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Chapter 2: Problem 43

What is meant by a weighted average?

Short Answer

Expert verified
Answer: 6.83

Step by step solution

01

Definition of Weighted Average

A weighted average is an average where each value in the dataset is multiplied by a weight, which represents the importance or frequency of the value, before summing the weighted values and dividing by the total weight. As opposed to a simple or arithmetic average, a weighted average takes into consideration the importance or relevance of each value in the calculation.
02

Formula for Weighted Average

The formula for weighted average is as follows: Weighted Average = \(\frac{\sum{i = 1}^{n}w_i \cdot x_i}{\sum{i = 1}^{n}w_i}\) Where: - \(w_i\) represents the weight assigned to value \(x_i\) - \(x_i\) represents each individual value in the dataset - \(n\) represents the total number of values in the dataset
03

Example of Weighted Average Calculation

Consider the following dataset with assigned weights: Dataset: {\(x_1=5, x_2=8, x_3=10\)} Weights: {\(w_1=3, w_2=2, w_3=1\)} Using the formula, we can calculate the weighted average as follows: Weighted Average = \(\frac{(w_1 \cdot x_1) + (w_2 \cdot x_2) + (w_3 \cdot x_3)}{(w_1 + w_2 + w_3)}\) Weighted Average = \(\frac{(3 \cdot 5) + (2 \cdot 8) + (1 \cdot 10)}{(3 + 2 + 1)}\) Weighted Average = \(\frac{(15) + (16) + (10)}{(6)}\) Weighted Average = \(\frac{41}{6}\) Weighted Average = \(6.83\) So, the weighted average of the given dataset is 6.83.

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Most popular questions from this chapter

There are three stable isotopes of magnesium. Their masses are \(23.9850,24.9858,\) and 25.9826 amu. If the average atomic mass of magnesium is 24.3050 amu and the natural abundance of the lightest isotope is \(78.99 \%,\) what are the natural abundances of the other two isotopes?

Suppose we know the atomic mass of each of the three stable isotopes of an element to six significant figures, and we know the natural abundances of the isotopes to the nearest \(0.01 \% .\) How well can we know the average atomic mass - that is, how many significant figures should be used to express its value?

Aluminum, silicon, and oxygen form minerals known as aluminosilicates. How many moles of aluminum are in 1.50 moles of the following? a. pyrophyllite,\(\mathrm{Al}_{2} \mathrm{Si}_{4} \mathrm{O}_{10}(\mathrm{OH})_{2}\) b. \(\operatorname{mica}, \mathrm{KAl}_{3} \mathrm{Si}_{3} \mathrm{O}_{10}(\mathrm{OH})_{2}\) c. albite, \(\mathrm{Na} \mathrm{AlSi}_{3} \mathrm{O}_{8}\)

The density of uranium \(\left(\mathrm{U} ; 19.05 \mathrm{g} / \mathrm{cm}^{3}\right)\) is more than five times as great as that of diamond \(\left(\mathrm{C} ; 3.514 \mathrm{g} / \mathrm{cm}^{3}\right) .\) If you have a cube ( \(1 \mathrm{cm}\) on a side) of each element, which cube contains more atoms?

How does mass spectrometry provide information on the molecular mass of a compound?
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