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Chapter 5: Problem 377

Find the median. $$99.2 , 101.9 , 98.6 , 99.5 , 100.8 , 99.8$$

Short Answer

Expert verified
The median is 99.65.

Step by step solution

01

Arrange the Data in Ascending Order

First, arrange the given data points in ascending order. This is crucial for identifying the median. The given data points are: 99.2, 101.9, 98.6, 99.5, 100.8, 99.8. Arranging them in ascending order, we get: 98.6, 99.2, 99.5, 99.8, 100.8, 101.9.
02

Identify the Median Position

The median is the middle value in a sorted list. If the number of observations (n) is odd, the median is the value at position \(\frac{n+1}{2}\). If n is even, the median is the average of the values at positions \(\frac{n}{2}\) and \(\frac{n}{2} + 1\). Here, n = 6, which is even.
03

Calculate the Median for Even Number of Data Points

Since we have an even number of data points (6), the median will be the average of the 3rd and 4th values in the sorted list. The 3rd value is 99.5 and the 4th value is 99.8. Calculate the average: \(\frac{99.5 + 99.8}{2} = 99.65\).

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Key Concepts

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

Ascending Order
When calculating the median, the first important step is to arrange the data points in ascending order. This means ordering the numbers from the smallest to the largest. For instance, if you are given a list of numbers like: 99.2, 101.9, 98.6, 99.5, 100.8, and 99.8, these should be arranged as 98.6, 99.2, 99.5, 99.8, 100.8, and 101.9.
Sorting the data is crucial because the median represents the middle value in a data set. Without arranging the data in order, it is impossible to accurately determine the median.
Sorting helps to pinpoint the exact center of your list, which is key to different statistical analyses, not just the calculation of the median.
Even Number of Data Points
When dealing with the median, whether the number of data points in your list is odd or even makes a significant difference. For this exercise, we have an **even number of data points**. There are 6 numbers in the sorted list: 98.6, 99.2, 99.5, 99.8, 100.8, and 101.9.
  • With an even number of data points, there isn't a single middle value.
  • This means we need to consider the middle two values to find the median.

Therefore, understanding whether you鈥檙e working with an odd or even set of numbers will inform how you calculate the median.
Median Position
The **median position** in a sorted list depends on the number of observations (n). For an even number of data points, the median position is determined by averaging the two middle numbers.
  • For an odd number, the median is the number located at position \(\frac{n+1}{2}\).
  • For an even number, like in our example with 6 data points, the median is found by taking the average of the values in the \( \frac{n}{2} \) position and the \( \frac{n}{2} + 1 \) position.

In our case, the third value is 99.5 and the fourth value is 99.8. So the median is calculated as:
\( \frac{99.5 + 99.8}{2} = 99.65 \)
This average gives us the center of the data set, or in other words, the median.

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Most popular questions from this chapter

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