Understanding Weighted Average — From Basics to Calculus

Last Updated on October 16, 2025 by Rajeev Bagra

When we think of averages, we usually imagine adding up all values and dividing by how many there are.
But in many real situations, not all values are equally important — and that’s where weighted averages come in.

In this post, you’ll understand:

What a weighted average means in plain English
The mathematical formula behind it
How the same concept appears beautifully in integral calculus

🧩 Weighted Average in Plain English

A weighted average is just like an ordinary average — but some numbers matter more than others.

Think of your semester marks:

Tests count for 60%
Assignments count for 30%
Attendance counts for 10%

If you scored:

80 in tests
70 in assignments
100 in attendance

You don’t just average them as (80 + 70 + 100)/3.
Instead, you multiply each by its weight (importance).

$ (80 \times 0.6) + (70 \times 0.3) + (100 \times 0.1) = 48 + 21 + 10 = 79 $

✅ Your weighted average score = 79.

📐 The Mathematical Formula

If you have several values $x_1, x_2, x_3, \dots, x_n$
and their respective weights $w_1, w_2, w_3, \dots, w_n$ ,
the weighted average $\bar{x}_w$ is:

$ \bar{x}w = \frac{\sum{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i} $

Steps:

Multiply each value by its weight.
Add all those products.
Divide by the sum of all weights.

⚖️ Example

Item	Value ( $x_i$ )	Weight ( $w_i$ )	$w_i x_i$
A	10	1	10
B	20	2	40
C	30	3	90

Now compute:

$ \bar{x}_w = \frac{10 + 40 + 90}{1 + 2 + 3} = \frac{140}{6} = 23.33 $

So, the weighted average is 23.33 — closer to 30, because 30 carried the largest weight.

🔢 Weighted Average in Integral Calculus

In calculus, we often deal with continuous data instead of discrete points.
When a function $f(x)$ changes smoothly over an interval,
we can define its weighted average using integrals.

If every point $x$ has a weight function $w(x)$ ,
the weighted average of $f(x)$ between $a$ and $b$ is:

$ \bar{f} = \frac{\int_a^b f(x) , w(x) , dx}{\int_a^b w(x) , dx} $

🧲 Example from Physics: Average Density of a Rod

Imagine a rod along the x-axis from $x = 0$ to $x = L$ ,
where density changes with position as $\rho(x)$ .

Then, the average density of the rod is:

$ \bar{\rho} = \frac{\int_0^L \rho(x) , dx}{L} $

This means:
add up the densities of all tiny pieces (using an integral),
then divide by the total length.

If you wanted the center of mass, the formula becomes:

$ x_{\text{center}} = \frac{\int_0^L x , \rho(x) , dx}{\int_0^L \rho(x) , dx} $

Here, each position $x$ is weighted by its mass density $\rho(x)$ .

🧠 Summary Table

Context	Formula	Meaning
Simple average	$\frac{\sum x_i}{n}$	All values equally important
Weighted average	$\frac{\sum w_i x_i}{\sum w_i}$	Some values count more
Continuous (calculus) weighted average	$\frac{\int f(x) w(x) dx}{\int w(x) dx}$	Smooth weighting across continuous data

✨ Final Thought

Weighted averages let us represent fairness in numbers —
whether it’s your grades, portfolio returns, or a rod’s center of mass.

They remind us that not every value has equal impact —
sometimes, what matters most should count more.

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Understanding Weighted Average — From Basics to Calculus

🧩 Weighted Average in Plain English