Critical Thinking. For Exercises 5–20, watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, (d) midrange, and then answer the given question.


Geography Majors The data listed below are estimated incomes (dollars) of students who graduated from the University of North Carolina (UNC) after majoring in geography. The data are based on graduates in the year 1984. The income of basketball superstar Michael Jordan (a 1984 UNC graduate and geography major) is included. Does his income have much of an effect on the measures of center? Based on these data, would the college have been justified by saying that the mean income of a graduate in their geography program is greater than \$250,000?


17,466 18,085 17,875 19,339 19,682 19,610 18,259 16,354 2,200,000

Verified step by step guidance

1. Step 1: Calculate the mean. The mean is the sum of all data values divided by the number of data values. Add all the incomes together, including Michael Jordan's income, and divide by the total number of data points (9 in this case). Use the formula: $\mathrm{Mean}=\frac{\sum x}{n}$, where $\sum x$ is the sum of all incomes and $n$ is the number of data points.
2. Step 2: Find the median. To find the median, first arrange the data in ascending order. The median is the middle value if the number of data points is odd, or the average of the two middle values if the number of data points is even. In this case, arrange the incomes and identify the middle value.
3. Step 3: Determine the mode. The mode is the value that appears most frequently in the dataset. Check if any income value is repeated, and if so, that value is the mode. If no value is repeated, the dataset has no mode.
4. Step 4: Calculate the midrange. The midrange is the average of the minimum and maximum values in the dataset. Use the formula: $\mathrm{Midrange}=\frac{\mathrm{Min}+\mathrm{Max}}{2}$, where $\mathrm{Min}$ is the smallest income and $\mathrm{Max}$ is the largest income.
5. Step 5: Analyze the effect of Michael Jordan's income. Compare the mean, median, and midrange to assess how much Michael Jordan's income skews the measures of center. Consider whether the mean is significantly higher than the median and midrange, which would indicate that his income has a large effect. Then, evaluate whether the mean income justifies the claim that the average income is greater than \$250,000.

Key Concepts

Measures of Central Tendency

Measures of central tendency, including mean, median, and mode, summarize a set of data by identifying the center point. The mean is the average of all values, the median is the middle value when data is ordered, and the mode is the most frequently occurring value. Understanding these measures is crucial for analyzing how data points, such as incomes, cluster around a central value.

Outliers

Outliers are data points that significantly differ from other observations in a dataset. In this context, Michael Jordan's income of $2,200,000 is an outlier that can skew the mean, making it much higher than the incomes of other geography graduates. Recognizing outliers is essential for accurately interpreting statistical measures and understanding their impact on data analysis.

Justification of Claims

Justifying claims about data, such as whether the mean income exceeds $250,000, requires careful analysis of the calculated measures of central tendency. This involves comparing the computed mean against the claim and considering the influence of outliers. A thorough justification ensures that conclusions drawn from the data are valid and supported by statistical evidence.