Question

The median of the following data is 52.5. Find the value of X and Y, if the sum of frequencies is 100.

Class Interval	0-10	10-20	20-30	30-40	40-50	50-60	60-70	70-80	80-90	90-100
Frequency	2	5	X	12	17	20	Y	9	7	4

Answer 1

Hint: We will first make the table of the frequencies and cumulative frequencies. We will make two equations by using the sum of frequencies and median of the given data. The formula for the median is $l + \dfrac{{\left( {\dfrac{n}{2}} \right) - c.f}}{f} \times h$, where $l$ is the lower value of median class, $n$ is the total number of frequency and c.f. is the cumulative frequency of previous class, $f$ is the frequency of median class and $h$ is the width of the class-interval.

Complete step-by-step answer:
We are given that the median of the data is 52.5 and the sum of frequency is 100.
First of all, we will make a table of frequency and cumulative frequency.

Class-Interval	Frequency $\left( f \right)$	Cumulative frequency(c.f.)
0-10	2	2
10-20	5	7
20-30	X	7+X
30-40	12	19+X
40-50	17	36+X
50-60	20	56+X
60-70	Y	56+X+Y
70-80	9	65+X+Y
80-90	7	72+X+Y
90-100	4	76+X+Y

Here, the sum of frequencies is 76+X+Y which is also equals to 100.
Hence,
$
  76 + {\text{X + Y = 100}} \\
   \Rightarrow {\text{X + Y = 24}} \\
$
We know that the median is calculated as $l + \dfrac{{\left( {\dfrac{n}{2}} \right) - c.f}}{f} \times h$, where $l$ is the lower value of median class, $n$ is the total number of frequency and c.f. is the cumulative frequency of previous class, $f$ is the frequency of median class and $h$ is the width of the class-interval.

Here, the sum of all frequencies is 100 which is equal to $n$
Then, $\dfrac{n}{2} = 50$ belongs to the class-interval 50-60
Hence, $l = 50$ and width of the class-interval is $60 - 50 = 10$, that is $h = 10$
The frequency of the median class is 20
Also, cumulative frequency of previous class is 36+X
Then we have
$
  52.5 = 50 + \dfrac{{50 - \left( {36 + {\text{X}}} \right)}}{{20}} \times 10 \\
   \Rightarrow 52.5\left( 2 \right) = 114 - {\text{X}} \\
   \Rightarrow 105 = 114 - {\text{X}} \\
   \Rightarrow {\text{X = 9}} \\
 $
On substituting the value of ${\text{X = 9}}$, we get
 $
  {\text{9 + Y = 24}} \\
   \Rightarrow {\text{Y = 15}} \\
$
Hence the value of X is 9 and the value of Y is 15.
Note: The median is a kind of average where the middle value of the given data. Students must know the formula of the median. Also, the cumulative frequency in the formula is till the previous median class.