Harmonic Mean (HM)
Harmonic Mean (HM)
The harmonic mean is a type of average, typically used when dealing with rates or ratios. It is particularly useful in situations where the average of rates is desired, and it is defined as the reciprocal of the arithmetic mean of the reciprocals of a set of numbers.
Definition
For a set of numbers $x_1, x_2, ..., x_n$, the harmonic mean (HM) is given by:
$$ HM = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + ... + \frac{1}{x_n}} $$
Alternatively, the harmonic mean can be expressed as:
$$ HM = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}} $$
Where:
- $n$ is the number of terms in the dataset
- $x_i$ are the individual non-zero data points
When to Use Harmonic Mean
The harmonic mean is most appropriate for situations where we are dealing with rates or ratios, and we want to find an average that takes into account the proportional contribution of each rate to a whole. It is commonly used in finance (e.g., average price of shares), physics (e.g., average speed), and other domains where averaging of ratios is meaningful.
Comparison with Arithmetic and Geometric Means
Here is a table that compares the harmonic mean (HM) with the arithmetic mean (AM) and the geometric mean (GM):
| Mean Type | Formula | When to Use |
|---|---|---|
| Arithmetic Mean (AM) | $AM = \frac{x_1 + x_2 + ... + x_n}{n}$ | When dealing with sums of quantities or when all values have the same importance. |
| Geometric Mean (GM) | $GM = \sqrt[n]{x_1 \cdot x_2 \cdot ... \cdot x_n}$ | When dealing with products of quantities or growth rates (e.g., compound interest). |
| Harmonic Mean (HM) | $HM = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + ... + \frac{1}{x_n}}$ | When dealing with rates or ratios, especially when the data points are inversely related to the quantity of interest. |
Properties of Harmonic Mean
- The harmonic mean is always less than or equal to the geometric mean, which is always less than or equal to the arithmetic mean (HM ≤ GM ≤ AM), except when all the numbers are equal, in which case all three means are equal.
- The harmonic mean is sensitive to the presence of small values in the dataset. A single small value can significantly decrease the harmonic mean.
- The harmonic mean is undefined if any of the data points are zero since division by zero is undefined.
Examples
Example 1: Average Speed
Suppose a person travels a certain distance at 30 km/h and returns over the same distance at 60 km/h. What is the average speed for the entire trip?
To find the average speed, we use the harmonic mean because we are dealing with rates (speeds) over the same distance.
$$ HM = \frac{2}{\frac{1}{30} + \frac{1}{60}} = \frac{2}{\frac{2}{60} + \frac{1}{60}} = \frac{2}{\frac{3}{60}} = \frac{2 \cdot 60}{3} = 40 \text{ km/h} $$
So, the average speed for the entire trip is 40 km/h.
Example 2: Stock Prices
An investor buys one share of stock at $100, one at $200, and one at $300. What is the average price per share?
Using the harmonic mean:
$$ HM = \frac{3}{\frac{1}{100} + \frac{1}{200} + \frac{1}{300}} = \frac{3}{\frac{0.01}{1} + \frac{0.005}{1} + \frac{0.00333}{1}} = \frac{3}{0.01833} \approx 163.64 $$
The average price per share, considering the proportional investment, is approximately $163.64.
Conclusion
The harmonic mean is a valuable tool for averaging rates and ratios. It is less commonly used than the arithmetic mean but is crucial in specific contexts where the data points are inversely related to the quantity of interest. Understanding when and how to use the harmonic mean can provide more accurate and meaningful averages in such situations.