Statistical data analysis would greatly benefit from accurately representing a collection of information. The weighted average formula is one such useful tool. It does not simply treat all points of data like an ordinary average but gives distinct values with different weights.
Such a method finds application in cases where specific numbers have more significance than others in a dataset. Hence, for instance, it could be applicable in examples like financial portfolios or school grades among others; thus if you want your decisions based on better knowledge and exactness mastering the weighted average formula is fundamental.
What is the Weighted Average?
The weighted average is a type of average where each value in a data set is multiplied by a predetermined weight before the final calculation. These weights are taken as the measures of relative importance or frequency with which each figure occurred within that particular observation series.
Whereas simple average considers all amounts equally, this type of measure reflects their significance and thus it may provide a better picture than others in some cases.
Formula for Weighted Average
Weighted Average=\frac{\Sigma(w_i\cdot x_i)}{\Sigma w_i}
where:
- w_i represents the weight assigned to each value x_i.
- x_i represents the individual values in the data set.
- \Sigma denotes the sum over all values.
Example
To illustrate the calculation of a weighted average, consider a student's final grade composed of homework, midterms, and finals, each with different weights. The table below shows the scores and corresponding weights for each component:
Component | Score (x_i) | Weight (w_i) | Weighted Score (w_i\cdot x_i) |
|---|
Homework | 85 | 0.20 | 17 |
|---|
Midterm | 90 | 0.30 | 27 |
|---|
Final | 80 | 0.50 | 40 |
|---|
To calculate the weighted average:
- Sum of Weighted Scores (\Sigma(w_i\cdot x_i)): 17+27+40=84
- Sum of Weights \Sigma w_i: 0.20+0.30+0.50=1
- Weighted Average: \frac{84}{1}=84
Therefore, the student's final weighted average grade is 84.
This tabular representation clearly shows how each component's score is multiplied by its respective weight and how the sum of these weighted scores is used to compute the weighted average.
Formula to Calculate Weighted Average in Excel
You can easily compute a weighted average in Excel by following these steps:
Utilize the "SUMPRODUCT" and "SUM" functions to determine the weighted average of a given dataset.
For instance, let's consider the price of 10 apples at 20 rupees and another fruit at 40 rupees pera 10 fruits. To calculate the weighted average, apply the formula "=SUMPRODUCT(A2:A3, B2:B3)/SUM(B2:B3)" where A2 and A3 represent the frequency and quantity of the items respectively, and B2 and B3 denote the respective values for the other variety of fruits in the dataset.
Conclusion
In statistics it is used to show values regarding things or describe weights based on their importance. Using the different weights attached to data, we can view a clearer picture especially in cases where some things matter more than others do.
The application of a formula for weighted average enables people to make decisions based on total appreciation and accurate data understanding ranging from academic grades calculations to financial portfolio valuation. Those who know how to use the weighted average effectively can improve their decision-making abilities and achieve optimal results in diverse disciplines as well as insights into them. Through the weighted average implementation one can get a more thorough and authentic analysis at a time when everything in question is accounted for significantly within its respective data set.
Solved Questions
Question 1: The different quantities and the associated weights with them are as given below. Find their weighted average.
Answer:
Sum of weights = ∑wi = 5 + 3 + 5 + 6 = 19
∑wi xi = 4×5 + 6×3 + 2×5 + 3×6
= 20 + 18 + 10 + 18
= 66
W = ∑wi×xi/ ∑wi
= 66 /19
Question 2: The different quantities and the associated weights with them are as given below. Find their weighted average.
