The Federal Reserve Bank of St. Louis reported two average personal incomes for 2018: \$33,706 and \$50,731. One of these averages is the mean and the other is the median. Which is the mean? Support your answer.

Verified step by step guidance

1. Understand the difference between the mean and the median: The mean is the arithmetic average of all data points, calculated by summing all values and dividing by the number of values. The median is the middle value when all data points are ordered from smallest to largest.
2. Recall that the mean is sensitive to extreme values (outliers), while the median is more robust and less affected by outliers.
3. Consider the context of personal incomes, which often have a right-skewed distribution due to a small number of very high incomes pulling the average up.
4. Since the mean is influenced by high incomes, it tends to be higher than the median in income data. Therefore, the larger value (\$50,731) is likely the mean, and the smaller value (\$33,706) is likely the median.
5. To support this conclusion, explain that the median represents the middle income, so half the population earns less than \$33,706 and half earns more, while the mean is pulled upward by high earners, resulting in the higher average of \$50,731.

Key Concepts

Mean

The mean is the arithmetic average of a data set, calculated by summing all values and dividing by the number of observations. It is sensitive to extreme values (outliers), which can skew the mean higher or lower than the typical data point.

Median

The median is the middle value in an ordered data set, dividing the data into two equal halves. It is resistant to outliers and skewed data, often providing a better measure of central tendency when the distribution is not symmetric.

Effect of Skewness on Mean and Median

In skewed distributions, the mean is pulled toward the tail, while the median remains near the center of the data. For income data, which is typically right-skewed, the mean is usually higher than the median due to a few high-income values.