Problem 14 - Algebra 2 and Trigonometry

Question

In \(9-14,\) find the median and the first and third quartiles for each set of data values. \(32,32,30,30,29,27,26,22,20,20,19,18,17\)

Step-by-Step Solution

Answer

The median is 26, the first quartile is 19.5, and the third quartile is 29.5.

Step 1: Order the Data

First, arrange the data values in ascending order. The given data set is: \(32, 32, 30, 30, 29, 27, 26, 22, 20, 20, 19, 18, 17\). When sorted in ascending order, it becomes: \(17, 18, 19, 20, 20, 22, 26, 27, 29, 30, 30, 32, 32\).

Step 2: Find the Median

The median is the middle value in the ordered data set. Since there are 13 values in the set, the median is the 7th value. Thus, the median is: \(26\).

Step 3: Locate the Quartiles

To find the first and third quartiles, we need to divide the data into four equal parts. The first quartile (\(Q_1\)) is the median of the first half, and the third quartile (\(Q_3\)) is the median of the second half. The first half (excluding the overall median) is: \(17, 18, 19, 20, 20, 22\), and the second half is: \(27, 29, 30, 30, 32, 32\).

Step 4: Calculate First Quartile (Q1)

The first quartile is the median of the first half of the data set. Since there are 6 values in this segment, the first quartile is the average of the 3rd and 4th values: \((19 + 20)/2 = 19.5\).

Step 5: Calculate Third Quartile (Q3)

The third quartile is the median of the second half of the data set. Since there are 6 values in this part as well, the third quartile is the average of the 3rd and 4th values: \((29 + 30)/2 = 29.5\).

Key Concepts

Median

To understand what a median is, think of it as the number that sits right smack in the middle of a sorted list of numbers. It's like the middle seat in a row of chairs. That way, the median is perfect for showing what is typical in a data set without being affected by really high or low numbers.
Say you line up all your data points in order, from smallest to biggest. The median is the one in the middle. If you have an odd number of data points, like in our example where there are 13 numbers, the middle one is right there for you to pick: it's the seventh number. But if you have an even number of values, the median is the average of the two in the middle. Simple and straightforward!

Quartiles

Quartiles help us dig a bit deeper into understanding how data is spread out. Imagine dividing your sorted data into four equal parts. The spots where you cut it are called quartiles.

• The first quartile ( Q_1 ) is a quarter of the way through the list.
• The second quartile (which is just the median, by the way) is halfway through.
• The third quartile ( Q_3 ) is three-quarters of the way through.

In the dataset example, the first quartile comes from the "lower half" of the data, and you find Q_1 by looking at the middle of those numbers. Similarly, Q_3 shows where the upper quarter is placed. This breakdown works well when examining how data is spread out.

Data Analysis

Analyzing data involves digging into the numbers to find patterns and insights. It's asking, "What story do these numbers tell me?" The process typically includes steps like organizing, summarizing, and interpreting data. Using median and quartiles as tools allows you to see how data stacks up and identify trends or anomalies. For instance, is most of the data bunched up at one end? Are there outliers? With this organized breakdown, making sense of information and making informed decisions becomes much easier. Data analysis helps to turn raw numbers into useful, actionable insights by highlighting not just the typical values but also the range and spread of the dataset.

Ordered Data Set

An ordered data set is simply a list of numbers arranged from smallest to largest. This order is key when it comes to finding the median and quartiles. It sets the stage for analyzing other statistical elements as well. Start with your raw data. Put them in order; this makes seeing the spread and overall layout much simpler. Without ordering, finding the middle or quartiles could be like locating a needle in a haystack. Think of it as organizing your work desk; everything falls into place and becomes more manageable. Having your data in order makes calculating the median and quartiles smooth and ensures nothing is missed or misplaced. It's the foundation of any good data analysis.