The problem you've shared involves two questions. Let's break them down:

Question 1:

The data represents the height measurements of 100 mangrove trees, with:

• The lowest data point = 5 m.
• The range of data = 49.
• The data is grouped into 10 classes with respective frequencies $6, 31, h, 22, 11, 6, 5, 4, 1, 1$.
• Standard deviation $= 9.85$.

The task is to find:

• The missing value $h$.
• Median, mode, interquartile range (IQR = $Q3 - Q1$).
• Percentile range $P90 - P10$.

Approach to solve:

1. Determine the missing value $h$:

   • The total sum of frequencies must equal 100.
   • Using this fact, we can find $h$.

2. Median:

   • The median is the value at the 50th percentile. You will locate the class containing the 50th data point.

3. Mode:

   • The mode is the class with the highest frequency.

4. Interquartile Range (IQR = $Q3 - Q1$):

   • You need to find the values at the 25th percentile ($Q1$) and 75th percentile ($Q3$) and subtract them.

5. Percentile range $P90 - P10$:

   • This is the difference between the values at the 90th percentile ($P90$) and 10th percentile ($P10$).

Question 2:

You are asked to prove that the variance calculated using the given formula: $\sigma^2 = \frac{\sum f x_i^2 - \frac{(\sum f x_i)^2}{\sum f}}{\sum f}$ produces the same result.

This requires a theoretical demonstration of the equivalence of this variance formula with the standard formula for variance.

Let me know if you would like to proceed step by step or need more details on any specific part!