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Chapter 3: Problem 8

The data shown are the total compensation (in millions of dollars) for the 50 top-paid CEOs for a recent year. Compare the averages, and state which one you think is the best measure. $$\begin{array}{lllll}17.5 & 18.0 & 36.8 & 31.7 & 31.7 \\\17.3 & 24.3 & 47.7 & 38.5 & 17.0 \\\23.7 & 16.5 & 25.1 & 17.4 & 18.0 \\\37.6 & 19.7 & 21.4 & 28.6 & 21.6 \\\19.3 & 20.0 & 16.9 & 25.2 & 19.8 \\\25.0 & 17.2 & 20.4 & 20.1 & 29.1 \\\19.1 & 25.2 & 23.2 & 25.9 & 24.0 \\\41.7 & 24.0 & 16.8 & 26.8 & 31.4 \\\16.9 & 17.2 & 24.1 & 35.2 & 19.1 \\\22.9 & 18.2 & 25.4 & 35.4 & 25.5\end{array}$$

Short Answer

Expert verified
The median is a better measure because it's less affected by outliers.

Step by step solution

01

Calculate the Mean

To find the mean of the total compensation, add all the values together and then divide by the total number of data points, which is 50.Mean = \( \frac{17.5 + 18.0 + 36.8 + ... + 25.5}{50} \).First, add all the compensation values together:\( 17.5 + 18.0 + 36.8 + 31.7 + 31.7 + 17.3 + 24.3 + 47.7 + 38.5 + 17.0 + 23.7 + 16.5 + 25.1 + 17.4 + 18.0 + 37.6 + 19.7 + 21.4 + 28.6 + 21.6 + 19.3 + 20.0 + 16.9 + 25.2 + 19.8 + 25.0 + 17.2 + 20.4 + 20.1 + 29.1 + 19.1 + 25.2 + 23.2 + 25.9 + 24.0 + 41.7 + 24.0 + 16.8 + 26.8 + 31.4 + 16.9 + 17.2 + 24.1 + 35.2 + 19.1 + 22.9 + 18.2 + 25.4 + 35.4 + 25.5 = 1154.9 \).Then divide by 50:Mean = \( \frac{1154.9}{50} = 23.098 \).
02

Calculate the Median

To find the median, first arrange the numbers in ascending order. The median will be the middle value(s) when the data set is ordered.Ordered data set: 16.5, 16.8, 16.9, 16.9, 17.0, 17.2, 17.2, 17.3, 17.4, 17.5, 18.0, 18.0, 18.2, 19.1, 19.1, 19.3, 19.7, 19.8, 20.0, 20.1, 20.4, 21.4, 21.6, 22.9, 23.2, 23.7, 24.0, 24.0, 24.1, 24.3, 25.0, 25.1, 25.2, 25.2, 25.4, 25.5, 25.9, 26.8, 28.6, 29.1, 31.4, 31.7, 31.7, 35.2, 35.4, 36.8, 37.6, 38.5, 41.7, 47.7.For 50 values, the median is the average of the 25th and 26th terms:Median = \( \frac{23.7 + 24.0}{2} = 23.85 \).
03

Compare the Measures

The mean is 23.098, and the median is 23.85. The median is less affected by extremely high or low values in the data set. The mean can be skewed by outliers, such as the higher compensations in this list.

