3 = Size of 3 × [ ( N + 1 ) / 4 ]th item
For Continues Distribution :
1) First Quartile = Q1 = Size of [ N / 4 ]th item.
2) Third Quartile= Q3 =Size of 3 × [ N / 4 ] th item
Quartiles Examples :
Computation of Quartiles for Individual Data :
1) Calculate the first, second and third quartile from the following values weights in kg of a group of students.
50, 51, 52, 50, 51, 53, 55, 54, 52, 51, 50.
Ans. :
Arranging data in ascending order :
50, 50, 50, 51, 51, 51, 52, 52, 53, 54, 55.
Here, N = 11.
First Quartile = Q1 = Size of ( N + 1 ) / 4 th item
Q1 = Size of ( 11 + 1 ) / 4 th item
Q1 = Size of 12 / 4 th item
Q1 = Size of 3rd item
Therefore, Q1 = 50 kg.
Second Quartile = Median = Q2
Q2 = Me = Size of ( N + 1 ) / 2 th item
Q2 = size of ( 11 + 1 ) / 2 th item
Q2 = size of 6th item
Therefore, Q2 = 51 kg.
Third Quartile = Q3 = Size of 3 × [ ( N + 1 ) /4 ]th item
Q3 = Size of 3 × [ ( 11 + 1 ) / 4 ] th item
Q3 = Size of 3 × 3 th item
Q3 = Size of 9th item
Therefore, Q3 = 53 kg.
Discrete Distribution :
Q2) From the following data , Calculate First Quartile, Second Quartile and Third Quartile.
Weight in kg. : 55, 56, 52, 51, 50, 53, 54.
No. of women : 20, 30, 25, 24, 28, 22, 32.
Ans. :
Step 1: Arrange the data in ascending order
Step 2: Prepare the column of cumulative frequency.
Calculations of Q1, Q2 and Q3.
Weight (X) No. of Women (f). Cumulative frequency
50 28 28
51 24 52
52 25 77
53 22 99
54 32 131
55 20 151
56 30 181
First Quartile = Q1 = size of ( N + 1 ) / 4 th item
= Size of ( 181 + 1 ) / 4 th item
= Size of ( 182 / 4 ) th item
Q1 = Size of 45.5 th item
The item 45.5 i.e. 46 lies between 28 and 52 and it is weight is 51.
Therefore, Q1 = 51 kg.
Second Quartile = Me = Q2 = Size of ( N + 1 ) / 2 th item
Q2 = Me. = Size of ( 181 + 1 ) / 2 th item
Q2 = size of 182 / 2 th item
Q2 = size of 91 th item
The item 91 lies between 77 and 91 and it is weight is 53.
Therefore, Q2 = Me = 53 kg.
Third Quartile = Q3 = Size of 3 [( N + 1 ) / 4] th item
Q3 = size of 3 ( 45.5 ) th item
Q3 = size of (136.5) th item
The item 136.5 i.e. 137 lies between 131 and 151 and it is weight is 55.
Therefore, Q3 = 55 kg.
Continuous Distribution :
Step 1: Arrange the data in ascending order
Step 2: Prepare the column of cumulative frequency.
Step 3: Find the class in which this item lies.
Step 4: Formula :
Q1 = Size of ( N / 4 ) th item
Q1 = L1 + { [ ( N / 4 ) - c.f. ] / f } × i
Where,
L1 = lower limit of the quartile class,
c.f. = cumulative frequency of the previous class,
f = frequency of the quartile class,
i= class interval of the quartile class.
Q2 = Me = Size of ( N / 2 ) th item
Q2 = L1 + { [ ( N / 2 ) - c.f. ] / f } × i
Q3 = Me = Size of 3( N / 4) th item
Q3 = L1 + { [3 ( N / 4 ) - c.f. ] / f }
Quartiles Examples :
Computation of Quartiles for Individual Data :
1) Calculate the first, second and third quartile from the following values weights in kg of a group of students.
50, 51, 52, 50, 51, 53, 55, 54, 52, 51, 50.
Ans. :
Arranging data in ascending order :
50, 50, 50, 51, 51, 51, 52, 52, 53, 54, 55.
Here, N = 11.
First Quartile = Q1 = Size of ( N + 1 ) / 4 th item
Q1 = Size of ( 11 + 1 ) / 4 th item
Q1 = Size of 12 / 4 th item
Q1 = Size of 3rd item
Therefore, Q1 = 50 kg.
Second Quartile = Median = Q2
Q2 = Me = Size of ( N + 1 ) / 2 th item
Q2 = size of ( 11 + 1 ) / 2 th item
Q2 = size of 6th item
Therefore, Q2 = 51 kg.
Third Quartile = Q3 = Size of 3 × [ ( N + 1 ) /4 ]th item
Q3 = Size of 3 × [ ( 11 + 1 ) / 4 ] th item
Q3 = Size of 3 × 3 th item
Q3 = Size of 9th item
Therefore, Q3 = 53 kg.
Discrete Distribution :
Q2) From the following data , Calculate First Quartile, Second Quartile and Third Quartile.
Weight in kg. : 55, 56, 52, 51, 50, 53, 54.
No. of women : 20, 30, 25, 24, 28, 22, 32.
Ans. :
Step 1: Arrange the data in ascending order
Step 2: Prepare the column of cumulative frequency.
Calculations of Q1, Q2 and Q3.
Weight (X) No. of Women (f). Cumulative frequency
50 28 28
51 24 52
52 25 77
53 22 99
54 32 131
55 20 151
56 30 181
First Quartile = Q1 = size of ( N + 1 ) / 4 th item
= Size of ( 181 + 1 ) / 4 th item
= Size of ( 182 / 4 ) th item
Q1 = Size of 45.5 th item
The item 45.5 i.e. 46 lies between 28 and 52 and it is weight is 51.
Therefore, Q1 = 51 kg.
Second Quartile = Me = Q2 = Size of ( N + 1 ) / 2 th item
Q2 = Me. = Size of ( 181 + 1 ) / 2 th item
Q2 = size of 182 / 2 th item
Q2 = size of 91 th item
The item 91 lies between 77 and 91 and it is weight is 53.
Therefore, Q2 = Me = 53 kg.
Third Quartile = Q3 = Size of 3 [( N + 1 ) / 4] th item
Q3 = size of 3 ( 45.5 ) th item
Q3 = size of (136.5) th item
The item 136.5 i.e. 137 lies between 131 and 151 and it is weight is 55.
Therefore, Q3 = 55 kg.
Continuous Distribution :
Step 1: Arrange the data in ascending order
Step 2: Prepare the column of cumulative frequency.
Step 3: Find the class in which this item lies.
Step 4: Formula :
Q1 = Size of ( N / 4 ) th item
Q1 = L1 + { [ ( N / 4 ) - c.f. ] / f } × i
Where,
L1 = lower limit of the quartile class,
c.f. = cumulative frequency of the previous class,
f = frequency of the quartile class,
i= class interval of the quartile class.
Q2 = Me = Size of ( N / 2 ) th item
Q2 = L1 + { [ ( N / 2 ) - c.f. ] / f } × i
Q3 = Me = Size of 3( N / 4) th item
Q3 = L1 + { [3 ( N / 4 ) - c.f. ] / f }
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