Problem 8 - Algebra 2

Question

Find the values at the 30 th and 90 th percentiles for each set of values. \(\begin{array}{llll}{6283} & {5700} & {6381} & {6274} & {6075} & {5993} & {5581}\end{array}\)

Step-by-Step Solution

Answer

The 30th percentile is 6278.5, and the 90th percentile is 6381.

Step 1: Arranging the data set

The first step is to arrange the data set in ascending order to get a clear understanding of the distribution of the values. The arranged data set is: 5581, 5700, 5993, 6075, 6274, 6283, 6381.

Step 2: Calculating the 30th percentile

To calculate the position for the 30th percentile, we use the formula: \( P = n * ((X/100) + 0.5) \) where n is the number of observations and X is the percentile to be calculated, in this case, 30. Hence, \( P = 7 * (30/100 + 0.5 ) = 5.1 \). Since the position for the percentile is between the 5th and 6th observations, we take the average of the 5th and 6th value by adding them together and dividing by 2 to get the 30th percentile. Hence, 30th percentile = (6274 + 6283) / 2 = 6278.5

Step 3: Calculating the 90th percentile

Similarly, to calculate the position for the 90th percentile, we use the same formula. Hence, \( P = 7 * (90/100 +0.5 ) = 9.8 \). Since there are only 7 observations, we take the highest observation, which is 6381 as the 90th percentile.

Key Concepts

Percentile Calculation

Percentile calculation is a method in statistics used to understand the relative standing of a particular value within a data set. It is useful for finding out where a specific point falls in an ordered data set. When calculating percentiles, we determine which value a certain percentage of the data falls below. For instance, if a score is at the 70th percentile, it means 70% of the data falls below that score.

To calculate the percentile, we usually apply a formula that combines the desired percentile with the total number of data points. The general formula is:

• P = n * ((X/100) + 0.5)

where:

• \( P \) is the position of the percentile,
• \( n \) is the total number of data points, and
• \( X \) is the desired percentile.

Once the position \( P \) is found, you may need to interpolate between two data points if \( P \) is not a whole number.

Ordered Data Set

Before performing any precise percentile calculation, it is crucial to order the data set. An ordered data set makes it easier to locate values and understand the distribution of data. It involves arranging the data points in a sequence, usually from smallest to largest.

Ordering a data set helps:

• Identify trends or patterns more effectively.
• Accurately determine positions for statistical calculations, like percentiles.

In the given exercise, the original data set was:
\( 6283, 5700, 6381, 6274, 6075, 5993, 5581 \).

Once ordered, the data became:
\( 5581, 5700, 5993, 6075, 6274, 6283, 6381 \).
This ordering was essential for calculating and locating the specific percentiles.

Statistical Distribution

A statistical distribution describes how scores or values in a data set are spread out. Understanding distributions helps in making sense of percentiles, as they show us where most data tends to cluster and how it spreads out from there. A symmetrical distribution has data points equidistant from the center, while skewed distributions show data that tail off towards one end.

Some common types of distribution are:

• Normal distribution, which is bell-shaped and symmetrical.
• Skewed distribution, where data may pile up on one side.

Percentiles give a snapshot of key points in this distribution, offering insight about thresholds or common data clustering. For instance, knowing the 30th and 90th percentiles in a data set helps gauge how it distributes around these points, which is useful for statistical analysis or decision-making.

Algebra 2 Concepts

When working with percentile calculations and distributions, many Algebra 2 concepts come into play. Algebra helps in understanding the relationships and operations necessary for calculating percentiles. It provides the foundational skills needed to manipulate formulas and solve equations, such as:

• Working with fractions and percentages to convert them into usable quantities.
• Using interpolation methods when a precise percentile location isn't a whole number.
• Addressing problems with inequalities and functions, which often appear in statistical analysis.

Algebra also supports the systematic arrangement and simplification of mathematical tasks, making it easier to work through step-by-step procedures in statistics. Understanding these algebraic tools refines the calculation processes, offering pathways to model and analyze real-world data accurately.