### Quartile

A quartile is the variate value which divides the value of given frequency into four equal parts. There are three quartiles, they are: -

1. Lower Quartile (Q1)
2. Middle Quartile (Q2)
3. Upper Quartile (Q3)

#### Quartiles can be calculated as following ways: -

Individual series
After arranging the data in ascending order, the quartiles are calculated by applying the following formula:
Lower quartile (Q1) = value of ($$\frac{N+1}{4}$$)thitem.

Upper Quartile (Q3) =value of 3($$\frac{N+1}{4}$$)th item.

Where, N = Total sum of frequency.

Discrete Series
When the data are arranged in ascending order, the quartile is calculated by the following formula:
Lower quartile (Q1) = value of ($$\frac{N+1}{4}$$)thitem

Upper quartile (Q3) = value of 3($$\frac{N+1}{4}$$)th item

Where N = Total sum of frequency

Continuous Series
For continuous series, the following formulae area used.
Lower quartile class = value of 3($$\frac{N}{4}$$)th item

Exact Lower Quartile (Q1) = L + $$\frac{\frac{N}{4} - C.F}{f}$$× i
where,
L = Lower limit of corresponding class
f = frequency
c.f = cumulative frequency
i = class size
N = Total sum of frequency

Upper quartile class = value of 3($$\frac{N}{4}$$)th item

Exact upper quartile (Q3) = L + $$\frac{\frac{3N}{4} - C.F}{f}$$× i

• Quartile is the variate value which divides the value of given frequency into four equal parts.
• There are three quartiles, they are: lower quartile, middle quartile and upper quartile

Solution:

Now,
Arranging in ascending order
42, 52, 62, 72, 82, 90, 100
Total number (N) = 7
We know that,
or, Q3 = 3$$\begin{pmatrix}\frac{n + 1}{4} \end{pmatrix}$$thitem
or, Q3 = 3$$\begin{pmatrix}\frac{7 + 1}{4} \end{pmatrix}$$thitem
or, Q3 = 3$$\begin{pmatrix}\frac{8}{4} \end{pmatrix}$$thitem
or, Q3 = 3 $$\times$$ 2thitem
or, Q3 = 6thitem
$$\therefore$$ The value of Q3is 90.

Solution:

Arrange the values in ascending order of magnitude:

18 20 20 23 23 23 24 27 29

Here, n =7

Lower quartile = value of$$\begin{pmatrix}\frac{n+1}{4} \end{pmatrix}$$thterm

= value of$$\begin{pmatrix}\frac{7+1}{4} \end{pmatrix}$$thterm

= value of$$\begin{pmatrix}\frac{8}{4} \end{pmatrix}$$thterm

= value of 2nditem

$$\therefore$$ Lower Quartile = 20

Upper Quartile = value of$$\begin{pmatrix}\frac{3(n+1)}{4} \end{pmatrix}$$thterm

= value of$$\begin{pmatrix}\frac{3(7+1)}{4} \end{pmatrix}$$thterm

= value of$$\begin{pmatrix}\frac{3 \times 8}{4} \end{pmatrix}$$thterm

= value of$$\begin{pmatrix}\frac{24}{4} \end{pmatrix}$$thterm

= value of 6thterm

$$\therefore$$ Upper Quartile = 27

Solution:

Here,
6, x+5, 12, 14, 17, 20, 21
Q1= 8
numbers = 7
We know that,
or, Q1 = $$\begin{pmatrix}\frac{n + 1}{4} \end{pmatrix}$$th item
or, Q1 = $$\begin{pmatrix}\frac{7 + 1}{4} \end{pmatrix}$$th item
or, Q1 = $$\begin{pmatrix}\frac{8}{4} \end{pmatrix}$$th item
or, Q1 = 2th item
or, Q1 =x + 5
Now,
or,Q1= x + 5
or, 8= x + 5
or, x = 8 - 5
$$\therefore$$ x = 5

Solution:

Here,
numbers = 7
Q3= 60
Now,
or, Q3= 3($$\frac{N + 1}{4}$$)th item
or, Q3= 3($$\frac{7 + 1}{4}$$)th item
or, Q3= 3($$\frac{8}{4}$$)th item
or, Q3= 3 $$\times$$ 2th item
or, Q3= 6th item
i.e.Q3= x + 25
Then,
or,Q3= x + 25
or, 60 = x + 25
or, x + 60 - 25
$$\therefore$$ x = 35

Solution:

Now,
Arranging in ascending order
50, 52, 56, 61, 64, 68, 72
numbers = 7
We know that,
or, Q1 = $$\begin{pmatrix}\frac{n + 1}{4} \end{pmatrix}$$thitem
or, Q1 = $$\begin{pmatrix}\frac{7 + 1}{4} \end{pmatrix}$$thitem
or, Q1 = $$\begin{pmatrix}\frac{8}{4} \end{pmatrix}$$thitem
or, Q1 = 2thitem
or,Q1 = 52
Again,
or, Q2= $$\begin{pmatrix}\frac{n + 1}{2} \end{pmatrix}$$thitem
or, Q2= $$\begin{pmatrix}\frac{7 + 1}{2} \end{pmatrix}$$thitem
or, Q2= $$\begin{pmatrix}\frac{8}{2} \end{pmatrix}$$thitem
or, Q2= 4thitem
or, Q2= 61
Similarly,
or, Q3= 3$$\begin{pmatrix}\frac{n + 1}{4} \end{pmatrix}$$thitem
or, Q3= 3$$\begin{pmatrix}\frac{7 + 1}{4} \end{pmatrix}$$thitem
or, Q3= 3$$\begin{pmatrix}\frac{8}{4} \end{pmatrix}$$thitem
or, Q3= 3 $$\times$$2thitem
or, Q3= 6thitem
or, Q3 = 68

$$\therefore$$ The value ofQ1, Q2 and Q3is 52, 61 and 68 respectively.

