Question

The median of the data \(5,6,7,8,9,10\) is _________ (1) 7 (2) 8 (3) \(7.5\) (4) \(8.5\)

Short answer

Answer: (3) 7.5

Step-by-step solution

Arrange the data in ascending order

The data is already arranged in ascending order: \(5,6,7,8,9,10\).

Determine if the dataset has an odd or even number of values

Our dataset has 6 values, which is an even number.

Find the middle two values

Since our dataset has an even number of values, we need to find the middle two values. They are \(7\) and \(8\).

Calculate the average of the middle two values

The median is the average of the middle two values, which is: \(\frac{7 + 8}{2} = \frac{15}{2} = 7.5\).

Select the correct answer

The median is \(7.5\), so the correct answer is (3) \(7.5\).

Key concepts

Data Arrangement

Understanding how to correctly arrange data is foundational to finding the median. The median is a measure of central tendency that indicates the middle value of a dataset when it is organized in a certain order. Traditionally, we arrange the dataset in ascending (least to greatest) or descending (greatest to least) order. This step is crucial because it sets the stage for locating the median. For example, if we have the values 5, 6, 7, 8, 9, and 10, we must first confirm they are in ascending order. Indeed, they are organized from smallest to largest: 5, 6, 7, 8, 9, 10. Only after this can we accurately proceed to identify the median.

It's important to be meticulous during this initial step to ensure data isn't out of order, as even a single misplacement can lead to an incorrect median and skew the measure of central tendency. For students, when double-checking your work, make sure to look over the data arrangement again to avoid any oversight.

Dataset Odd or Even Number

Identifying whether a dataset has an odd or even number of values is the next critical step in finding the median. This determination affects how the median is calculated:

• If the dataset contains an odd number of values, the median will be the middle number after arranging the data correctly.
• If the dataset contains an even number of values, the median will be computed as the average of the two middle numbers.

For the example dataset (5, 6, 7, 8, 9, 10), we count six numbers in total which is an even number. Thus, we know to look for the middle two values upon which the median will be based. When assessing datasets with a large number of values, be careful to count accurately — a single miscount can lead to misidentifying the dataset as odd or even and significantly alter the median calculation. A helpful tip for students is to use tally marks when counting numbers to ensure accuracy in this step.

Average Calculation

The final step in finding the median for an even-numbered dataset is the average calculation of the middle two values. To find this average, add the two middle numbers together and then divide the sum by two. This process provides the central value, or median, of the entire dataset. Taking our example of the dataset 5, 6, 7, 8, 9, and 10, the middle two numbers are 7 and 8. Adding these together gives us 15, and dividing by 2 yields 7.5. Thus, the median is 7.5, which represents the halfway mark in our ordered set of numbers.

Key Point in Average Calculation:

In contexts where the precision of numeric results is important, such as statistics or science-related exercises, consider how to deal with decimal points or fractions. Always follow the instruction or the standard practice in your field of study. When working on homework problems or preparing for exams, also be sure to understand whether rounding is appropriate, and if so, to what extent.