The arithmetic mean is the familiar average. You calculate the mean by adding up the values and dividing by the number of values. For example, the mean of 1, 4 and 7 is (1 + 4 + 7)/3 = 12/3 = 4.
The arithmetic mean measures the central tendency -- that is, where the middle of a set of values is. While the arithmetic mean is widely used and can be very useful, it is not always appropriate.
Influenced by Outliers
Outliers are points that are surprising, or far from the rest of the points. Outliers highly influence the arithmetic mean. For instance, if you wanted to find the average height of American men and, just by chance, your sample included a guy who was 7 feet 6 inches tall, the mean would be high. A better choice here might be the median, or a trimmed mean.
Not Appropriate with Highly Skewed Data
Skewed data has a long tail on one side. That is, the values on one side of the mean are much farther from the mean than those on the other side. A familiar example is income. According to the Census Bureau, the mean household income in the United States in 2004 was $60,056. But while the lowest income is around $0 (which is $60,000 below the mean), the highest is well over $120,000 (which is $60,000 above the mean). Indeed, 1,699,000 households made more than $250,000. This is why income is usually reported as median, rather than mean.
Not Good for Rates
Suppose you travel 120 miles and back. On the first part of your trip, you travel at 40 miles per hour. On the second part of your trip you travel at 60 mph. But the average rate is not the arithmetic mean (40 + 60)/2 = 50 miles per hour. The trip out takes three hours. The trip back takes two hours. So, you traveled 240 miles in five hours, for an average of 240/5, or 48 mph. Here you should use the harmonic mean.
Not Good for Averaging Averages
This is similar to the previous problem, but it appears differently. When you average averages, the arithmetic mean is not right. For instance, suppose you want to find the average height of people in your college class. You find that the average height of women is 5 feet 4 inches and the average for men is 5 feet 10 inches. But, unless there are the same number of men and women, the overall average will not be 5 feet 7 inches.
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