Solution:

 Marks No. of students(f) cumulative frequency (c.f.) 10 2 2 20 4 6 30 6 12 40 12 24 50 8 32 60 6 38 70 1 39 N = 39

We have,

Q1= value of $$\begin{pmatrix}\frac{n + 1}{4} \end{pmatrix}$$th term

= value of $$\begin{pmatrix}\frac{39 + 1}{4} \end{pmatrix}$$th term

= value of 10th term

In c.f. column just greater than 10 is 12 and its coressponding value is 30.

$$\therefore$$ Q1 = 30

Q2 =value of $$\begin{pmatrix}\frac{n + 1}{2} \end{pmatrix}$$th term

= value of $$\begin{pmatrix}\frac{39 + 1}{2} \end{pmatrix}$$th term

= value of 20th term

In c.f. column just greater than 20 is 24and its coressponding value is 40.

$$\therefore$$ Q2=40

Q3 =value of $$\begin{pmatrix}\frac{3(n + 1)}{4} \end{pmatrix}$$th term

=value of $$\begin{pmatrix}\frac{3(39 + 1)}{4} \end{pmatrix}$$th term

=value of $$\begin{pmatrix}\frac{3 * 40}{4} \end{pmatrix}$$th term

=value of $$\begin{pmatrix}\frac{3(39 + 1)}{4} \end{pmatrix}$$th term

=value of 30th term

In c.f. column just greater than 30 is 32 and its coressponding value is 50.

$$\therefore$$ Q3= 50

Solution:

 Marks Frequency (f) c.f 15 5 5 25 10 15 35 16 31 45 7 38 55 5 43 N = 43

Now,
or,Q1= $$\begin{pmatrix}\frac{N + 1}{4} \end{pmatrix}$$th item
or,Q1= $$\begin{pmatrix}\frac{43 + 1}{4} \end{pmatrix}$$th item
or,Q1= $$\begin{pmatrix}\frac{44}{4} \end{pmatrix}$$th item
or,Q1= 11th item

In c.f column, c.f is just greater than 11 is 15 and its corresponding value is 25. So, Q1 = 25
Again,
or, Q3 = 3$$\begin{pmatrix}\frac{n + 1}{4} \end{pmatrix}$$
or, Q3 = 3$$\begin{pmatrix}\frac{43 + 1}{4} \end{pmatrix}$$
or, Q3 = 3$$\begin{pmatrix}\frac{44}{4} \end{pmatrix}$$
or, Q3 = 3 $$\times$$ 11thitem
or, Q3 = 33th item

In c.f column, c.f is just greater than 33 is 38 and its corresponding value is 45.
So, Q3 is 45.

Solution:

 x f c.f 0 -10 6 6 10 - 20 8 14 20 - 30 10 24 30 - 40 7 31 40 - 50 5 36 N = 36

Now,
or, Q1 = $$\begin{pmatrix}\frac{N}{4} \end{pmatrix}$$th item
or, Q1 = $$\begin{pmatrix}\frac{36}{4} \end{pmatrix}$$th item
or, Q1 = 9th item
In c.f column, c.f is just greater than 9 is 14 and its corresponding value is 10 - 20.
So, Q1 lies in class 10 - 20
Then,
l = 10, c.f = 6, f = 8, and i = 10

or, Q1 =L + $$\frac{\frac{N}{4}-C.F}{f}$$ × i
or,Q1 = 10 + $$\frac{\frac{36}{4}-6}{8}$$ × 10
or,Q1 = 10 + $$\frac{9 - 6}{8}$$ × 10
or,Q1 = 10 + $$\frac{3}{8}$$ × 10
or,Q1 = 10 + $$\frac{3}{4}$$ × 5
or,Q1 = 10 + 0.75× 5
or,Q1 = 10 +3.75
or,Q1 = 13.75
$$\therefore$$ The value of Q1is 13.75.

