To find the Upper $P_{80}$ (80th percentile) of the sorted list of ages, follow these steps:

Step 1: Compute the position in the sorted list for the 80th percentile

The formula for the position of the $P_{k}$ percentile in a data set of size $n$ is:

$P_k = \frac{k}{100} \times (n + 1)$

where:

• $k$ is the percentile you are looking for (in this case, 80),
• $n$ is the number of data points (36 ages in this case).

$P_{80} = \frac{80}{100} \times (36 + 1) = 0.80 \times 37 = 29.6$

Step 2: Interpret the position

• Since the 29.6th position is between the 29th and 30th data points, we will interpolate between these two points to find the value.

Step 3: Locate the 29th and 30th data points

From the sorted list, the 29th and 30th data points are:

• 29th data point: 59
• 30th data point: 61

Step 4: Interpolation

To interpolate between these two points:

$P_{80} = 59 + 0.6 \times (61 - 59)$

$P_{80} = 59 + 0.6 \times 2 = 59 + 1.2 = 60.2$

Final Answer:

The 80th percentile, $P_{80}$, is approximately 60.2.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How would the result change if the data set had more ages?
2. What is the difference between the median and the 80th percentile in this set?
3. How do we handle percentiles when the position is an integer?
4. What would happen if you had to calculate the 90th percentile instead?
5. Can percentiles help in identifying outliers in a data set?

Tip: When calculating percentiles, always ensure that the data is sorted before applying the formula.