Question

If the median of 60 observations, given below is 28.5. Find the values of ‘x’ and ‘y’.
$\begin {gathered}\boxed{\boxed {\begin{array}{ |c |c|} \sf{\bf{\underline{Class \: Interval}}}&\sf{\bf{\underline{Frequency}}} \\ \\ \sf{0 - 10} & \sf{5} \\ \\ \sf{10 - 20}& \sf{x} \\ \\ \sf{20 - 30}& \sf{20} \\ \\ \sf{30-40}&\sf{15} \\ \\ \sf{40-50}&\sf{y} \\ \\ \sf{50-60}&\sf{5}\end{array}}}\end{gathered}$
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Answer 1

$\sf\small\underline{Given:-}$

$\tt{\implies Median\:of\:60\: observation=28.5}$

$\tt{\implies Sum\:of\: frequency=60}$

$\sf\small\underline{To\:Find:-}$

$\tt{\implies The\: value\:of\:x\: and\:y=?}$

$\sf\small\underline{Solution:-}$

By Applying formula we have to set up equation then calculate the value of x , y by solving equations.

$\sf\small\underline{As\:per\: referred\: attachment:-}$

$\tt{\implies M=l+\dfrac{N/2-F}{f}\times\:i}$

$\tt{\implies M=28.5\:\:,l=20\:\:,i=10}$

$\tt{\implies f=20\:\:,F=5+x\:\:,N/2=30}$

• By substituting value we get:-

$\tt{\implies 28.5=20+\dfrac{30-5-x}{20}\times\:10}$

$\tt{\implies 28.5=20+\dfrac{25-x}{20}\times\:10}$

$\tt{\implies 28.5=20+\dfrac{250-10x}{20}}$

$\tt{\implies 28.5-20=\dfrac{250-10x}{20}}$

$\tt{\implies 8.5*20=250-10x}$

$\tt{\implies 170=250-10x}$

$\tt{\implies 10x=250-170}$

$\tt{\implies 10x=80}$

$\tt{\implies x=8}$

$\bf{\implies sum\:of\: frequency=60}$

$\tt{\implies 45+x+y=60}$

$\tt{\implies x+y=60-45}$

$\tt{\implies x+y=15----(i)}$

• Putting the value of x= 8 in eq (i)

$\tt{\implies 8+y=15}$

$\tt{\implies y=15-8}$

$\tt{\implies y=7}$

$\sf\small\underline\pink{Hence,\:the\: missing\: frequency\:=\:8\:and\:7:-}$

Answer 2

Answer:

x=8, y=7

Step-by-step explanation:

$\sf \: Median = 28.5 \\ \tt \: \: n = ∑f_{(1) } = 60 \\ \tt \: \red{n = \frac{60}{2} = 30} \\ \sf \: median \: class = 20 - 30 \\ \green{ \tt \: l = 20,h = 10,f = 20,c.f = 5 + x} \\ \sf \:So, Median = l( \frac{ \frac{n}{2} - cf}{f}) h \\ \leadsto \tt \: 28.5 = 20 + \frac{25 - x}{2} \\ \leadsto \tt28.5 - 20 = \frac{25 - x}{2} \\ \leadsto \tt8.5 \times 2 = 25 - x \\ \leadsto \tt \purple {x = 25 - 17 = \bold{8}} \\ \\ \tt \: and \: so, \: 45 + x + y = 60 \\ \tt \: \leadsto \: y = 60 - (45 + 8) \\ \tt \leadsto \: \blue{y = 60 - 53 = \bold{7}}$