Question 1

What is a weighted mean in statistics?

Accepted Answer

A weighted mean (or weighted average) is a statistical measure that calculates the average of a set of values, where each value is multiplied by a corresponding weight before averaging. Unlike a simple arithmetic mean, the weighted mean gives more importance to values with higher weights. The formula is: Weighted Mean = Σ(value × weight) ÷ Σ(weight).

Question 2

How do you calculate weighted mean?

Accepted Answer

To calculate weighted mean: 1) Multiply each value by its corresponding weight, 2) Sum all the weighted values to get the total weighted sum, 3) Sum all the weights to get the total weight, 4) Divide the total weighted sum by the total weight. Formula: Weighted Mean = (v₁w₁ + v₂w₂ + ... + vₙwₙ) ÷ (w₁ + w₂ + ... + wₙ).

Question 3

What is the difference between weighted mean and arithmetic mean?

Accepted Answer

Arithmetic mean treats all values equally, while weighted mean assigns different importance (weights) to each value. When all weights are equal, the weighted mean equals the arithmetic mean. Weighted mean is used when some data points are more reliable, frequent, or important than others, such as in grade calculations where different assignments have different point values.

Question 4

When should you use weighted mean?

Accepted Answer

Use weighted mean when: 1) Different data points have different levels of importance or reliability, 2) Some values occur more frequently than others, 3) Calculating grades where assignments have different point values, 4) Portfolio returns where investments have different amounts, 5) Survey results where responses have different sample sizes, 6) Any situation where equal weighting would be inappropriate.

Question 5

What are common examples of weighted mean calculations?

Accepted Answer

Common examples include: 1) Grade point averages where different courses have different credit hours, 2) Investment portfolio returns weighted by amount invested, 3) Survey results weighted by sample size, 4) Price indices weighted by consumption patterns, 5) Employee performance ratings weighted by department size, 6) Test scores weighted by question difficulty or importance.

Question 6

How do you interpret weighted mean results?

Accepted Answer

Interpret weighted mean by considering: 1) The result represents the average value adjusted for the importance of each data point, 2) Higher-weighted values have more influence on the final result, 3) The result will be closer to values with higher weights, 4) If weights vary significantly, the result may differ substantially from the arithmetic mean, 5) The weighted mean provides a more accurate representation when weights reflect true importance.

Question 7

What are the properties of weighted mean?

Accepted Answer

Key properties: 1) If all weights are equal, weighted mean equals arithmetic mean, 2) The result is always between the minimum and maximum values, 3) Multiplying all weights by a constant doesn't change the result, 4) The weighted mean is sensitive to changes in high-weight values, 5) It satisfies the linearity property: weighted mean of (a×values) = a × weighted mean of values.

Question 8

How do you handle zero or negative weights in weighted mean?

Accepted Answer

Zero weights: Values with zero weight don't contribute to the calculation. Negative weights: Generally avoided as they can lead to counterintuitive results. If negative weights are necessary, ensure the total weight is positive. Best practice: Use only positive weights, and if a value should have no influence, assign it a very small positive weight instead of zero.

Question 9

What is the weighted mean formula in mathematical notation?

Accepted Answer

The weighted mean formula is: μ_w = Σ(xᵢ × wᵢ) / Σ(wᵢ), where μ_w is the weighted mean, xᵢ are the individual values, wᵢ are the corresponding weights, and the summation is over all i from 1 to n. This formula ensures that values with higher weights contribute more to the final average.

Question 10

How is weighted mean used in real-world applications?

Accepted Answer

Real-world applications include: 1) Education: GPA calculations with credit hours as weights, 2) Finance: Portfolio returns weighted by investment amounts, 3) Economics: Price indices weighted by consumption patterns, 4) Research: Survey results weighted by sample sizes, 5) Quality control: Product ratings weighted by customer importance, 6) Performance evaluation: Employee ratings weighted by department size or importance.

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