Problem 20 - Thinking Mathematically

Question

In Exercises 13-20, find the median for each group of data items. \(1.4,2.1,1.6,3.0,1.4,2.2,1.4,9.0,9.0,1.8\)

Step-by-Step Solution

Answer

The median of the given data set is 1.95.

Step 1: Arrange the data items in ascending order

The first step is to rearrange the given data items in increasing order: \(1.4, 1.4, 1.4, 1.6, 1.8, 2.1, 2.2, 3.0, 9.0, 9.0\)

Step 2: Find the middle data items

The data set has 10 items, an even number, implying that there are two middle numbers. In this case, the 5th and 6th data items (after arranging them in ascending order) are the middle numbers. These are \(1.8\) and \(2.1\).

Step 3: Calculate the Median

Since there are an even number of data items, the median is calculated as the average of the 5th and 6th data items. The formula for the average of two numbers is \((a + b) / 2\). Substituting \(1.8\) for \(a\) and \(2.1\) for \(b\), we get \((1.8 + 2.1) / 2 = 1.95\).

Key Concepts

Median Calculation

Understanding how to find the median of a set of numbers is a fundamental skill in basic statistics. The median represents the middle value of an ordered dataset, or the average of the two middle values if the dataset has an even number of observations.

In the exercise provided, we follow three key steps: First, we order the numbers from the least to the greatest. Here, the numbers are:
1.4, 1.4, 1.4, 1.6, 1.8, 2.1, 2.2, 3.0, 9.0, 9.0.

Next, because there are an even total number—ten—we identify the fifth and sixth numbers as the middle values. The median is then found by averaging these two middle numbers, in our case 1.8 and 2.1.

The calculation is straightforward:
\[(1.8 + 2.1) / 2 = 1.95\].

This result, 1.95, is the median of our dataset, which can be interpreted as the central tendency of the group of numbers.

Data Organization

When faced with a list of numbers, organizing the data is the first critical step before any form of analysis. Well-organized data can provide insights and help in identifying patterns, trends, or outliers—which could be hidden in a jumbled list.

In context to our median calculation exercise, organizing data implies arranging the numbers in ascending or descending order. This helps us quickly identify the dataset's range, quartiles, and median.

For example, when we order the given list of numbers:
1.4, 2.1, 1.6, 3.0, ...

We obtain a neatly organized sequence from lowest to highest. Such organization makes it easier to count the number of data points and accurately determine the central point for the median. It's also beneficial for spotting any anomalies, as was the case with two 9.0 values that stand out from the other single-digit figures in our list—an indicator that they could be outliers.

Descriptive Statistics

Descriptive statistics includes the summary and organization of data to describe the basic features of a dataset. It provides simple quantitative descriptions and can include measures of central tendency (like the mean, median, and mode) and measures of variability (like standard deviation and variance).

The median is a crucial measure within descriptive statistics because it is less affected by outliers and skewed data—this trait makes it a reliable representation of the 'middle' value in many real-world situations.

Standard techniques for descriptive analysis involve visual representation through graphs, charts, and tables, but also concise numerical summaries such as the median. By understanding and applying descriptive statistics correctly, we ensure that we can describe data patterns in a way that is both accurate and meaningful, thus enabling better decision-making based on the data under analysis.