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Measures of central tendency — Statistics STD 11 Commerce — Question

Gujarat BoardEnglish MediumSTD 11 CommerceStatisticsMeasures of central tendency5 Marks
Question
The data about marks scored by $55$ students from a school are given below :
Marks $0 -10$ $10 – 20$ $20 -30$ $30 – 40$ $40 – 50$ $50 – 60$ $60 – 70$
No. of students $4$ $7$ $11$ $14$ $9$ $7$ $3$
$(i)$ If $30%$ students failed in the examination, what are the passing marks ?
$(1)$ If top $5%$ students are to be selected for scholarship, find the lowest marks for scholarship ?
✓

Answer

Marks No. of students $f$ Cumulative
frequency $cf$
$0 – 10$ $4$ $4$
$10 – 20$ $7$ $11$
$20 – 30$ $11$ $22$
$30 – 40$ $14$ $36$
$40 – 50$ $9$ $45$
$50 – 60$ $7$ $52$
$60 – 70$ $3$ $55$
Total $n = 55$ $-$
$(i)$ if $30 \%$ of students are failed, their marks are $D_3$ or less.
Now, $D _3$ class $=$ Class that includes $3\left(\frac{ n }{10}\right)$ th observation
$=$ Class that includes $3\left(\frac{55}{10}\right)$
$=$ Class that Includes $3(5.5)$
$=$ Class that includes $16.5$ th observation
Referring to column cf, $D_3$ class $=20-30$
Now, $D_3=L+\frac{3\left(\frac{ m }{10}\right)- cf }{ f } \times c$
Putting $L=20,3\left(\frac{ n }{10}\right)=16.5$, cf $=11, f=11$ and $c=10$ in the formula.
$D_3=20+\frac{16.5-11}{11} \times 10=20+\frac{5.5 \times 10}{11}=20+\frac{55}{11}=20+5=25$ marks
Hence, maximum marks of the student who fails will be $25.$
$\therefore$ The marks required for passing is $26.$
$(ii)$ Scholarship is to be given to the $5 \%$ of students obtaining maximum marks.
Therefore, minimum marks required for scholarship is $P _{95}$.
$P _{95}$ class $=$ Class that includes $95\left(\frac{ n }{100}\right)$ th observation
$=$ Class that includes $95\left(\frac{55}{100}\right)$
$=$ Class that includes $95(0.55)$
$=$ Class that includes $52.25^{\text {th }}$ observation
Referring to column cf, $P _{95}$ class $=60-70$
Now, $P_{95}=L+\frac{95\left(\frac{ n }{200}\right)- cf }{ f } \times C$
Putting $L=60,95\left(\frac{ n }{100}\right)=52.25$, cf $=52, f=3$ and $c=10$ in the formula.
$P_{95}=60+\frac{52.25-52}{3} \times 10$
$=60+\frac{0.25 \times 10}{3}=60+\frac{2.5}{3}=60+0.83=60.83$ marks
Hence, the minimum marks of the $5 \%$ students getting scholarship is $60.83$ marks.

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