How to Use Geometric and Harmonic Means in Statistical Analysis
The average, or mean, is one of the most popular descriptive measures in statistics. When most people use the term "mean," they are referring to the arithmetic mean, in which we add the numbers in a set and then divide the sum by the number of values or observations in the set. Some analytical situations, such as average rate of return or average speed, may require greater precision than the arithmetic mean allows. For these situations, the geometric and harmonic means are available for our use.
Things You'll Need
- A computer or scientific calculator
- A set of data
- Statistics book or manual for reference
Instructions
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Geometric and Harmonic Means: What They Are and When to Use Them
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Suppose you have a set of data that cannot best be summarized by the regular arithmetic mean. Let's first decide whether you should use the geometric or harmonic mean. In contrast to adding the numbers in a set, n, and dividing by n to get an arithmetic mean, the geometric mean multiplies the values in n, then takes the nth root. In situations involving rates of growth or rates of return, such as interest rates, use the geometric mean.
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Application of the geometric mean is similar to the principle of compound interest. For example, say you want the average rate of growth in a worker's wages for a three-year period. Suppose that worker's wages increase 5 percent in year 1, 3 percent in year 2, and 4 percent in year 3. The arithmetic mean of these 3 numbers is 4. The problem, though, is that each increase is cumulative and multiplied by that year's base wage. To calculate a geometric mean, you would multiply (1.05 x 1.03 x 1.04) and then take the 1/3 power, which gives you 3.91 as an average rate of growth. In this example, the geometric mean is only slightly lower than the arithmetic mean, but it illustrates how with larger sets of data and larger variation in rates of growth or rates of return, the arithmetic mean could greatly overestimate the average rate of growth. Financial data may require more exact estimation than the arithmetic mean offers.
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Just as the arithmetic mean is not the appropriate measure for average rates of growth or return, such as interest or return on investments, neither is it the appropriate measure for average quantities, such as rates of speed. In those situations, the harmonic mean is the best average measure. The harmonic mean, often used in physics and related fields, involves taking the number of elements or observations in a set and dividing it by the sum of reciprocals.
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Let's say you want to calculate the average rate of speed for a driving trip. Suppose I drove 35 miles per hour from point A to point B, then drove 70 miles per hour going back from B to A. The arithmetic mean of these two numbers is 52.5, but it doesn't account for the fact that I drove 35 mph for twice as long as I drove 70 mph. The harmonic mean would take the sum of the reciprocals of 35 and 70, which is 3/70. The number of elements in this example is 2, which divided by 3/70 gives us a harmonic mean of 46.66, well below the arithmetic mean, which greatly exaggerates the average rate of speed.
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Tips & Warnings
Of the three means discussed in this article, the arithmetic mean is always the greatest, while the harmonic mean is always the lowest. The geometric mean is the value in the middle. The harmonic mean is especially sensitive to abnormally small values in a set of observations.
Make sure data are entered accurately; otherwise, your means will be wrong, no matter which one you use. Garbage in, garbage out.
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want better solution.