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Key Concepts

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

Understanding the Mean
The mean, typically known as the average, is a fundamental concept in statistics. It helps us understand the central tendency of a data set. To find the mean, you add all the numbers together and then divide by how many numbers there are. This gives a single number that represents the center of the data.
  • Add up all values: this gives the total sum.
  • Divide the total sum by the number of items.
For the CEO compensation example, the sum of compensations is 1154.9 million dollars, and there are 50 CEOs. So, the mean is calculated as:\[\text{Mean} = \frac{1154.9}{50} = 23.098\] million dollars. The mean is a useful measure, but it can sometimes be impacted by extremely high or low values, which can skew the perception of typical compensation.
The Importance of the Median
The median provides a different measure of central tendency that can be more informative in certain cases. It represents the middle value in an ordered data set. This makes it less susceptible to distortion from outliers just like in our CEO compensation example.
  • Arrange data in numerical order.
  • Find the middle number when there is an odd set, or average the two middle numbers when there is an even set.
Given there are 50 CEO compensations, the median is the average of the 25th and 26th numbers in the ordered list. Here, these values are 23.7 and 24.0 million dollars. So,\[\text{Median} = \frac{23.7 + 24.0}{2} = 23.85\] million dollars. The median often tells us more about what a typical value is when a dataset has extreme values.
Spotting Outliers
Outliers are data points that differ significantly from other observations. They can skew results, especially the mean. However, the median stays relatively unaffected by them. In the case of CEO salaries, higher compensations can be outliers. These might arise due to extraordinary business successes or unique company situations.
  • Outliers can distort the mean, making it higher or lower than most typical values.
  • Outliers have less impact on the median, so it remains a good measure of central tendency in skewed data.
Recognizing these outliers helps in understanding how they affect data analysis and which measure provides a more accurate picture of the data set's central tendency.
The Process of Data Analysis
Data analysis is a systematic approach for interpreting data. It involves several steps to obtain meaningful information, particularly in comparing statistical measures.
  • Collection: Gathering the required data effectively.
  • Processing: Organizing data, often in ascending order, for ease of analysis.
  • Analyzing: Using statistical measures like mean and median.
  • Interpreting: Drawing conclusions from these statistical measures.
In our example, data analysis involves calculating and comparing mean and median CEO compensations, to assess which measure better represents the central tendency, considering outliers in the dataset.
Analyzing CEO Compensation
CEO compensation analysis involves evaluating how much CEOs earn and using statistics to understand trends. In this context, comparing the mean and median of salaries tells us different information about pay trends.
  • The mean gives a straightforward average but can be skewed by very high salaries, making it seem like typical earnings are higher than they actually are.
  • The median, providing the midpoint that isn't affected by extreme values, often offers a more reliable insight into typical CEO compensation.
For assessing true average compensation levels, understanding these metrics is crucial. Especially in high-variance fields like executive pay where a few very high compensations can heavily influence the mean.

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Most popular questions from this chapter

A local fast-food company claims that the average salary of its employees is \(\$ 13.23\) per hour. An employee states that most employees make minimum wage. If both are being truthful, how could both be correct?

Harmonic Mean The harmonic mean (HM) is defined as the number of values divided by the sum of the reciprocals of each value. The formula is $$\mathrm{HM}=\frac{n}{\Sigma(1 / X)}$$ For example, the harmonic mean of \(1,4,5,\) and 2 is $$\mathrm{HM}=\frac{4}{1 / 1+1 / 4+1 / 5+1 / 2} \approx 2.051$$ This mean is useful for finding the average speed. Suppose a person drove 100 miles at 40 miles per hour and returned driving 50 miles per hour. The average miles per hour is not 45 miles per hour, which is found by adding 40 and 50 and dividing by 2 . The average is found as shown. Since Time \(=\) distance \(\div\) rate then Time \(1=\frac{100}{40}=2.5\) hours to make the trip Time \(2=\frac{100}{50}=2\) hours to return Hence, the total time is 4.5 hours, and the total miles driven are \(200 .\) Now, the average speed is $$\text { Rate }=\frac{\text { distance }}{\text { time }}=\frac{200}{4.5} \approx 44.444 \text { miles per hour }$$ This value can also be found by using the harmonic mean formula $$\mathrm{HM}=\frac{2}{1 / 40+1 / 50} \approx 44.444$$ Using the harmonic mean, find each of these. a. A salesperson drives 300 miles round trip at 30 miles per hour going to Chicago and 45 miles per hour returning home. Find the average miles per hour. b. A bus driver drives the 50 miles to West Chester at 40 miles per hour and returns driving 25 miles per hour. Find the average miles per hour. c. A carpenter buys \(\$ 500\) worth of nails at \(\$ 50\) per pound and \(\$ 500\) worth of nails at \(\$ 10\) per pound. Find the average cost of 1 pound of nails.

The average weekly earnings in dollars for various industries are listed below. Find the percentile rank of each value. $$ \begin{array}{lllllllll} 804 & 736 & 659 & 489 & 777 & 623 & 597 & 524 & 228 \end{array} $$ For the same data, what value corresponds to the 40 th percentile?

A recent survey of a new diet cola reported the following percentages of people who liked the taste. Find the weighted mean of the percentages. $$\begin{array}{ccc}\text { Area } & \text { \% Favored } & \text { Number surveyed } \\\\\hline 1 & 40 & 1000 \\\2 & 30 & 3000 \\\3 & 50 & 800\end{array}$$

The mean of the waiting times in an emergency room is 80.2 minutes with a standard deviation of 10.5 minutes for people who are admitted for additional treatment. The mean waiting time for patients who are discharged after receiving treatment is 120.6 minutes with a standard deviation of 18.3 minutes. Which times are more variable?
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