Simple, Weighted, combined arithmetic mean | Geometric Mean | Harmonic Mean | Statistics

The weighted arithmetic mean – measures of central tendency – statistics is a type of average that takes into account the importance, or weight, of each value in a dataset. It is calculated by multiplying each value by its corresponding weight, summing up these products, and then dividing by the sum of the weights. The formula for the weighted arithmetic mean is:

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In this formula:

The weighted arithmetic mean is particularly useful when certain values in the dataset are more significant or contribute more to the overall average than others.

For example, suppose you have a set of grades for a student, and each grade has a different weight based on the number of credit hours for the corresponding course. The weighted mean can be calculated as follows:

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This formula gives more importance to the grades with higher credit hours, reflecting the fact that those courses contribute more to the overall academic performance.

It’s important to note that in the case of equal weights (all ��wi​ values are the same), the weighted mean formula reduces to the regular arithmetic mean formula.

Combined Arithmetic Mean

The term “Combined Arithmetic Mean” is not a standard statistical concept, and it’s not clear if it refers to a specific mathematical operation or formula. Arithmetic mean (or average) is a well-defined concept, but the term “combined” in this context may require more context or clarification.

If you are referring to a situation where you have two or more sets of data and you want to calculate the overall arithmetic mean, you typically calculate the mean for each set separately and then combine them by taking a weighted average if the sets have different sizes.

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This formula gives more weight to the mean of the set with a larger sample size.

If your use of “combined arithmetic mean” refers to a different concept or operation, please provide more context or details, and I’ll do my best to assist you.

Geometric Mean – Statistics

The geometric mean is a measure of central tendency that is calculated by multiplying together a set of numbers and then taking the nth root (where n is the number of values in the set) of the product. It is often used in situations where the values being averaged represent rates of growth or proportions.

It’s important to note that the geometric mean is sensitive to the values in the dataset. It is particularly useful in situations where relative magnitudes or ratios are important, such as calculating average rates of return in finance or average growth rates in various scientific and economic contexts.

For example, if you have a set of values {2, 4, 8}, the geometric mean would be calculated as follows:​

Harmonic Mean​

The harmonic mean is another measure of central tendency, and it is calculated by dividing the number of observations by the reciprocal of each number in the dataset and then taking the reciprocal of the result. The harmonic mean is particularly useful in situations where rates or ratios are involved.