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Measures of Central Tendency — Economics STD 11 Commerce — Question

CBSE BoardEnglish MediumSTD 11 CommerceEconomicsMeasures of Central Tendency6 Marks
Question
Calculate the upper and lower quartiles for the following frequency distribution.
Class Interval
Frequency (f)
13-25
6
25-37
11
37-49
23
49-61
7
61-73
3
Total
50
✓

Answer

Calculation of Upper and Lower Quartiles
Class Interval
Frequency (f)
Cumulative Frequency (cf)
13-25
6
6
25-37
11
17
37-49
23
40
49-61
7
47
61-73
3
50
 
$\text{n}=\Sigma\text{f}=50$
 
Here, $n = 50$
$\therefore$ Lower Quartile number $(q_1) =\text{Size of }\Big(\frac{\text{n}}{4}\Big)\text{th item}$
$=\Big(\frac{50}{4}\Big)\text{th item}=12.5\text{th item}$
cf just greater than 12.5 is 17 and the corresponding class is 25-37.
So, $l_1 = 25$
$cf = 6, f = 11$ and, $c = 12$
$\therefore\ \text{Q}_1=\text{l}_1+\frac{\frac{\text{n}}{4}-\text{cf}}{\text{f}}\times\text{c}$
$=25+\frac{12.5-6}{11}\times12$
$=25+\frac{6.5}{11}\times12=25+7.09$
$\Rightarrow\ \text{Q}_1=32.09$
Now,
Upper Quartile number $(q_3) =\text{Size of }3\Big(\frac{\text{n}}{4}\Big)\text{th item}$
$= 37$.5th item
cf just greater than 37.5 is 40 and corresponding class is $37-49.$
So, $l_1 = 37, cf = 17$
$f = 23$
$c = 12$
$\therefore\ \text{Q}_3=\text{l}_1+\frac{\frac{3\text{n}}{4}-\text{cf}}{\text{f}}\times\text{c}$
$=37+\frac{37.5-17}{23}\times12$
$=37+\frac{20.5}{23}\times12$
$=37+10.70\Rightarrow\ \text{Q}_3=47.70$

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