Final answer:
The weighted harmonic mean is calculated by taking the weighted reciprocal of each number in a set, summing these, dividing by the sum of weights, and taking the reciprocal of the result.
Step-by-step explanation:
The weighted harmonic mean is a type of average that is calculated by taking the reciprocal of the arithmetic mean of the reciprocals of a set of numbers, each multiplied by a specified weight. To calculate the weighted harmonic mean, you follow these steps:
- Multiply each value of the dataset by its corresponding weight.
- Find the reciprocal of these weighted values.
- Sum all of these reciprocals.
- Take the sum of the weights of the dataset.
- Divide this sum of weights by the sum of the reciprocals obtained in step 3.
- Finally, take the reciprocal of this quotient to get the weighted harmonic mean.
An example from the periodic table might illustrate this concept. Different isotopes have different masses and different abundances (or weights). The periodic table lists the average mass of an element based on the relative abundance of each isotope. This average mass is a weighted average and very similar in concept to the weighted harmonic mean.
If you were to calculate a weighted harmonic mean instead, you would use the isotopic weights as the weights in the calculation and the isotopic masses as the values in the dataset.