Q.The mean and standard deviation of six observations are 8 and 4, respectively. If each observation is multiplied by 3, find the new mean and new standard deviation of the resulting observations.

Effect of Scaling Variance – First Encounter

Let’s start with a simple question: what happens to the spread of a dataset if you multiply every number by 2? Or by 0.5? Or by –3?

If you have a set of marks: 40, 50, 60, 70, 80, the variance is some number. Now imagine the teacher doubles every mark: 80, 100, 120, 140, 160. The marks are now twice as far apart from each other. The spread has clearly changed — but by how much?

That’s exactly what the effect of scaling tells you: a precise rule for how variance changes when you multiply (or divide) every observation by a constant.

Intuition first

Variance measures average squared distance from the mean. If you scale all values by a factor $k$, two things happen:

1. The mean also gets scaled by $k$ (because mean is linear).
2. Each individual distance from the mean also gets scaled by $k$.

But variance squares those distances. So if each distance is multiplied by $k$, each squared distance is multiplied by $k^2$. The average of those squared distances — the variance — therefore gets multiplied by $k^2$.

The precise statement

Let $X$ be a random variable (or a dataset) with variance $\text{Var}(X)$. Let $k$ be any real constant. Then:

That’s it. No extra terms, no dependence on the mean. Just $k^2$ times the original variance.

What about adding a constant?

This is a separate but related idea: if you add a constant $c$ to every value, the spread doesn’t change — the whole distribution just shifts. So:

Combining both: for a linear transformation $Y = aX + b$,

The constant $b$ has no effect on variance.

A quick example

Suppose the variance of heights (in cm) is 25. Convert to metres: divide by 100, i.e., multiply by $0.01$.

That’s a tiny number — but it’s correct, because metres are a larger unit, so the spread in metres is much smaller numerically.

Common mistake to avoid