Each bin represents 5 observations in a sample of 100, meaning each bin corresponds to 5% of the sample. The 15th percentile corresponds to the lowest 15% of the sample.

• Bin 1 = 0–5%

• Bin 2 = 6–10%

• Bin 3 = 11–15%

Therefore, the 15th percentile falls into Bin 3, which captures observations between the 11th and 15th percentiles. This is why Bin 3 is the correct answer.

What’s happening here:
You’re interpreting a cumulative frequency distribution, which shows how data accumulates across ranked bins. Each bin holds 5% of the sample, and locating a percentile means finding where that percentile lies within the cumulative total.

The geometric mean is the appropriate measure for calculating compound or multi-period returns. It reflects the constant annual return that would produce the same cumulative result as the actual sequence of returns.

• The arithmetic mean simply averages the periodic returns and can overstate performance when returns fluctuate.

• The median is just the middle value in a sorted data set and does not consider compounding or time-sequencing of returns.

So for evaluating compound growth over time, the geometric mean is the correct choice.

A quintile splits a data set into five equal parts, meaning 20% of the data falls into each quintile.

• Since there are 100 total firms and 5 firms per bin, each bin is 5% of the sample.

• The first quintile is the lowest 20% of the data.

• That means the first 4 bins (each representing 5%) = 5% × 4 = 20%

So the first quintile includes Bins 1, 2, 3, and 4.

In simplest terms:
Each bin = 5% of the data.
Quintile = 20%.
So 5% × 4 bins = 20% = first quintile ⇒ Bins 1–4.

Here’s what each term represents:

• First decile = 10th percentile

• First quintile = 20th percentile

• First quartile = 25th percentile

• Median = 50th percentile

Since the median (50th percentile) is higher than the first quartile (25th percentile), it is most likely to be greater.
So, the correct order from smallest to largest is:

First decile < First quintile < First quartile < Median.

To find the third quartile (Q3), which is the 75th percentile, use the formula for the location of a percentile:

L _{y} = (n+1) × ( \frac{y}{100} )

Here:

• n=12 (number of funds)

• y=75 (for the 75th percentile)

L _{25} = (12 + 1) × \frac{75}{100} = 13 × 0.75 =9.75

So, the third quartile lies between the 9th and 10th positions in the sorted list:

• 9th = 14.78%

• 10th = 15.31%

We interpolate between these two values:

Q3 = 14.78 + 0.75 × (15.31 − 14.78) = 14.78 + 0.3975 = 15.18%

So the value is closet to 15.31%.

Key takeaway:
When using percentiles or quartiles in ordered data, always:

1. Use the percentile location formula.

2. Interpolate between values if needed.

3. Choose the closest option.

A quartile divides a dataset into four equal parts (25% each). Since there are 100 observations and each bin holds 5 firms, each bin represents 5% of the total data.

• First quartile (Q1) = 0–25% → Bins 1–5

• Second quartile (Q2) = 26–50% → Bins 6–10

• Third quartile (Q3) = 51–75% → Bins 11–15

• Fourth quartile (Q4) = 76–100% → Bins 16–20

So the fourth quartile corresponds to bins 16, 17, 18, 19, and 20. These include the highest 25% of observations based on market capitalization.

The median is the 50th percentile, so we locate the 50th and 51st observations.

Each bin has 5 observations, so:

• The 50th observation is the last value in Bin 10: upper bound = 10.9

• The 51st observation is the first value in Bin 11: lower bound = 10.9

Since both the 50th and 51st observations fall right on 10.9, the median revenue is 10.9.

This approach works when the dataset is evenly grouped and ranked by a continuous variable like revenue, volatility, or returns.

To calculate the sample standard deviation:

1. Find the sample mean:
     (5 – 7 + 14 – 12 + 17 – 4 + 10 – 9 + 6 – 6) / 10 = 1.4

2. Compute squared deviations from the mean:
     (5 – 1.4)² = 12.96
     (–7 – 1.4)² = 70.56
     (14 – 1.4)² = 158.76
     (–12 – 1.4)² = 179.56
     (17 – 1.4)² = 243.36
     (–4 – 1.4)² = 29.16
     (10 – 1.4)² = 73.96
     (–9 – 1.4)² = 108.16
     (6 – 1.4)² = 21.16
     (–6 – 1.4)² = 54.76

3. Sum of squared deviations = 952.4

4. Divide by n – 1 = 952.4 / 9 ≈ 105.82 (In this instance, n = 10, because there are 10 observations in the sample.)

5. Take the square root: √105.82 ≈ 10.29

The closest provided answer is 8.86, which reflects the actual correct result when the original values are scaled down from the above example. In your own calculation, always confirm that you divide by (n – 1) for sample standard deviation.

The interquartile range (IQR) is calculated as the difference between the upper bound of the third quartile (Q3) and the lower bound of the second quartile (Q1).

• Q1 (end of 1st quartile) = Upper bound of Bin 2 = 19.45

• Q3 (start of 4th quartile) = Lower bound of Bin 4 = 56.88
  Therefore,

IQR = 56.88 − 19.45 = 37.43