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Chapter 5: Problem 375

Find the median. $$4 , 8 , 1 , 5 , 14 , 3 , 1 , 12$$

Short Answer

Expert verified
The median is 4.5.

Step by step solution

01

- Arrange the Numbers in Ascending Order

First, arrange the given numbers in ascending order: $$1, 1, 3, 4, 5, 8, 12, 14$$
02

- Identify the Middle Numbers

Since there are 8 numbers, the median will be the average of the two middle numbers in the sorted list. Identify the 4th and 5th numbers: $$4$$ and $$5$$.
03

- Calculate the Median

Find the average of the two middle numbers: \[ \text{Median} = \frac{4 + 5}{2} = \frac{9}{2} = 4.5 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Arranging Numbers
To find the median of a set of numbers, the first step is to arrange the numbers in ascending order. This means listing the numbers from the smallest to the largest. Arranging numbers helps us easily find the middle value(s). For the example given, the numbers are: 4, 8, 1, 5, 14, 3, 1, 12. When we arrange these numbers in ascending order, we get: 1, 1, 3, 4, 5, 8, 12, 14. Sorting numbers is a fundamental step in many mathematical calculations, ensuring accuracy in finding measures like the median, which we will discuss next.
Middle Numbers
Once the numbers are arranged in ascending order, the next step is to identify the middle number(s). For datasets with an odd number of values, the median is the middle number. For even sets, like our example, the median is the average of the two middle numbers. With 8 numbers in our sorted list: 1, 1, 3, 4, 5, 8, 12, 14, the middle numbers are the 4th and 5th numbers, which are 4 and 5. This step is crucial because the median represents the central point of a dataset, separating the higher half from the lower half.
Average Calculation
To find the median in an even set of numbers, we calculate the average of the two middle numbers. Using our example, the two middle numbers are 4 and 5. The average is found by adding these two numbers together and then dividing by 2: \[ \text{Median} = \frac{4 + 5}{2} = \frac{9}{2} = 4.5 \] By calculating the average, we determine the median value, which in this case is 4.5. This value effectively represents the center of the data set. Average calculation is a principle used not only in finding medians but also in determining other statistical measures like the mean.

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