Problem 9 - Algebra 2

Question

Find the values at the 30 th and 90 th percentiles for each set of values. \(\begin{array}{lllllllll}{7} & {12} & {3} & {14} & {17} & {20} & {5} & {3} & {17} & {4} & {13} & {2} & {15} & {9} & {15} & {18} & {16} & {9} & {1} & {6}\end{array}\)

Step-by-Step Solution

Answer

The 30th percentile of the given set of values is 6 and the 90th percentile is 17.

Step 1: Arrange the Values

Firstly, arrange the given numbers in ascending order: \(1, 2, 3, 3, 4, 5, 6, 7, 9, 9, 12, 13, 14, 15, 15, 16, 17, 17, 18, 20 \).

Step 2: Find the Index of the Percentiles

The number of values is 20. To find the 30th percentile index, multiply 30% with 20, resulting in 6 (it means the 6th value in the ordered list). To find the 90th percentile index, multiply 90% with 20, resulting in 18.

Step 3: Determine the Values of the Percentiles

The 6th value in the ordered list is 6, hence the 30th percentile is 6. The 18th value in the ordered list is 17, hence the 90th percentile is 17.

Key Concepts

Working with Ordered Data

When dealing with the concept of percentiles, it's crucial to first understand ordered data. Simply put, ordered data is a list of values arranged from the smallest to the largest. This organization is essential because percentile calculations rely on understanding the position of a value in a list. For example, given a chaotic list of numbers, like \( \{7, 12, 3, 14, 17, 20, 5, 3, 17, 4, 13, 2, 15, 9, 15, 18, 16, 9, 1, 6\} \), the first step is to arrange these numbers in ascending order: \(1, 2, 3, 3, 4, 5, 6, 7, 9, 9, 12, 13, 14, 15, 15, 16, 17, 17, 18, 20\). This ordered list helps in easily identifying the position of each value, and thus aids in other calculations. Remember, without ordering, it would be impossible to accurately determine what value corresponds to a particular percentile. The process of ordering, while seemingly simple, is foundational to properly understanding percentile calculations. It ensures we are working from a clear and logical base.

Understanding the Percentile Index

The percentile index is a critical concept in calculating percentiles. It determines the position of the percentile in the ordered list of data. We derive it by multiplying the desired percentile by the total number of values in the dataset. For example, to find the 30th percentile index for a list containing 20 values, we calculate:- \(30\% \times 20 = 0.30 \times 20 = 6\) This calculation tells us that the 30th percentile corresponds to the 6th value in the ordered list.Similarly, for the 90th percentile:- \(90\% \times 20 = 0.90 \times 20 = 18\) Here, the 90th percentile corresponds to the 18th value in the ordered list. These indices help us pinpoint the exact position in the dataset that corresponds to a specific percentile. When counting, always ensure you're starting from the smallest value, as ordering plays a crucial role in determining accuracy. These indices simplify finding percentiles by directly relating a percentage to a specific position in the data array.

Finding Percentile Values

Once we have determined the percentile indices, the next step is to identify the corresponding percentile values in the ordered data. This simply means picking out the values from the sorted list that match the indices we calculated earlier. Using our example: - **30th Percentile:** From our ordered list, the 6th value is 6. Thus, the 30th percentile value is 6. - **90th Percentile:** The 18th value in our list is 17, making the 90th percentile value 17. Identifying these values allows us to understand the distribution of our data set better.

• If you randomly pick a value from the list, there is a 30% chance it will be less than or equal to the 30th percentile value, which is 6.
• Similarly, 90% of the values are less than or equal to the 90th percentile value, 17.

Grasping the concept of percentile values helps in making informed decisions based on data, whether in academics, business, or everyday applications.