## Weighted Mean

### Definition of Weighted Mean

The weighted mean, alternatively termed as a weighted average, presents a methodology for computing the average of a set of values. In this method, each value is assigned a weight, a measure of its importance or frequency within the dataset. To calculate the weighted mean, each value is multiplied by its corresponding weight, the products are then summed, and the resulting total is divided by the sum of the weights:

Weighted Mean = (Σ (value × weight) ) / Σ (weight)

The weighted mean proves instrumental when working with data containing values with unequal significance or when certain values better represent the population.

### Examples of Weighted Mean Applications

The scope of applications for the weighted mean extends to a variety of fields:
Grades and Coursework: The education sector frequently employs the weighted mean to determine a student's overall course grade, given different assignments or exams carry distinct weights towards the final grade.
Finance and Investment: In the realm of finance, the weighted mean assists in calculating elements like the weighted average cost of capital (WACC) or determining portfolio returns based on the proportion of each investment in the portfolio.
Survey Data Analysis: Analysts and researchers working with survey data use the weighted mean to account for discrepancies in response rates or representation across demographic groups.

Several benefits arise from using a weighted mean over a simple arithmetic mean:
Incorporates Importance: The weighted mean acknowledges the relative importance or frequency of each dataset value, hence offering a more precise depiction of the overall population or distribution.
Reduced Bias: Weight assignment to different values in the weighted mean can curtail potential biases that might result from uneven sampling or inconsistent response rates.
More Accurate Estimates: When some values prove more representative or reliable than others, the weighted mean can yield more precise estimates of central tendency than the arithmetic mean.

### Limitations of Weighted Mean

Despite its advantages, the weighted mean also comes with certain limitations:
Subjectivity in Assigning Weights: Allocating appropriate weights to each value can be subjective, often necessitating expert knowledge or assumptions regarding the values' relative importance.
Sensitive to Extreme Weights: The weighted mean can be heavily swayed by extreme weights, potentially resulting in skewed results if the weights do not correctly denote the true importance or frequency of the values.

### Weighted Mean vs. Arithmetic Mean

The weighted mean and the arithmetic mean differentiate primarily based on how they address relative importance or frequency:
Weighted Mean: The weighted mean accounts for the relative importance or frequency of each dataset value, rendering it more suitable when values possess varying degrees of significance or representation.
Arithmetic Mean: The arithmetic mean treats all values with equal importance by summing them and dividing by the total count. This method is preferable when values hold equal significance or when no details are accessible regarding the relative importance or frequency of values.
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