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Chapter 5: Problem 373

Find the median. $$65 , 56 , 35 , 34 , 44 , 39 , 55 , 52 , 45$$

Short Answer

Expert verified
The median is 45.

Step by step solution

01

- Arrange the Numbers

Start by arranging the numbers in ascending order. The given numbers are: 65, 56, 35, 34, 44, 39, 55, 52, 45. When arranged in ascending order, they become: 34, 35, 39, 44, 45, 52, 55, 56, 65.
02

- Identify the Position of the Median

Since there are 9 numbers in the list, the median will be the number found in the \(\frac{9+1}{2}=5\)-th position when the numbers are arranged in ascending order.
03

- Determine the Median

Count to the 5th position in the arranged list. The 5th number is 45. Therefore, the median is 45.

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Key Concepts

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

Arranging Numbers in Ascending Order
To find the median of a set of numbers, the first step is to arrange them in ascending order. Ascending order means organizing the numbers from the smallest to the largest. This step is essential because the median is the middle number in an ordered set. If the numbers are not ordered, it鈥檚 impossible to identify the correct median. Let鈥檚 take an example:
Given the numbers: 65, 56, 35, 34, 44, 39, 55, 52, 45.
We arrange them in ascending order, which gives us: 34, 35, 39, 44, 45, 52, 55, 56, 65.
Now the numbers are sorted from the smallest to the largest, making it easier to find the median.
Calculating Median
The median is the middle number in a data set. If we have an odd number of observations, the median is the number exactly in the middle. If the number of observations is even, the median is calculated by taking the average of the two middle numbers.
Using our example from the previous section where we have arranged the numbers: 34, 35, 39, 44, 45, 52, 55, 56, 65, we see that we have 9 numbers in total.
For an odd set of numbers, we use the formula: \(\frac{n+1}{2}\), where n is the number of observations. Here n = 9, so: \(\frac{9+1}{2}=5\).
The median is the 5th number in the ordered list, which is 45. Therefore, the median of the set is 45.
Position of Median in a Data Set
Understanding the position of the median is crucial to finding the correct median value. For any ordered data set, the position of the median depends on whether you have an odd or even number of observations.
  • **Odd Number of Observations**: Use the formula: \(\frac{n+1}{2}\). This formula will give you the position of the median directly.
  • **Even Number of Observations**: Find the average of the numbers at positions \(\frac{n}{2}\) and \(\left(\frac{n}{2}\right)+1\). The median is the average of these two middle numbers.
Let's apply this to our example:
We have 9 numbers (which is odd), so the position of the median is the 5th number. Counting to the 5th position in our sorted list (34, 35, 39, 44, 45, 52, 55, 56, 65), we see the 5th number is 45.
If there were 10 numbers, we would take the 5th and 6th positions, add those numbers, and then divide by 2 to find the median.

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