When we look at data, we are often not interested in every individual fluctuation. We want to see the bigger picture: is this growing? Is this declining? Where is it heading? The moving average is one of the simplest tools we have to answer that question. Before we define it formally, let us build the intuition from the ground up.

Consider those three data points: 10, 20, 10

20 is like a bump or big increase in our data. Now average those three values.

Real-World Analogies

Think about how you pay for services in everyday life.

A post-paid phone plan is a perfect analogy for an annuity immediate. You make calls throughout the month, and only at the end of the period does the carrier tally up your usage and send you a bill. You consume first, pay later.

Conversely, a prepaid internet plan mirrors an annuity due. You load credit before the service begins. If you want internet access for the month, you pay upfront

This article introduces the accumulation function and effective interest rates through concrete examples, showing how we measure investment growth over time. Breaking down total growth into period-by-period factors allows us to better understand how an investment performs and compare growth rates across different time periods.

Starting with Numbers

Let me define a simple amount function at different time periods:

$$A(0) = 100, \quad A(1) = 110, \quad A(2) = 121, \quad A(3) = 133.1, \quad A(4) = 146.41$$

What’s the growth factor between $t=0$ and $t=1$? What number do you need to multiply $A(0)$ by to get to $A(1)$?

As a health-conscious person, you’re interested in tracking your weight. You decide to step on the scale every day and record the result. After three days, your data might look like this:

$$w(1) = 150, \quad w(2) = 151, \quad w(3) = 149$$

where the numbers in parentheses represent the day number, and the values represent your weight in pounds.

This creates a weight function, denoted $w(d)$. This function takes a day number $d$ as input and returns your weight on that particular day as output. In other words, $w(d)$ tells you your total weight on day $d$.

When learning hypothesis testing, have you ever wondered why sometimes we use one-tail tests instead of two-tail tests? One of the biggest sources of confusion is figuring out which one to use. Let me show you what goes wrong when you pick the wrong test.

Let’s say we have two classes of students and we compute their mean GPA. Our question is: Is there any significant difference in mean GPA between class A and class B? Notice the key word here is “difference.” We want to know if they’re different, not specifically if one is higher than the other. We’ll test this at the 5 percent significance level.

Consider the function $f: \mathbb{N} \to \mathbb{R}$ where

$$ f(x) = x^2 $$

This is a simple quadratic equation you’re acquainted with, but note that our domain are only natural numbers, meaning $1, 2, 3, 4, \ldots$. The graph of this function looks like this:

But this may not be the graph of the most common version of this quadratic. Indeed, it’s more commonly defined as a function from $f: \mathbb{R} \to \mathbb{R}$. While the natural numbers are said to be countable, the real numbers are uncountable.