A quartile is the variate value which divides the value of given frequency into four equal parts. There are three quartiles, they are: -
Individual series
After arranging the data in ascending order, the quartiles are calculated by applying the following formula:
Lower quartile (Q_{1}) = value of (\(\frac{N+1}{4}\))^{th }item.
Upper Quartile (Q_{3}) =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 (Q_{1}) = value of (\(\frac{N+1}{4}\))^{th }item
Upper quartile (Q_{3}) = 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 (Q_{1}) = 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 (Q_{3}) = L + \(\frac{\frac{3N}{4} - C.F}{f}\)× i
.
Solution:
Now,
Arranging in ascending order
42, 52, 62, 72, 82, 90, 100
Total number (N) = 7
We know that,
or, Q_{3} = 3\(\begin{pmatrix}\frac{n + 1}{4} \end{pmatrix}\)^{th}item
or, Q_{3} = 3\(\begin{pmatrix}\frac{7 + 1}{4} \end{pmatrix}\)^{th}item
or, Q_{3} = 3\(\begin{pmatrix}\frac{8}{4} \end{pmatrix}\)^{th}item
or, Q_{3} = 3 \(\times\) 2^{th}item
or, Q_{3} = 6^{th}item
\(\therefore\) The value of Q_{3}is 90.
Find the lower quartile and upper quartile of the following data set of scores:
18 20 23 20 23 27 24
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\(\begin{pmatrix}\frac{n+1}{4} \end{pmatrix}\)^{th}term
= value of\(\begin{pmatrix}\frac{7+1}{4} \end{pmatrix}\)^{th}term
= value of\(\begin{pmatrix}\frac{8}{4} \end{pmatrix}\)^{th}term
= value of 2^{nd}item
\(\therefore\) Lower Quartile = 20
Upper Quartile = value of\(\begin{pmatrix}\frac{3(n+1)}{4} \end{pmatrix}\)^{th}term
= value of\(\begin{pmatrix}\frac{3(7+1)}{4} \end{pmatrix}\)^{th}term
= value of\(\begin{pmatrix}\frac{3 \times 8}{4} \end{pmatrix}\)^{th}term
= value of\(\begin{pmatrix}\frac{24}{4} \end{pmatrix}\)^{th}term
= value of 6^{th}term
\(\therefore\) Upper Quartile = 27
Solution:
Here,
6, x+5, 12, 14, 17, 20, 21
Q_{1}= 8
numbers = 7
We know that,
or, Q_{1} = \(\begin{pmatrix}\frac{n + 1}{4} \end{pmatrix}\)^{th} item
or, Q_{1} = \(\begin{pmatrix}\frac{7 + 1}{4} \end{pmatrix}\)^{th} item
or, Q_{1} = \(\begin{pmatrix}\frac{8}{4} \end{pmatrix}\)^{th} item
or, Q_{1} = 2^{th} item
or, Q_{1} =x + 5
Now,
or,Q_{1}= x + 5
or, 8= x + 5
or, x = 8 - 5
\(\therefore\) x = 5
Solution:
Here,
numbers = 7
Q_{3}= 60
Now,
or, Q_{3}= 3(\(\frac{N + 1}{4}\))^{th} item
or, Q_{3}= 3(\(\frac{7 + 1}{4}\))^{th} item
or, Q_{3}= 3(\(\frac{8}{4}\))^{th} item
or, Q_{3}= 3 \(\times\) 2^{th} item
or, Q_{3}= 6^{th} item
i.e.Q_{3}= x + 25
Then,
or,Q_{3}= x + 25
or, 60 = x + 25
or, x + 60 - 25
\(\therefore\) x = 35
In the data set below, what is the lower quartile?
1, 2, 4, 4, 5, 6, 6, 6, 7, 7, 8
In the data set below, what is the upper quartile?
1, 2. 4, 4, 5, 6, 6, 6 , 7, 7, 8
In the data set below, what is the lower quartile?
29, 34, 38, 41, 51, 57, 65, 65, 81, 83
In the data set below, what is the upper quartile?
17, 23, 32, 36, 52, 54, 67, 83, 98, 99
In the data set below, what is the lower quartile?
2, 2, 2, 2, 3 , 5, 5, 5, 7, 8, 9, 9
In the data set below, what is the upper quartile?
2, 2, 2, 2, 3 , 5, 5, 5, 7, 8, 9, 9
In the data set below, what is the upper quartile?
20, 38, 52, 54, 60, 87, 93
In the data set below, what is the lower quartile?
10, 27, 27, 29, 44, 6727
In the data set below, what is the lower quartile?
19, 37, 41, 43, 83, 85, 94
In the data set below, what is the upper quartile?
19, 37, 41, 43, 83, 85, 94
In the data set below, what is the lower quartile?
23, 26, 26, 30, 34, 42, 49, 60, 96, 96, 96
In the data set below, what is the upper quartile?
59, 86, 86, 88, 96, 96
In the data set below, what is the lower quartile?
2, 4, 5, 6, 6, 7, 8, 8, 8, 9
In the data set below, what is the upper quartile?
2, 4, 5, 6, 6, 7, 8, 8, 8, 9
In the data set below, what is the upper quartile?
1, 1, 1, 2, 2, 4, 4, 5, 5, 6
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