Consider the following for the four (04) items that follow: The...

The cumulative frequencies for the given classes are:

$160-162$: $12$
$162-164$: $12 + 15 = 27$
$164-166$: $27 + 24 = 51$
$166-168$: $51 + 13 = 64$

Total number of students, $N = 64$
$\dfrac{N}{2} = \dfrac{64}{2} = 32$

The cumulative frequency just greater than $32$ is $51$, which corresponds to the class $164-166$. Thus, the median class is $164-166$.

Lower limit of the median class, $L = 164$
Cumulative frequency of the class preceding the median class, $cf = 27$
Frequency of the median class, $f = 24$
Class width, $h = 2$

Using the formula for median:
$\text{Median} = L + \left( \dfrac{\dfrac{N}{2} - cf}{f} \right) \times h$

$\text{Median} = 164 + \left( \dfrac{32 - 27}{24} \right) \times 2$

$\text{Median} = 164 + \left( \dfrac{5}{24} \right) \times 2$

$\text{Median} = 164 + \dfrac{5}{12}$

$\text{Median} \approx 164 + 0.416 = 164.41 \text{ cm}$

Answer: $164.41$ cm