Question

2. The following
following data have been arranged in
cending order. If their median is 63, find the
value of x.
34, 37, 53, 55, X, X + 2, 77, 83, 89 and 100.
​

Answer 1

Solution :-

Data which is arranged in ascending order = 34, 37, 53, 55, x, x + 2, 77, 83, 89, 100

Number of observations n = 10 (even)

So, Median = Average of (n/2)th observation and (n/2 + 1)th observation

⇒ Median = Average of (10/2)th observation and (10/2 + 1)th observation

⇒ Median = Average of 5th and 6th observations

⇒ Median = Average of x and (x + 2)

⇒ Median = (x + x + 2)/2

⇒ Median = (2x + 2)/2

⇒ Median = x + 1

Also, Median = 63

⇒ x + 1 = 63

⇒ x = 63 - 1

⇒ x = 62

Therefore the value of x is 62.

Answer 2

Given: Data arranged in ascending order and the median.

To find: The value of x.

Answer:

Data: 34, 37, 53, 55, x, x + 2, 77, 83, 89 and 100.

Median: 63

Formula to find the median:

$\tt When\ n\ (the\ number\ of\ observations)\ is\ an\ odd\ number,\\ \\\\\implies\ \bigg(\dfrac{n\ +\ 1}{2}\bigg)th\ term.\\\\\\When\ n\ is\ an\ even\ number,\\\\\\\implies\ \dfrac{\bigg[\bigg(\dfrac{n}{2}\bigg)\ +\ \bigg(\dfrac{n}{2}\ +\ 1\bigg)\bigg]}{2}\ th\ term.$

Since the number of observations (n) is an even number (10), the second formula will be used.

Substituting the respective values in the formula,

$\tt 63\ =\ \dfrac{\bigg[\bigg(\dfrac{10}{2}\bigg)\ +\ \bigg(\dfrac{10}{2}\ +\ 1\bigg)\bigg]}{2}\ th\ term\\ \\\\63\ =\ \dfrac{\bigg[5th\ +\ 6th\bigg]}{2}\ term\\\\\\63\ =\ \dfrac{x\ +\ x\ +\ 2}{2}\\\\\\63\ =\ \dfrac{\bigg[2x\ +\ 2\bigg]}{2}\\\\\\63\ =\ x\ +\ 1\\\\\\63\ -\ 1\ =\ x\\\\\\62\ =\ x$

Therefore, x = 62.