Solution:

 Class Frequency(f) c.f. 0-5 6 6 5-10 8 14 10-15 10 24 15-20 7 31 20-25 5 36 25-30 4 40 N =40

Now,

Q1 Class = value of ($$\frac{N}{4}$$)th item

= value of ( $$\frac{40}{4}$$)th item

= value of 10th item

In c.f. column, c.f. just greater than 10 is 14 and its corresponding class is (5-10).

$$\therefore$$ Q1 lies in the class 5-10

Here,

$$\frac{N}{4}$$ = 10,l= 5, c.f. = 6, f = 8 and i = 5

$$\therefore$$Q1= l + $$\frac{\frac{N}{4}− c.f}{f}$$ × i

= 5 + $$\frac{10− 6}{8}$$ ×5

=5 + $$\frac{4}{8}$$ ×5

= 5 + 2.5

= 7.5

Again,

Q3 Class = value of($$\frac{3N}{4}$$)th item

= value of($$\frac{3 * 40}{4}$$)th item

=value of30th item

In c.f. column, c.f.just greater than 30 is 31 and its corresponding class is 15-20.

Here,

$$\frac{3N}{4}$$ = 30,l= 5, c.f. = 24, f = 7and i = 5

$$\therefore$$Q3= l + $$\frac{\frac{3N}{4}− c.f}{f}$$ × i

= 5 + $$\frac{30− 24}{7}$$ ×5

= 5 + 4.28

= 9.28

Solution:

 Class frequency c.f 5 - 10 4 4 10 - 15 6 10 15 - 20 10 20 20 - 25 15 35 25 - 30 10 45 30 - 35 6 51 35 - 40 5 56 N = 56

Now,
or, Q3= 3($$\frac{N}{4}$$)th item
or, Q3= 3($$\frac{56}{4}$$)th item
or, Q3= 3 $$\times$$ 14th item
or, Q3= 42th item
In c.f column, c.f is just greater than 42 is 45 and its corresponding value is 25 - 30.
Now,
l = 25, c.f = 35, f = 10, and i = 5
Then,
or, Q3= L + $$\frac{\frac{3N}{4}-c.f}{f}$$× i
or, Q3= 25 + $$\frac{42 - 35}{10}$$× 5
or, Q3= 25 + $$\frac{7}{2}$$
or, Q3= 25 +3.5
or, Q3= 28.5
$$\therefore$$ The value of Q3 is 28.5.

Solution:

 Marks Students (f) c.f. 30-40 5 5 40-50 12 17 50-60 25 42 60-70 15 57 70-80 8 65 80-90 3 68 N =68

Now,

Q1 Class = value of ($$\frac{N}{4}$$)th item

= value of ( $$\frac{68}{4}$$)th item

= value of 17th item

In c.f. column, c.f. just greater than 17 is 42 and its corresponding class is (50-60).

$$\therefore$$ Q1 lies in the class 50-60

Here,

$$\frac{N}{4}$$ = 17, l= 50, c.f. = 17, f = 25 and i =10

$$\therefore$$Q1= l + $$\frac{\frac{N}{4}− c.f}{f}$$ × i

= 50 + $$\frac{17− 17}{25}$$ ×10

=50 + $$\frac{0}{25}$$ ×10

= 50 +0

= 50

Again,

Q2Class = value of ($$\frac{N}{2}$$)th item

= value of ( $$\frac{68}{2}$$)th item

= value of 34th item

In c.f. column, c.f. just greater than 34 is 42 and its corresponding class is (50-60).

$$\therefore$$ Q2lies in the class 50-60

Here,

$$\frac{N}{2}$$ = 34, l = 50, c.f. = 17, f = 25 and i =10

$$\therefore$$Q1= l + $$\frac{\frac{N}{2}− c.f}{f}$$ × i

= 50 + $$\frac{34− 17}{25}$$ ×10

=50 + $$\frac{17}{25}$$ ×10

= 50 +6.8

= 56.8

Q3 Class = value of($$\frac{3N}{4}$$)th item

= value of($$\frac{3 * 68}{4}$$)th item

=value of 51th item

In c.f. column, c.f.just greater than 51 is 57 and its corresponding class is 60-70.

Here,

$$\frac{3N}{4}$$ = 51, l= 60, c.f. = 42, f = 15 and i = 10

$$\therefore$$Q3= l + $$\frac{\frac{3N}{4}− c.f}{f}$$ × i

= 60 + $$\frac{51− 42}{15}$$ × 10

= 60 + 6

= 66

Solution:

 x f c.f 50 8 8 60 16 24 70 12 36 80 4 40 90 3 43 N = 43

Now,
or, Q1 =($$\frac{N+1}{4}$$)th item
or, Q1 =($$\frac{43+1}{4}$$)th item
or, Q1 =($$\frac{44}{4}$$)th item
or, Q1 =11th item
In c.f column, c.f is just greater than 11 is 24 and its corresponding value is 60.
So,Q1 = 60
Again,
or, Q3 =3($$\frac{N+1}{4}$$)th item
or, Q3 =3($$\frac{43+1}{4}$$)th item
or, Q3 =3($$\frac{44}{4}$$)th item
or, Q3 =3 $$\times$$ 11th item
or, Q3 =33th item
In c.f column, c.f is just greater than 33 is 36 and its corresponding value is 70.
i.e.Q3 = 70

0%

2
6
4
7

7
5
8
6

41
38
34
29

99
83
98
67

3
5
2
7

8
9
5
7

93
87
60
54

29
27
10
44

43
85
37
19

83
85
94
43

32
30
26
34

96
86
88
59

5
6
2
4

8
9
7
6

6
1
4
5