Problem 10 - Algebra 2 and Trigonometry

Question

In \(9-14,\) find the median and the first and third quartiles for each set of data values. \(34,35,35,36,38,40,42,43,43,43,44,46,48,50\)

Step-by-Step Solution

Answer

Median = 42.5, Q1 = 36, Q3 = 46.

Step 1: Arrange the Data

Ensure that the data set is in ascending order. The given data is: \[34, 35, 35, 36, 38, 40, 42, 43, 43, 43, 44, 46, 48, 50\]Since it is already sorted, we can move to the next step.

Step 2: Find the Median

The median is the middle number of the sorted data set. Since there are 14 numbers (an even set), the median is the average of the 7th and 8th numbers in the data set:\[\text{Median} = \frac{42 + 43}{2} = 42.5\]

Step 3: Identify the Lower Half

To find the quartiles, we first split the data into two halves. The lower half includes the first seven data points:\[34, 35, 35, 36, 38, 40, 42\]

Step 4: Find the First Quartile (Q1)

The first quartile is the median of the lower half. There are 7 numbers:\[ \text{Q1} = 36 \]since 36 is the 4th number in this subset.

Step 5: Identify the Upper Half

The upper half includes the last seven data points:\[43, 43, 43, 44, 46, 48, 50\]

Step 6: Find the Third Quartile (Q3)

The third quartile is the median of the upper half. There are 7 numbers:\[ \text{Q3} = 46 \]since 46 is the 4th number in this subset.

Key Concepts

Median

The median of a data set is an important statistic that represents the middle value when all data points are ordered from the smallest to largest. If the number of data points is odd, the median is the middle number. However, if the data set has an even number of points, such as in this exercise with 14 numbers, the median is calculated as the average of the two middle numbers.

In our example, the data set is:

• 34
• 35
• 35
• 36
• 38
• 40
• 42
• 43
• 43
• 43
• 44
• 46
• 48
• 50

To find the median, average the 7th and 8th values in the sorted list: that is, \ \( \text{Median} = \frac{42 + 43}{2} = 42.5 \ \). This means that half of the data points are below 42.5 and half are above.

Data Set

A data set is a collection of numbers or values that relate to a particular subject. In this example, our data set consists of 14 numbers. Presenting your data set in a clear and logical order is the first step towards any statistical analysis, like calculating the median or quartiles.

When working with data sets, remember to:

• Arrange the data in ascending order for easier calculation of medians and quartiles.
• Check for any outliers or anomalies, as they can affect statistical calculations.
• Understand what each number represents to provide context to your analysis results.

Working with data sets is a fundamental skill in statistics, and organizing your data properly can make solving problems more straightforward and error-free.

First Quartile (Q1)

The first quartile, or Q1, represents the value below which 25% of the data falls. To find Q1 in a sorted data set, you focus on the lower half of data points. This helps to understand the distribution of data in the lower segment.

In our example, the lower half of the data includes:

• 34
• 35
• 35
• 36
• 38
• 40
• 42

Since this half contains 7 numbers, Q1 is the 4th number, effectively dividing the lower segment into two equal parts. Therefore, we find: \ \( \text{Q1} = 36 \ \). This value, 36, shows that 25% of the data values are less than or equal to 36.

Third Quartile (Q3)

The third quartile, or Q3, marks the point below which 75% of the data falls, essentially demarcating the upper quarter of the data distribution. To determine Q3, we look exclusively at the upper half of the data set.

Considering our data set, the upper half is composed of the following numbers:

• 43
• 43
• 43
• 44
• 46
• 48
• 50

This subdivision, like the lower half, also consists of 7 numbers. Consequently, Q3 is positioned at the 4th value, which is 46. Thus, \ \( \text{Q3} = 46 \ \), indicating that 75% of the population in the data set falls below this value.