Jaslyn King

When you’re working with data, one of the first questions you end up asking is, “How far do these values drift from the center?”

Variance is one way to answer that, and the part that often feels unintuitive is the decision to square every deviation from the mean.

Here’s a grounded way to think about why this is useful.

Why we need something more than raw distances

If you subtract the mean from each value, you immediately run into a problem:

Positive and negative distances cancel each other out.

Not because the values are close together, but because the math collapses them.

So the real question becomes:

How do we turn all distances into a meaningful measure of spread?

Squaring solves this in a clean, practical way:

Everything becomes positive
Larger deviations naturally stand out
And importantly, squared functions behave smoothly — something modeling algorithms rely on

It’s less about elegance and more about stability. Squaring creates a version of “distance” that math can actually work with.

What about simpler alternatives?

You might wonder why we don’t just use the average of absolute deviations.

That measure is intuitive and easy to interpret.

The limitation is behind the scenes: absolute values introduce sharp corners in the math. They’re harder to optimize, especially for the calculus-based methods that power things like regression.

Squared deviations avoid that issue. They give us a surface that’s continuous and predictable — which is exactly what statistical modeling needs.

What squaring reveals

Squared deviations also show up naturally across different fields:

In physics, they’re tied to energy and power
In machine learning, loss functions often rely on squared error
In ANOVA, variation splits cleanly because of how squares behave

It’s not just a statistical quirk — squaring creates a structure that a lot of mathematical tools depend on.

And when you take the square root, you arrive at the standard deviation — the version people actually use because it sits back in familiar units.

Why this matters for people working with real data

You don’t need to love math to appreciate the idea.

Variance gives you a sense of:

How steady (or erratic) a process is
How much noise sits around your signals
whether changes in the data actually mean something

Squared measures aren’t there to complicate things — they’re there to give you a stable foundation. When variation is captured cleanly, everything from forecasting to troubleshooting becomes easier to reason about.

Takeaway

A simple way to remember this:

Variance explains how stability is built
Standard deviation gives you the human-friendly version of that story

Understanding why we square is really understanding how modern analytics make sense of variation. Once that clicks, a lot of other pieces fall into place.