Question

* 2. The median of the following data is 50
Find the values of p and q if the sum of
all the frequencies is 90
Marks
Frequency
20-30
p
30-40
15
40-50
25
50-60
20
60-70
70-80
8
80-90​

Answer 1

✦CORRECT QUESTION :-

◮The median of the following data is 50 .Find the value of p and q ,if the sum of all frequencies is 90.

$\begin{gathered}\boxed{\begin{array}{c|c} \underline{\sf{Classes}}&\sf{ \underline{Frequency}} \\ \sf{} & \sf{} & \sf{} \\ 20-30&p\\30 - 40&15 \\ 30-40 & 25 \\ 40-50 & 10\\ 50-60 &20 \\ 60 - 70&q \\ 70 - 80&8 \\ 80 - 90&10 \\\boxed{ \sf \: \: total \: \: }& \longrightarrow \boxed{ \sf90} \: \: \: \: \: \: \end{array}}\end{gathered}$

✦FORMULA REQUIRED :-

$\bigstar \boxed{ \sf median = l + \frac{ \frac{n}{2} - c.f }{f} \times h}$

✦SOLUTION:-

⠀⠀⠀⠀•Let n= sum of frequencies

⠀⠀⠀⠀•c.f=Cumulative frequency

⠀⠀⠀⠀•l=lower limit of the class

⠀⠀⠀⠀•h=height of the class

⠀⠀⠀⠀•f=frequency of the preceding class

⠀⠀⠀⠀•m=median of the frequencies.

Let's draw Cumulative frequency distributive table

$\begin{gathered}\boxed{\begin{array}{c|c|c } \underline{\sf{Classes}}&\sf{ \underline{Frequency}}& \sf \underline{cumulative \: frequency}\\ \sf{} & \sf{} & \sf{} \\ 20-30&p&p\\30 - 40&15 &15 + p \\ 40-50 & 25&40 + p\\ 50-60 &20 &60 + p\\ 60 - 70&q &60 + p + q\\ 70 - 80&8 &68 + p + q\\ 80 - 90&10& \underline{ \: \: \: \: \: \: \: \: \: \: \: 78 + p + q \: \: \: \: \: \: \: \: \: } \\\boxed{ \sf \: \: total \: \: }& \longrightarrow \boxed{ \sf90} \: \: \: \: \: \: \end{array}}\end{gathered}$

Now , Given that,

n=90,

n/2=90/2=45

⠀⠀⠀⠀⠀⠀

Which lies in the interval 50-60

Here, l=50, f=20

cf=40+p and h=10

⠀⠀⠀⠀⠀⠀

$\sf median = 50 + \dfrac{45 - 40 - p}{20} \times 10$

It is given that median of the following frequencies is 50

⠀⠀⠀⠀⠀⠀

$\sf \therefore50 = 50 + \dfrac{45 - 40 - p}{20}$

$\implies \sf \cancel{50} - \cancel{ 50} = \dfrac{5 - p}{2}$

$\implies \sf2 \times 0 = 5 - p$

$\implies \sf 5 - p = 0$

$\implies \sf p = 5$

⠀⠀⠀⠀⠀⠀

Also we have

$\implies \sf78 + p + q = 90$

⠀⠀⠀⠀⠀⠀

Putting p=5 in above equation we get,

$\implies \sf78 + 5 + q = 90$

$\implies \sf q = 90 - 83$

$\implies \sf q = 7$

⠀⠀⠀⠀⠀⠀

Hence ,the required value of p and q are 5 and 7 respectively.