Question

The median of the following data is $525.$find the values of x and y if the total frequency is $100$

Class interval	Frequency
$ 0 - 100 \\ 100 - 200 \\ 200 - 300 \\ 300 - 400 \\ 400 - 500 \\ 500 - 600 \\ 600 - 700 \\ 700 - 800 \\ 800 - 900 \\ 900 - 1000 \\ $	$ 2 \\ 5 \\ x \\ 12 \\ 17 \\ 20 \\ y \\ 9 \\ 7 \\ 4 \\ $

Answer 1

Hint: First of all we will find the cumulative frequency and then will use the median formula finding the class of the range for the given value for the median. Find the correlation between the known and unknown terms and get the values for “x” and “y”.

Complete step-by-step answer:
Cumulative frequency can be defined as the sum of all the previous frequencies up to the current point.

Class interval	Frequency	Cumulative frequency
$ 0 - 100 \\ 100 - 200 \\ 200 - 300 \\ 300 - 400 \\ 400 - 500 \\ 500 - 600 \\ 600 - 700 \\ 700 - 800 \\ 800 - 900 \\ 900 - 1000 \\ $	$ 2 \\ 5 \\ x \\ 12 \\ 17 \\ 20 \\ y \\ 9 \\ 7 \\ 4 \\ $	$ 2 \\ 7 \\ 7 + x \\ 19 + x \\ 36 + x \\ 56 + x \\ 56 + x + y \\ 65 + x + y \\ 72 + x + y \\ 76 + x + y \\ $
	$N = 100$

We can observe that, $76 + x + y = 100$
Make the variables on one side and constants on the opposite side. When you move any term from one side to the opposite side then the sign of the term also changes.
$x + y = 100 - 76$
Simplify the expressions finding the difference of the terms.
$x + y = 24$ ….. (A)
Given that: Median$ = 525$which lies between the class $500 - 600$
Median class $ = 500 - 600$
Apply, Median $ = l + \dfrac{{\dfrac{n}{2} - c.f.}}{f} \times h$
Place the known values in the above expression –
$525 = 500 + \dfrac{{\dfrac{{100}}{2} - (36 + x)}}{{20}} \times 100$
Simplify the above expression –
$25 = (50 - 36 - x)5$
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
 $\dfrac{{25}}{5} = (14 - x)$
Common factors from the numerator and denominator cancel each other.
$5 = (14 - x)$
Make “x” the subject
$
  x = 14 - 5 \\
  x = 9 \;
 $
$x + y = 24$
Place the value of “x” in the above expression –
$9 + y = 24$
Simplify the above equation –
$
  y = 24 - 9 \\
  y = 15 \;
 $
Hence, the required solutions are $x = 9$ and $y = 15 $
So, the correct answer is “ $x = 9$ and $y = 15 $”.

Note: Remember the difference between mean and median. Mean can be defined as the sum of all the numbers upon the total number of the numbers. Mean can be expressed as the data which is affected by the extreme values of the sequences while in the median extreme values do not affect the extreme values. Be careful about the sign convention while simplifying and while moving terms from one side to the opposite side.