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The following data give the 2006-07 team salaries for 20 teams of the English Premier League, arguably the best-known soccer league in the world. The salaries are given in the order in which the teams finished during the \(2006-07\) season. The salaries are in millions of British pounds (note that the approximate value of 1 British pound was \(\$ 1.95\) during the \(2006-07\) season, so the team salaries range from \(\$ 34.3\) million to \(\$ 259\) million). (Source: \(B B C\), May \(28,2008 .\) ) \(\begin{array}{lrlllll}92.3 & 132.8 & 77.6 & 89.7 & 43.8 & 38.4 & 30.7 \\ 29.8 & 36.9 & 36.7 & 43.2 & 38.3 & 62.5 & 36.4 \\ 44.2 & 35.2 & 27.5 & 22.4 & 34.3 & 17.6 & \end{array}\) Find the mean and median for these data.

Step by step solution

01

Preparation

First, gather all the team salaries. This was provided in the problem statement. The following are the team salaries in millions of pounds: 92.3, 132.8, 77.6, 89.7, 43.8, 38.4, 30.7, 29.8, 36.9, 36.7, 43.2, 38.3, 62.5, 36.4, 44.2, 35.2, 27.5, 22.4, 34.3, 17.6.

02

Calculate the mean

Sum all the team salaries and divide by the total number of teams to find the average salary. This can be done using the formula \[ Mean = \frac{Sum \ of \ Salaries}{Number \ of \ Teams} \]

03

Find the median

To find the median, arrange the team salaries in ascending order and determine the middle number. If there happens to be an even number of teams, then take the average of the two middle numbers.

04

Calculate and verify

Calculate the Mean and Median using steps 2 and 3. Verify if the calculations are correct.

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Key Concepts

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

Mean Calculation

The mean, often called the average, provides a central value for a set of numbers. It's obtained by adding all the figures in a data set and then dividing by the number of items in the dataset.
For the Premier League team salaries, you need to sum all the salaries in millions of pounds and divide by the number of teams, which is 20.

Here's how you can calculate it step by step:

• Add all the salaries together: 92.3 + 132.8 + 77.6 + 89.7 + 43.8 + 38.4 + 30.7 + 29.8 + 36.9 + 36.7 + 43.2 + 38.3 + 62.5 + 36.4 + 44.2 + 35.2 + 27.5 + 22.4 + 34.3 + 17.6.
• This results in a total sum.
• Then, you divide this sum by 20, which is the total number of teams.

This calculation will provide you with the average salary across the teams, offering a sense of the general trend or center point of the dataset.

Median Calculation

Unlike the mean, the median is the number in the middle of a sorted list. It can give a better sense of the central tendency when there are outliers or skewed data.
The steps to find the median are straightforward.

Here's what you'll need to do:

• First, list all salaries in ascending order. Whether you use a calculator or do it by hand, this helps you visualize the middle point more clearly.
• For an odd number of data points, the median is the middle number. But here, with 20 salaries, you have an even set.
• To find the median in an even set, identify the two middlemost numbers in the sorted list.
• Calculate the mean of these two numbers by adding them together and dividing by 2.

This approach to finding the median helps balance the impact of any extremely high or low salaries, which might skew other measures of central tendency.

Statistical Analysis

Statistical analysis involves collecting and scrutinizing every data sample in a set of items from which samples can be drawn. Understanding the mean and median is foundational to statistical analysis.

These concepts help in:

• Describing the central tendency of the data, which provides a summary of the data's overall trend.
• Identifying outliers, which are data points significantly different from others in the dataset, influencing the mean and essential for accurate data interpretation.
• Determining data distribution, aiding in visualizing how data points are spread and skewed, important for deeper statistical studies.

Statistical analysis is essential for making informed decisions based on data, highlighting trends, and understanding the behavior of data in fields like economics, science, and much more.
Mastering these basic concepts can enhance the ability to perform more complex analytical tasks later on in your study of statistics.