Question

Ten of the 50 largest businesses in the United States, randomly selected from the Fortune \(500,\) are listed as follows along with their revenues (in millions of dollars) \(^{4}\) : $$\begin{array}{lr|lr}\hline \text { Company } & \text { Revenues } & \text { Company } & \text { Revenues } \\\\\hline \text {GeneralMotors } & \$ 166,380 & \text { Target } & \$ 69,495 \\\\\text { IBM } & 79,919 & \text { Morgan Stanley } & 37,949 \\\\\text { Bank of America } & 93,662 & \text { Johnson \& Johnson } & 71,890 \\\\\text { Home Depot } & 94,595 & \text { Apple } & 215,639 \\\\\text { Boeing } & 94,571 & \text { Exxon Mobil } & 205,004 \\\\\hline\end{array}$$ a. Draw a stem and leaf plot for the data. Are the data skewed? b. Calculate the mean revenue for these 10 businesses. Calculate the median revenue. c. Which of the two measures in part b best describes the center of the data? Explain.

Short answer

Answer: The median revenue is more appropriate to describe the center of the data because the data is positively skewed, and the median is less sensitive to outliers, making it more representative of the typical revenue rate for large businesses.

Step-by-step solution

Draw a Stem and Leaf Plot

To create a stem and leaf plot, we need to sort the revenues in ascending order and create stems based on the grouping of the tens of thousands in revenue. Then, place the leaves accordingly. Here are the sorted revenues: \(37,949, 69,495 ,71,890, 79,919, 93,662, 94,571, 94,595, 166,380, 205,004, 215,639\) Next, we make a stem-and-leaf plot: $$ \begin{array}{c|c} \hline \textbf{Stem (1000s)} & \textbf{Leaf (10000s)} \\ \hline 3 & 7 \\ 6 & 9 \\ 7 & 1, 9 \\ 9 & 3, 4, 4 \\ 16 & 6 \\ 20 & 5, 15 \\ \hline \end{array} $$ The data seems to be positively skewed, as there is a tail extending towards the right.

Calculate Mean and Median Revenue

To calculate the mean revenue, sum all revenues and divide by the total number of companies (10): Mean revenue = \(\frac{37,949 + 69,495 + 71,890 + 79,919 + 93,662 + 94,571 + 94,595 + 166,380 + 205,004 + 215,639}{10} = \$112,520.4\) To find the median revenue, locate the middle value when the revenues are ordered from least to greatest. Since there are 10 companies, the median will be the average of the 5th and 6th values: Median revenue = \(\frac{93,662+94,571}{2} = \$94,116.5\)

Determine Best Central Measure and Explain

Given the positively skewed distribution of the data observed in the stem and leaf plot, the median revenue would be a better measure of central tendency in this case. The mean can be greatly influenced by a few extremely large values (like Apple's and Exxon Mobil's revenues in this case), whereas the median is less sensitive to outliers. Therefore, the median revenue of \(94,116.5\) million is more representative of the typical revenue rate for these large businesses.

Key concepts

Stem and Leaf Plots

When dealing with large data sets, visual representations can be incredibly valuable. A stem and leaf plot is a simple, yet effective, way to organize data while maintaining the original values. Imagine this plot as a hybrid between a bar graph and a list: the 'stem' represents the first part of the number, commonly the highest order digits, while the 'leaf' lists the remaining digits.

To create a stem and leaf plot, you first need to sort your data in ascending or descending order. Next, determine a consistent base unit for your stems, like tens or hundreds, and list these as your stems. The remaining digits of each data point become the leaves. For example, if your data point is 45 and your stem unit is tens, the stem would be 4, and the leaf would be 5.

One strength of the stem and leaf plot is its ability to preserve the original data points, which allows you to determine the shape of the distribution and observe individual values. In the exercise, the plot reveals a right-sided tail, indicating that the data might be positively skewed, which takes us to the concept of skewed data.

Measures of Central Tendency

Central tendency represents a central value for a set of data. The most commonly used measures are the mean, median, and mode. The mean is the numerical average of a data set and is sensitive to extreme values. It's calculated by adding all the numbers up and dividing by the total count. On the other hand, the median is the middle number of an ordered data set, or the average of the two middle numbers in case of an even set. It is more robust against outliers than the mean.

For a data set with no extreme values, the mean and median may be quite close. However, when outliers are present, the median often better reflects the central tendency of the 'typical' data point because it isn't as influenced by extremely high or low values, unlike the mean. In educational contexts, ensuring students understand both concepts is vital for their analytical skills, as different scenarios call for different measures.

Skewed Data

Skewed data refer to data distributions that are not symmetrical. When data is positively skewed, also known as right-skewed, it means there's a long tail in the positive direction on a graph. Negative skewness, or left-skewed data, occurs when there's a long tail in the negative direction. This concept is crucial because it affects the choice of central tendency measure.

For a symmetric distribution, the mean and median will be similar. However, in the case of skewed distributions, the mean is pulled towards the tail, and hence can be misleading. It is thus important, when analyzing data, to consider the shape of the distribution. If data are skewed, the median may offer a better representation of the central location of the data set. In education, teaching students to accurately assess the skewness of data helps them develop a more nuanced understanding of statistical analysis.

Considering the aforementioned company revenue data, the presence of high revenue companies creates a positively skewed distribution, suggesting that the median is a more accurate reflection of the central tendency for this particular data set.