Problem 15 - Thinking Mathematically

Question

In Exercises 13-20, find the median for each group of data items. \(91,95,99,97,93,95\)

Step-by-Step Solution

Answer

The median of the group of data items \(91,95,99,97,93,95\) is 95.

Step 1: Order the Data

Organize the numbers from least to greatest. The ordered list of numbers is \(91, 93, 95, 95, 97, 99\).

Step 2: Find the Median

Since this set of numbers has an even amount (6), the median will be the average of the two middle numbers. The two middle numbers from the ordered list are 95 and 95.

Step 3: Calculate the Average

To calculate the average of two numbers, add them together and divide by 2. In this case, the calculation becomes: median = \(\frac{(95 + 95)}{2}\)

Step 4: Complete the Calculation

Upon computing the operation, the result is 95. Therefore, the median is 95.

Key Concepts

Data Organization

Data organization is a crucial first step when you're working with any set of numbers, including finding measurements of central tendency, like the median. To visualize the process, imagine you have a set of numbers: 91, 95, 99, 97, 93, 95. It's a bit like having a mixed jumble of socks; to make sense of what you have, it's best to sort them. Similarly, with numbers, you organize them from smallest to largest, giving you a neatly ordered list: 91, 93, 95, 95, 97, 99.

Organizing data helps to clearly see the relationships between data points, identify any outliers, and determine other statistical measures with ease. It's like tidying up your room to find that missing sock; once things are in order, it all makes more sense!

Average Calculation

Calculating an average is a basic arithmetic skill, yet it forms the backbone of more complex statistical analysis. When we have an even number of data points in a set and need to find the median, we don't have a single middle number to declare as our median. Instead, we find the average of the two middle numbers. To draw a parallel, think of it as finding the halfway point between two landmarks.

If our landmarks are two instances of the number 95, as in our example, finding the midway point is straightforward. You add the two numbers up—95 plus 95—which gives you 190, and then divide by the number of data points, which is 2 in this case, resulting in 95. Thus, your average, and in this instance also your median, is 95.

Descriptive Statistics

Descriptive statistics is like a storyteller that gives us a summary of the data we have. It includes organizing, summarizing, and presenting data in a way that's understandable. When we find the median, we're searching for the value that sits right in the middle of our data set, once it's organized. This is a very telling piece of the story since it represents the middle ground of our data.

In the given problem, the median is 95, telling us that half of our data points are below 95 and half are above. This can be particularly telling in larger data sets as an indicator of distribution - whether the data is skewed towards higher or lower values, or if it is symmetrically distributed. Understanding the median, along with other measures like mean and mode, paints a fuller picture of our data's story.