Answer:
Sum of weights = ∑wi = 40 + 30 = 70
∑wi×xi
= 70×40 + 50×30
= 2800 + 1500
= 4300
W = ∑wi×xi/ ∑wi
= 4300/70
= 430/7
Question 3: The data values and the associated weights with them are as follows:
| DATA VALUE | WEIGHT |
| 80 | 6 |
| 45 | 2 |
| 60 | 3 |
| 20 | 9 |
Answer:
Sum of weights = ∑wi = 6 + 2 + 3 + 9 = 20
∑wi×xi = 80×6 + 45×2 + 60×3 + 20×9
= 480 + 90 + 180 + 180
= 930
W = ∑wi×xi/ ∑wi
W = 930/20
= 93/2
Question 4: The different quantities and the associated weights with them are as follows:
| Quantities | Weights |
| 3 | 5 |
| 6 | 3 |
| 8 | 6 |
| 4 | 3 |
| 2 | 7 |
Answer:
Sum of weights = ∑wi = 5 + 3 + 6 + 3 + 7 = 24
∑wi×xi = 3×5 + 6×3 + 8×6 + 4×3 + 2×7
= 15 + 18 + 48 + 12 + 14
= 107
W = ∑wi×xi/ ∑wi
= 107/24
Question 5: The quantities and the associated weights with them are as follows:
| Quantity Value | Weight |
| 20 | 2 |
| 30 | 3 |
| 40 | 1 |
| 50 | 2 |
| 60 | 4 |
| 70 | 2 |
Answer:
Sum of weights = ∑wi = 2 + 3 + 1 + 2 + 4 + 2 = 14
∑wi×xi = 20×2 + 30×3 + 40×1 + 50×2 + 60×4 + 70×2
= 40 + 90 + 40 + 100 + 240 + 140
= 650
W = ∑wi×xi/ ∑wi
= 650/14
= 325/7
Related Topics:
Practice Problems - Weighted Average Formula
1. A student's final grade is based on three components: assignments (worth 25%), midterm exam (worth 35%), and final exam (worth 40%). The student scored 80 on assignments, 75 on the midterm, and 90 on the final exam. Calculate the weighted average of the student's scores.
2. A company evaluates its employee performance based on three criteria: productivity (40%), teamwork (30%), and punctuality (30%). If an employee scores 85 in productivity, 90 in teamwork, and 80 in punctuality, what is their weighted average performance score?
3. In a portfolio, an investor holds three stocks with the following values and weights: Stock A (value: $50, weight: 40%), Stock B (value: $30, weight: 35%), and Stock C (value: $20, weight: 25%). Calculate the weighted average value of the portfolio.
4. A school calculates GPA based on grades in four subjects with the following weights: Mathematics (weight: 30%), Science (weight: 25%), English (weight: 25%), and History (weight: 20%). If a student has grades of 88 in Mathematics, 92 in Science, 85 in English, and 90 in History, what is their GPA?
5. A restaurant rates its food items based on taste (weight: 40%), presentation (weight: 30%), and value for money (weight: 30%). If a dish scores 9 for taste, 8 for presentation, and 7 for value for money, what is the weighted average rating for the dish?
6. An athlete's overall performance score is based on three events: sprint (weight: 50%), long jump (weight: 30%), and high jump (weight: 20%). The athlete scores 85 in sprint, 90 in long jump, and 80 in high jump. Calculate the weighted average performance score.
7. A product's overall rating is determined by three factors: durability (weight: 40%), design (weight: 35%), and cost (weight: 25%). If the product scores 7 in durability, 8 in design, and 6 in cost, what is the weighted average rating?
8. A university's admission committee considers three criteria for student selection: academic performance (weight: 60%), entrance test (weight: 30%), and extracurricular activities (weight: 10%). If a student has an academic score of 90, an entrance test score of 85, and an extracurricular score of 80, calculate their weighted average admission score.
Similar Reads
Maths Mathematics, often referred to as "math" for short. It is the study of numbers, quantities, shapes, structures, patterns, and relationships. It is a fundamental subject that explores the logical reasoning and systematic approach to solving problems. Mathematics is used extensively in various fields
5 min read
Basic Arithmetic
What are Numbers?Numbers are symbols we use to count, measure, and describe things. They are everywhere in our daily lives and help us understand and organize the world.Numbers are like tools that help us:Count how many things there are (e.g., 1 apple, 3 pencils).Measure things (e.g., 5 meters, 10 kilograms).Show or
15+ min read
Arithmetic OperationsArithmetic Operations are the basic mathematical operations—Addition, Subtraction, Multiplication, and Division—used for calculations. These operations form the foundation of mathematics and are essential in daily life, such as sharing items, calculating bills, solving time and work problems, and in
9 min read
Fractions - Definition, Types and ExamplesFractions are numerical expressions used to represent parts of a whole or ratios between quantities. They consist of a numerator (the top number), indicating how many parts are considered, and a denominator (the bottom number), showing the total number of equal parts the whole is divided into. For E
7 min read
What are Decimals?Decimals are numbers that use a decimal point to separate the whole number part from the fractional part. This system helps represent values between whole numbers, making it easier to express and measure smaller quantities. Each digit after the decimal point represents a specific place value, like t
10 min read
ExponentsExponents are a way to show that a number (base) is multiplied by itself many times. It's written as a small number (called the exponent) to the top right of the base number.Think of exponents as a shortcut for repeated multiplication:23 means 2 x 2 x 2 = 8 52 means 5 x 5 = 25So instead of writing t
