### Theory:

Problem $$1$$
Find the median of the following numbers.

$$45$$, $$57$$, $$87$$, $$23$$, $$35$$, $$46$$, $$94$$

Let us now find the solution.

Step $$1$$: Arrange the numbers either in ascending order or descending order.

Here we have arranged the numbers in ascending order.

$$23$$, $$35$$, $$45$$, $$46$$, $$57$$, $$87$$, $$94$$

Step $$2$$: Find the total number of terms.

The total number of terms $$= 7$$

Step $$3$$: Find the median.

Since the total number of terms is odd:

$$\text{Median} =$$ ${\left(\frac{N\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}1}{2}\right)}^{\mathit{th}}$ $$\text{term}$$

$$\text{Median} = \Big( \frac{7+ 1}{2} \Big)^{th}$$ $$\text {term}$$

$$\text{Median} = 4^{th}$$ $$term$$

$$\text{Median} = 46$$

Problem $$2$$
Find the median of the following numbers.

$$54$$, $$25$$, $$75$$, $$65$$, $$97$$, $$31$$, $$55$$, $$84$$

Let us now find the solution.

Step $$1$$: Arrange the numbers in ascending order or descending order.

$$25$$, $$31$$, $$54$$, $$55$$, $$65$$, $$75$$, $$84$$, $$97$$

Step $$2$$: Find the total number of terms.

The total number of terms $$= 8$$

Step $$3$$: Find the median.

Since the total number of terms is even:

$$\text{Median} =$$ $$\frac{1}{2}$$$$\Bigg[$$${\left(\frac{N}{2}\right)}^{\mathit{th}}$$$\text{term}$$$$+$$$\phantom{\rule{0.147em}{0ex}}{\left(\frac{N}{2}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}1\right)}^{\mathit{th}}$$$\text{term}$$$$\Bigg]$$

$$\text{Median}$$ $$=$$ $$\frac{1}{2}$$ $$\Bigg[$$ $$\Big(\frac{8}{2}\Big)^{th}$$ $$\text{term}$$ $$+$$ $$\Big(\frac{8}{2} + 1 \Big)^{th}$$ $$\text {term})$$ $$\Bigg]$$

$$\text{Median}$$ $$=$$ $$\frac{1}{2}$$ $$[4^{th} \text{term} + 5^{th} \text{term}]$$

$$\text{Median} = \frac{55 + 65}{2}$$

$$\text{Median} = \frac{120}{2}$$

$$\text{Median} = 60$$