Calculate The Weighted Mean Of The Following Data

Calculating the Weighted Mean: A Comprehensive Guide

The weighted mean, unlike the simple arithmetic mean, assigns different weights to different data points. This is crucial when some data points are more significant or reliable than others. This comprehensive guide will delve into the intricacies of calculating the weighted mean, exploring various scenarios, applications, and interpretations. We'll also touch upon the differences between weighted and simple means and when each is most appropriate.

Understanding the Weighted Mean

The weighted mean is a type of average where each data point is assigned a weight reflecting its relative importance. The weights are numbers that represent the influence each data point has on the final average. A higher weight signifies a greater impact. The formula for calculating the weighted mean is:

Weighted Mean = Σ (xᵢ * wᵢ) / Σ wᵢ

Where:

• xᵢ represents each individual data point.
• wᵢ represents the weight assigned to the corresponding data point xᵢ.
• Σ denotes the summation (adding up all values).

This formula essentially sums up the product of each data point and its weight, then divides by the sum of all weights. This ensures that data points with higher weights contribute more significantly to the final weighted average.

When to Use a Weighted Mean

The weighted mean is particularly useful in situations where:

• Data points have varying levels of importance or reliability: For instance, in calculating a student's final grade, different assignments might carry different weights (e.g., exams worth 60%, homework 20%, and projects 20%). The weighted mean accurately reflects the relative importance of each component.

• Data points are clustered differently: Imagine surveying customer satisfaction across different demographics. If one demographic group is significantly larger than another, a simple mean would be skewed. A weighted mean, weighting by the size of each demographic group, provides a fairer representation of overall satisfaction.

• Data points represent different sample sizes: In statistical analysis, different samples might have different sample sizes. Using a weighted mean with the sample sizes as weights ensures that larger samples have a proportionally greater influence on the overall result.

Calculating the Weighted Mean: Step-by-Step Examples

Let's illustrate the calculation process with a few examples:

Example 1: Grade Calculation

A student's grades in a course are as follows:

Assignment	Grade (xᵢ)	Weight (wᵢ)
Midterm Exam	85	30%
Final Exam	92	40%
Homework Assignments	88	30%

1. Convert percentages to decimal weights:

• Midterm Exam weight: 0.30
• Final Exam weight: 0.40
• Homework Assignments weight: 0.30

2. Calculate the weighted sum (Σ (xᵢ * wᵢ)):

• (85 * 0.30) + (92 * 0.40) + (88 * 0.30) = 25.5 + 36.8 + 26.4 = 88.7

3. Calculate the sum of weights (Σ wᵢ):

• 0.30 + 0.40 + 0.30 = 1.00

4. Calculate the weighted mean:

• Weighted Mean = 88.7 / 1.00 = 88.7

Therefore, the student's weighted average grade is 88.7.

Example 2: Customer Satisfaction Survey

A company conducts a customer satisfaction survey across three regions:

Region	Satisfaction Score (xᵢ)	Number of Customers (wᵢ)
North	8.2	500
South	7.8	300
West	9.1	700

1. Calculate the weighted sum (Σ (xᵢ * wᵢ)):

• (8.2 * 500) + (7.8 * 300) + (9.1 * 700) = 4100 + 2340 + 6370 = 12810

2. Calculate the sum of weights (Σ wᵢ):

• 500 + 300 + 700 = 1500

3. Calculate the weighted mean:

• Weighted Mean = 12810 / 1500 = 8.54

The weighted average customer satisfaction score across all regions is 8.54.

Example 3: Investment Portfolio Return

An investor has the following portfolio:

Investment	Return (xᵢ)	Amount Invested (wᵢ)
Stock A	12%	$5,000
Stock B	8%	$3,000
Bond C	5%	$2,000

1. Calculate the weighted sum (Σ (xᵢ * wᵢ)):

• (0.12 * 5000) + (0.08 * 3000) + (0.05 * 2000) = 600 + 240 + 100 = 940

2. Calculate the sum of weights (Σ wᵢ):

• 5000 + 3000 + 2000 = 10000

3. Calculate the weighted mean:

• Weighted Mean = 940 / 10000 = 0.094 = 9.4%

The weighted average return on the investor's portfolio is 9.4%.

Weighted Mean vs. Simple Mean: Key Differences

The simple arithmetic mean is calculated by summing all data points and dividing by the number of data points. It treats all data points equally. The weighted mean, as discussed, assigns different weights to each data point.

The choice between using a weighted mean or a simple mean depends entirely on the nature of the data and the research question. If all data points are equally important and there are no underlying factors influencing their relative significance, then a simple mean is appropriate. However, if there are varying levels of importance, reliability, or sample sizes associated with the data points, the weighted mean offers a more accurate and nuanced representation.

Potential Pitfalls and Considerations

• Choosing appropriate weights: Selecting the correct weights is critical. Incorrect or arbitrary weights can lead to misleading results. The weights should reflect the true relative importance or influence of each data point.

• Data quality: The accuracy of the weighted mean depends on the accuracy of both the data points and the assigned weights. Errors in either will affect the final result.

• Interpretation: Always carefully interpret the weighted mean within the context of the data and the chosen weights. Don't overgeneralize or make unwarranted assumptions based solely on the calculated average.

Advanced Applications of the Weighted Mean

The weighted mean finds applications in various fields beyond the simple examples provided above:

• Finance: Portfolio performance analysis, calculating average returns, and risk assessment.

• Statistics: Estimating population parameters, combining data from multiple sources with differing reliabilities, and meta-analysis.

• Economics: Calculating weighted averages of economic indicators, such as GDP or inflation, across different regions or sectors.

• Engineering: Calculating the center of gravity, determining average material properties, and analyzing experimental results.

• Image processing: Applying weighted averages to pixel values for image enhancement and filtering.

Conclusion

Calculating the weighted mean provides a powerful tool for analyzing data where not all data points are created equal. By understanding the underlying principles and applying the formula correctly, researchers and analysts can obtain a more accurate and insightful representation of their data, leading to better decision-making in diverse fields. Remember to carefully select your weights and interpret the results within the context of your data. The weighted mean, when used appropriately, surpasses the limitations of the simple mean, offering a more robust and representative measure of central tendency.