Question

For the data in Exercises \(4-6,\) calculate the median and the upper and lower quartiles. \(n=6\) measurements: 1,7,4,5,2,9

Short answer

Question: Determine the median, lower quartile (Q1), and upper quartile (Q3) for the given dataset: 1, 7, 4, 5, 2, 9. Answer: The median (Q2) of the dataset is 4.5, the lower quartile (Q1) is 1.5, and the upper quartile (Q3) is 8.

Step-by-step solution

Arrange the data in ascending order

First, we need to arrange the data in ascending order. It will help us find the positions of quartiles easily. The sorted dataset will be: 1, 2, 4, 5, 7, 9.

Find the position of the median (Q2)

Since there are six measurements, the position of Q2 (the median) will be halfway between the third and fourth values. We need to take the average of these values to get the median.

Calculate the median (Q2)

The values at the third and fourth positions are 4 and 5. As the median is the average of these values, we get the median as: $$ Q2 = \frac{4 + 5}{2} = \frac{9}{2} = 4.5 $$

Determine the positions of the lower quartile (Q1) and upper quartile (Q3)

To find the positions of Q1 and Q3, we need to divide the dataset into four equal parts. For Q1 (25% of the data), the position will be halfway between the first and second values. Similarly, for Q3 (75% of the data), the position will be halfway between the fifth and sixth values.

Calculate the lower quartile (Q1) and upper quartile (Q3)

For Q1, we need to take the average of the first and second values (1 and 2): $$ Q1 = \frac{1 + 2}{2} = \frac{3}{2} = 1.5 $$ For Q3, we need to take the average of the fifth and sixth values (7 and 9): $$ Q3 = \frac{7 + 9}{2} = \frac{16}{2} = 8 $$ The median (Q2) of the dataset is 4.5, the lower quartile (Q1) is 1.5, and the upper quartile (Q3) is 8.

Key concepts

Median

The median is a valuable measure of central tendency in statistics. It represents the middle value in a data set, effectively dividing it into two equal halves. To find the median, you must first organize your data in ascending order. In this context, the median helps provide a balanced understanding of a dataset's center. For the given example, with six measurements, once organized, the data is: 1, 2, 4, 5, 7, 9.
Since there is an even number of elements, the median is the average of the third and fourth numbers. Thus, the median calculation is as follows:

• Locate the middle positions: third and fourth values (4 and 5)
• Calculate their average: \[Q2 = \frac{4+5}{2} = 4.5\]

This gives us a median value of 4.5 for the data set.

Data Arrangement

Arranging data correctly is crucial for calculating statistics like the median and quartiles. This process involves organizing a dataset in ascending or descending order. A structured arrangement simplifies the identification of specific points in the data, such as quartiles.
For the provided data set of measurements: 1, 7, 4, 5, 2, 9, arranging it properly gives us: 1, 2, 4, 5, 7, 9. This array helps you see at a glance where the divisions for median and quartiles occur.
Well-organized data makes further calculations effortless and more precise, reducing the risk of errors when performing statistical analyses.

Quartile Calculation

Quartiles are essential metrics that divide your data set into four equal parts. These include the lower quartile (Q1), median quartile (Q2), and upper quartile (Q3). They offer insights into data spread and variability.

• **Lower Quartile (Q1):** Represents the value separating the lowest 25% from the remaining data. For six sorted measurements (1, 2, 4, 5, 7, 9), Q1 lies between the first and second values. \[Q1 = \frac{1+2}{2} = 1.5\]
• **Upper Quartile (Q3):** This quartile cuts off the highest 25% of data. It is the average of the fifth and sixth values: \[Q3 = \frac{7+9}{2} = 8\]

These quartiles help to identify the spread of data and understand outliers in your dataset.

Descriptive Statistics

Descriptive statistics are statistics that summarize or describe a data set's features. They offer insights into the data's center, spread, and other aspects without drawing conclusions beyond the data given.
Measures like the median and quartiles are key components. They articulate data characteristics in a digestible manner, highlighting central tendencies and variability.
Beyond median and quartiles, descriptive statistics can include mean, mode, range, variance, and standard deviation. These tools collectively provide a comprehensive picture of the data's behavior and composition, enabling informed decision-making in various real-world contexts.