Note on Quartile

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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)

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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 (\(\frac{N+1}{4}\))th item.

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 (\(\frac{N+1}{4}\))th item

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.

 

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Very Short Questions

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



0%
  • In the data set below, what is the lower quartile?

    1, 2, 4, 4, 5, 6, 6, 6, 7, 7, 8

    2
    6
    4
    7
  • In the data set below, what is the upper quartile?

    1, 2. 4, 4, 5, 6, 6, 6 , 7, 7, 8

    8
    7
    5
    6
  • In the data set below, what is the lower quartile?

    29, 34, 38, 41, 51, 57, 65, 65, 81, 83

    34
    38
    29
    41
  • In the data set below, what is the upper quartile?

    17, 23, 32, 36, 52, 54, 67, 83, 98, 99

    67
    98
    99
    83
  • In the data set below, what is the lower quartile?
    2, 2, 2, 2, 3 , 5, 5, 5, 7, 8, 9, 9

    7
    2
    3
    5
  • In the data set below, what is the upper quartile?
    2, 2, 2, 2, 3 , 5, 5, 5, 7, 8, 9, 9

    9
    5
    8
    7
  • In the data set below, what is the upper quartile?

    20, 38, 52, 54, 60, 87, 93

    87
    54
    93
    60
  • In the data set below, what is the lower quartile?

    10, 27, 27, 29, 44, 6727

    10
    29
    44
    27
  • In the data set below, what is the lower quartile?

    19,   37,   41,   43,   83,  85,   94

     

     

     

     

    37
    43
    19
     85
  • In the data set below, what is the upper quartile?

    19,   37,   41,   43,   83,  85,   94

     

     

     

     

     

    83
    94
    85
    43
  • In the data set below, what is the lower quartile?

    23, 26, 26, 30, 34, 42, 49, 60, 96, 96, 96

    26
    32
    30
    34
  • In the data set below, what is the upper quartile?
    59, 86, 86, 88, 96, 96

    86
    59
    96
    88
  • In the data set below, what is the lower quartile?

    2, 4, 5, 6, 6, 7, 8, 8, 8, 9

    5
    2
    6
    4
  • In the data set below, what is the upper quartile?

    2, 4, 5, 6, 6, 7, 8, 8, 8, 9

    8
    7
    9
    6
  • In the data set below, what is the upper quartile?

    1, 1, 1, 2, 2, 4, 4, 5, 5, 6

    6
    4
    5
    1
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