To estimate the median using interpolation from the grouped data, we need to follow these steps:

1. Find the median class: Since the total frequency (number of animals) is 60, the median will be in the class that contains the $\frac{60}{2} = 30$-th animal.

2. Calculate cumulative frequencies: We will sum the frequencies to determine where the 30th animal lies:

   • Cumulative frequency up to the first class: $5$
   • Cumulative frequency up to the second class: $5 + 8 = 13$
   • Cumulative frequency up to the third class: $13 + 19 = 32$

   So, the 30th animal lies in the third class interval $80 < x \leq 90$, since the cumulative frequency reaches 32 after this class.

3. Apply the interpolation formula: The formula for estimating the median in a grouped frequency table is:

   $\text{Median} = L + \left( \frac{\frac{n}{2} - F}{f} \right) \times h$

   Where:

   • $L = 80$ (the lower boundary of the median class),
   • $n = 60$ (the total frequency),
   • $F = 13$ (the cumulative frequency before the median class),
   • $f = 19$ (the frequency of the median class),
   • $h = 90 - 80 = 10$ (the width of the median class).

4. Substitute values into the formula:

   $\text{Median} = 80 + \left( \frac{30 - 13}{19} \right) \times 10$ $\text{Median} = 80 + \left( \frac{17}{19} \right) \times 10$ $\text{Median} = 80 + \left( 0.8947 \right) \times 10$ $\text{Median} = 80 + 8.947 = 88.9$

Thus, the estimated median is 88.9 cm, correct to 1 decimal place.

Would you like more details on any step? Here are 5 related questions to expand this topic:

1. How would the interpolation method change if the total frequency were an odd number?
2. Why do we use cumulative frequency to determine the median class?
3. How does the size of the class interval ($h$) affect the median estimate?
4. Can the median be outside the boundaries of the median class? Why or why not?
5. What is the difference between estimating the median from grouped data versus raw data?

Tip: Always ensure the cumulative frequency is correctly calculated to avoid errors in identifying the median class.