Problem 11 - Algebra 2 and Trigonometry

Question

In \(9-14,\) find the median and the first and third quartiles for each set of data values. \(23,27,15,38,12,17,22,39,28,20,27,18,25,28,30,29\)

Step-by-Step Solution

Answer

Median is 26, Q1 is 19, and Q3 is 28.5.

Step 1: Organize the Data

First, sort the data values in ascending order. The data set is: 12, 15, 17, 18, 20, 22, 23, 25, 27, 27, 28, 28, 29, 30, 38, 39.

Step 2: Find the Median

The median is the middle value of the data set. Since there are 16 data points (an even number), the median is the average of the 8th and 9th values: These values are 25 and 27.Therefore, the median is: \[\text{Median} = \frac{25 + 27}{2} = 26\]

Step 3: Determine the First Quartile (Q1)

To find the first quartile, calculate the median of the lower half of the data (the first 8 values): 12, 15, 17, 18, 20, 22, 23, 25.The first quartile is the average of the 4th and 5th values, which are 18 and 20:\[Q1 = \frac{18 + 20}{2} = 19\]

Step 4: Determine the Third Quartile (Q3)

To find the third quartile, calculate the median of the upper half of the data (the last 8 values): 27, 27, 28, 28, 29, 30, 38, 39.The third quartile is the average of the 4th and 5th values, which are 28 and 29:\[Q3 = \frac{28 + 29}{2} = 28.5\]

Key Concepts

Understanding the Median

The median is a central property in statistics that helps us understand the middle value of a data set. To find the median, you need to organize your data in ascending order. This makes it easy to find the middle point.
In a data set with an odd number of values, the median is simply the middle value.
However, in a data set with an even number of values, the median is calculated by taking the average of the two middle numbers.
In our given exercise, we have 16 data points. After sorting, the 8th and 9th numbers are 25 and 27.
Thus, the median is calculated as \[ \frac{25 + 27}{2} = 26 \]
This median value tells us that half of our data points are less than 26, and the other half are more than or equal to 26.

Data Organization for Quartile Calculation

Arranging data systematically is crucial for finding quartiles. Quartiles divide a data set into four equal parts, and proper arrangement makes it easier to locate these boundaries.
Before calculating quartiles, always sort your data in ascending order. For our given example, the data was sorted from smallest to largest as:

• 12, 15, 17, 18, 20, 22, 23, 25
• 27, 27, 28, 28, 29, 30, 38, 39

Data organization helps not just in finding quartiles, but also enhances overall data analysis. Once organized, you can easily locate the median, first quartile (Q1), and third quartile (Q3), as these rely on a well-ordered data set.

Exploring the Interquartile Range

The Interquartile Range (IQR) measures the spread of the middle 50% of your data. It's a valuable tool for identifying the variability within a data set.
The IQR is calculated as the difference between the third quartile (Q3) and the first quartile (Q1). In our exercise case, we've found:

• First Quartile (Q1) to be 19
• Third Quartile (Q3) to be 28.5

Thus, the IQR is computed as: \[ IQR = 28.5 - 19 = 9.5 \]
This range highlights where the central majority of the data resides, telling us how data varies around the median, and is useful in spotting outliers.

Demystifying Data Analysis with Quartiles

Data analysis aims to draw insights and make sense of numerical information. Quartiles aid in breaking down data to understand its distribution better.
By using the median and quartiles, we get a clearer picture of how data points are spread. Here's what each tells us:

• The median defines a central point dividing data into two equal parts.
• Quartiles divide the data further, giving us values marking the 25% (Q1) and 75% (Q3) thresholds.
• The Interquartile Range (IQR) shows how the bulk of the data varies, free from the extreme values.

These elements are powerful in spotting trends and patterns, recognizing outliers, and ensuring a comprehensive understanding of the data's structure and spread. Data analysis using quartiles makes interpreting large data sets manageable and meaningful.