Q.The variance of 20 observations is 5. If each observation is multiplied by 2, find the new variance 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

Why this matters

This rule is used everywhere:

• Standardisation (z-scores): you subtract the mean and divide by the standard deviation. The variance of the result becomes 1.
• Units conversion: changing from cm to m, or rupees to lakhs.
• Understanding regression coefficients: if you rescale a predictor, its coefficient changes, but the model’s predictions don’t — because variance scales accordingly.

One-line summary

Scaling a variable by $k$ multiplies its variance by $k^2$; adding a constant does nothing to variance.

The Effect of Scaling on Variance is a key property covered in the NCERT Class 11 Mathematics chapter on Statistics, matching searches like "effect of scale change on variance formula" or "statistics important questions class 11 maths". This k-squared scaling rule is also the foundation behind standardisation (z-scores) and shows up in JEE Main and other competitive exam statistics questions.