9 min read
PercentageIn mathematics, a percentage is a figure or ratio that signifies a fraction out of 100, i.e., A fraction whose denominator is 100 is called a Percent. In all the fractions where the denominator is 100, we can remove the denominator and put the % sign.For example, the fraction 23/100 can be written a
5 min read
Algebra
Variable in MathsA variable is like a placeholder or a box that can hold different values. In math, it's often represented by a letter, like x or y. The value of a variable can change depending on the situation. For example, if you have the equation y = 2x + 3, the value of y depends on the value of x. So, if you ch
5 min read
Polynomials| Degree | Types | Properties and ExamplesPolynomials are mathematical expressions made up of variables (often represented by letters like x, y, etc.), constants (like numbers), and exponents (which are non-negative integers). These expressions are combined using addition, subtraction, and multiplication operations.A polynomial can have one
9 min read
CoefficientA coefficient is a number that multiplies a variable in a mathematical expression. It tells you how much of that variable you have. For example, in the term 5x, the coefficient is 5 — it means 5 times the variable x.Coefficients can be positive, negative, or zero. Algebraic EquationA coefficient is
8 min read
Algebraic IdentitiesAlgebraic Identities are fundamental equations in algebra where the left-hand side of the equation is always equal to the right-hand side, regardless of the values of the variables involved. These identities play a crucial role in simplifying algebraic computations and are essential for solving vari
14 min read
Properties of Algebraic OperationsAlgebraic operations are mathematical processes that involve the manipulation of numbers, variables, and symbols to produce new results or expressions. The basic algebraic operations are:Addition ( + ): The process of combining two or more numbers to get a sum. For example, 3 + 5 = 8.Subtraction (−)
3 min read
Geometry
Lines and AnglesLines and Angles are the basic terms used in geometry. They provide a base for understanding all the concepts of geometry. We define a line as a 1-D figure that can be extended to infinity in opposite directions, whereas an angle is defined as the opening created by joining two or more lines. An ang
9 min read
Geometric Shapes in MathsGeometric shapes are mathematical figures that represent the forms of objects in the real world. These shapes have defined boundaries, angles, and surfaces, and are fundamental to understanding geometry. Geometric shapes can be categorized into two main types based on their dimensions:2D Shapes (Two
2 min read
Area and Perimeter of Shapes | Formula and ExamplesArea and Perimeter are the two fundamental properties related to 2-dimensional shapes. Defining the size of the shape and the length of its boundary. By learning about the areas of 2D shapes, we can easily determine the surface areas of 3D bodies and the perimeter helps us to calculate the length of
10 min read
Surface Areas and VolumesSurface Area and Volume are two fundamental properties of a three-dimensional (3D) shape that help us understand and measure the space they occupy and their outer surfaces.Knowing how to determine surface area and volumes can be incredibly practical and handy in cases where you want to calculate the
10 min read
Points, Lines and PlanesPoints, Lines, and Planes are basic terms used in Geometry that have a specific meaning and are used to define the basis of geometry. We define a point as a location in 3-D or 2-D space that is represented using coordinates. We define a line as a geometrical figure that is extended in both direction
14 min read
Coordinate Axes and Coordinate Planes in 3D spaceIn a plane, we know that we need two mutually perpendicular lines to locate the position of a point. These lines are called coordinate axes of the plane and the plane is usually called the Cartesian plane. But in real life, we do not have such a plane. In real life, we need some extra information su
6 min read
Trigonometry & Vector Algebra
Trigonometric RatiosThere are three sides of a triangle Hypotenuse, Adjacent, and Opposite. The ratios between these sides based on the angle between them is called Trigonometric Ratio. The six trigonometric ratios are: sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec).As give
4 min read
Trigonometric Equations | Definition, Examples & How to SolveTrigonometric equations are mathematical expressions that involve trigonometric functions (such as sine, cosine, tangent, etc.) and are set equal to a value. The goal is to find the values of the variable (usually an angle) that satisfy the equation.For example, a simple trigonometric equation might
9 min read
Trigonometric IdentitiesTrigonometric identities play an important role in simplifying expressions and solving equations involving trigonometric functions. These identities, which include relationships between angles and sides of triangles, are widely used in fields like geometry, engineering, and physics. Some important t
10 min read
