Problem 13 - Algebra 2 and Trigonometry

Question

In \(9-14,\) find the median and the first and third quartiles for each set of data values. \(75,72,69,68,66,65,64,63,63,61,60,59,59,58,56,54,52,50\)

Step-by-Step Solution

Answer

Median = 62, Q1 = 58, Q3 = 66.

Step 1: Organize the Data

First, ensure that the data is in ascending order. Since the given data is already in ascending order: \[ 50, 52, 54, 56, 58, 59, 59, 60, 61, 63, 63, 64, 65, 66, 68, 69, 72, 75 \]

Step 2: Determine the Median

The median is the middle value of an ordered data set. Since there are 18 data points, the median is the average of the 9th and 10th values. Find the 9th value: 61 Find the 10th value: 63 Calculate the median: \[ \text{Median} = \frac{61 + 63}{2} = 62 \]

Step 3: Find the First Quartile (Q1)

The first quartile is the median of the first half of the data. The first half consists of the first 9 data points: \[ 50, 52, 54, 56, 58, 59, 59, 60, 61 \] The median of this half is the 5th value: \[ Q1 = 58 \]

Step 4: Find the Third Quartile (Q3)

The third quartile is the median of the second half of the data. The second half consists of the last 9 data points: \[ 63, 63, 64, 65, 66, 68, 69, 72, 75 \] The median of this half is the 5th value: \[ Q3 = 66 \]

Key Concepts

Understanding the Median

The median is a key concept in statistics that helps us understand the center of a data set. Unlike the mean, which adds up all the values and divides them by the number of values, the median is the exact middle value once the data is sorted. To find the median:

• Arrange your data in ascending order.
• If the number of data points is odd, the median is the middle number.
• If the number of data points is even, as in our data set with 18 numbers, the median is the average of the two middle numbers. In this case, the 9th and 10th values are 61 and 63. So, we calculate the mean to find the median: \[ \text{Median} = \frac{61 + 63}{2} = 62 \]

By using the median, you get a measure of central tendency that is resistant to outliers or skewed data. This makes it a robust choice for understanding the central point of your data set.

Exploring Quartiles

Quartiles further divide your data set into four equal parts, giving you a detailed view of its distribution. They are crucial for understanding the spread and any skewness in your data.Let's break them down:

• First Quartile (Q1) - This is the median of the first half of the data. It marks the 25th percentile, meaning 25% of the data is at or below this point. For the data set given, \[ Q1 = 58 \]
• Third Quartile (Q3) - This represents the median of the second half of the data and marks the 75th percentile. For our data, \[ Q3 = 66 \]

Knowing the quartiles allows you to gauge how data values are distributed around the central median. It can also help in identifying outliers and measuring the spread of the data.

Importance of Data Organization

Organizing data is a fundamental step in statistics that simplifies the process of analyzing data sets. Before any calculations can be meaningful, the data must be arranged in a logical order. Here’s why this is crucial:

• It allows you to easily find the median and quartiles, as demonstrated in the example.
• An organized data set reveals patterns, trends, and anomalies, making it easier for further analysis like spotting outliers or calculating measures of spread such as variance or standard deviation.
• Having data in ascending order is essential for visual representations like box plots or stem-and-leaf plots.

By organizing data efficiently, we set the stage for deeper insights and a clearer understanding of the statistical properties of the data. Proper data organization is therefore not just a preliminary step, but a foundational process for effective data analysis.