Question

If the median of the following distribution is 36.51, then find the value of p and q.

Answer 1

Solution:

We know that Median of grouped data

$= l + ( \frac{ \frac{n}{2} - cf}{f} ) h$
since Median = 36.51

Median class is :32.5-39.5

l = 32.5

f= 68

cf=35+p


h=6

$36.51 = 32.5 + ( \frac{ \frac{n}{2} - 35 - p }{68} ) \times 6 \\ \\ \frac{4}{6} = ( \frac{ \frac{n}{2} - 35 - p }{68} )$
Since total observation are not given,so put the value of n and solve for p and hence q.

Answer 2

Answer:

Step-by-step explanation:

Given median of the following distribution is $M=36.51$

Now make a table

Class interval           frequency$f_{1}$                  cumulative frequency

   19-25                                  35                                                    35

   26-32                                  p                                                  35+p

   33-39                                 68                                               35+p+68

   40-46                                  q                                              35+p+68+q

   47-53                                 35                                          35+p+68+q+35

   54-60                                  4                                         35+p+68+q+35+4

So 33-39 is the median class.

we know that median (M)

$M=l+\frac{\frac{n}{2}-c_{f}}{f}\times h$

Where,

$l=\text{ Lower limit of median class}= 33$

$h=\text{Class interval}=39-33=6$

$n=\sum{f}=60\\\\\therefore\frac{n}{2}=\frac{60}{2}=30$

$c_{f}=\text{cumulative frequency of the class before median class}= 35+p$

$f=\text{ frequency of the median class}=68$

therefore.

$M=l+\frac{\frac{n}{2}-c_{f}}{f}\times h\\\\\Rightarrow36.51=33+\frac{30-(35+p)}{68}\times6\\\\\Rightarrow36.51-33=\frac{(30-35-p)}{68}\times6\\\\\Rightarrow3.51=\frac{(-5-p)}{68}\times6\\\\\Rightarrow3.51\times68={(-5-p)}\times6\\\\\Rightarrow\frac{3.51\times68}{6}={(-5-p)}\\\\\Rightarrow39.78={(-5-p)}\\\\\Rightarrow{p}=-5-39.78\\\\\Rightarrow{p}=-44.78$

We know that

$\sum{f}=60= 35+p+68+q+35+4\\\\\Rightarrow60=138+p+q\\\\\Rightarrow{q}=60-138-p\\\\\Rightarrow{q}=60-138-(-44.78)\\\\\Rightarrow{q}=60-138+44.78\\\\\Rightarrow{q}=-33.22$