Trigonometric FunctionsTrigonometric Functions, often simply called trig functions, are mathematical functions that relate the angles of a right triangle to the ratios of the lengths of its sides.Trigonometric functions are the basic functions used in trigonometry and they are used for solving various types of problems in
6 min read
Inverse Trigonometric Functions | Definition, Formula, Types and Examples Inverse trigonometric functions are the inverse functions of basic trigonometric functions. In mathematics, inverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. The inverse trigonometric functions are the inverse functions of basic trigonometric function
11 min read
Inverse Trigonometric IdentitiesInverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. These functions are the inverse functions of basic trigonometric functions, i.e., sine, cosine, tangent, cosecant, secant, and cotangent. It is used to find the angles with any trigonometric ratio. Inv
9 min read
Calculus
Introduction to Differential CalculusDifferential calculus is a branch of calculus that deals with the study of rates of change of functions and the behaviour of these functions in response to infinitesimal changes in their independent variables.Some of the prerequisites for Differential Calculus include:Independent and Dependent Varia
6 min read
Limits in CalculusIn mathematics, a limit is a fundamental concept that describes the behaviour of a function or sequence as its input approaches a particular value. Limits are used in calculus to define derivatives, continuity, and integrals, and they are defined as the approaching value of the function with the inp
12 min read
Continuity of FunctionsContinuity of functions is an important unit of Calculus as it forms the base and it helps us further to prove whether a function is differentiable or not. A continuous function is a function which when drawn on a paper does not have a break. The continuity can also be proved using the concept of li
13 min read
DifferentiationDifferentiation in mathematics refers to the process of finding the derivative of a function, which involves determining the rate of change of a function with respect to its variables.In simple terms, it is a way of finding how things change. Imagine you're driving a car and looking at how your spee
2 min read
Differentiability of a Function | Class 12 MathsContinuity or continuous which means, "a function is continuous at its domain if its graph is a curve without breaks or jumps". A function is continuous at a point in its domain if its graph does not have breaks or jumps in the immediate neighborhood of the point. Continuity at a Point: A function f
11 min read
IntegrationIntegration, in simple terms, is a way to add up small pieces to find the total of something, especially when those pieces are changing or not uniform.Imagine you have a car driving along a road, and its speed changes over time. At some moments, it's going faster; at other moments, it's slower. If y
3 min read
Probability and Statistics
Basic Concepts of ProbabilityProbability is defined as the likelihood of the occurrence of any event. It is expressed as a number between 0 and 1, where 0 is the probability of an impossible event and 1 is the probability of a sure event.Concepts of Probability are used in various real life scenarios : Stock Market : Investors
7 min read
Bayes' TheoremBayes' Theorem is a mathematical formula used to determine the conditional probability of an event based on prior knowledge and new evidence. It adjusts probabilities when new information comes in and helps make better decisions in uncertain situations.Bayes' Theorem helps us update probabilities ba
13 min read
Probability Distribution - Function, Formula, TableA probability distribution is a mathematical function or rule that describes how the probabilities of different outcomes are assigned to the possible values of a random variable. It provides a way of modeling the likelihood of each outcome in a random experiment.While a Frequency Distribution shows
13 min read
Descriptive StatisticStatistics is the foundation of data science. Descriptive statistics are simple tools that help us understand and summarize data. They show the basic features of a dataset, like the average, highest and lowest values and how spread out the numbers are. It's the first step in making sense of informat
5 min read
What is Inferential Statistics?Inferential statistics is an important tool that allows us to make predictions and conclusions about a population based on sample data. Unlike descriptive statistics, which only summarize data, inferential statistics let us test hypotheses, make estimates, and measure the uncertainty about our predi
7 min read
Measures of Central Tendency in StatisticsCentral tendencies in statistics are numerical values that represent the middle or typical value of a dataset. Also known as averages, they provide a summary of the entire data, making it easier to understand the overall pattern or behavior. These values are useful because they capture the essence o
11 min read
Set TheorySet theory is a branch of mathematics that deals with collections of objects, called sets. A set is simply a collection of distinct elements, such as numbers, letters, or even everyday objects, that share a common property or rule.Example of SetsSome examples of sets include:A set of fruits: {apple,
3 min read
Practice