The standard deviation of a data is ______ of any change in orgin, but is ___ on the change of scale.

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Q. The standard deviation of a data is ___________ of any change in orgin, but is _____ on the change of scale.

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Concept understanding — Effect of Scaling Variance

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:

The mean also gets scaled by $k$ (because mean is linear).
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 $k^2$ .

Note

Scaling by $k$ multiplies the variance by $k^2$ , not by $k$ . This is because variance is in squared units of the original data.

The precise statement

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

\text{Var}(kX) = k^2 \cdot \text{Var}(X)

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:

\text{Var}(X + c) = \text{Var}(X)

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

\text{Var}(Y) = a^2 \cdot \text{Var}(X)

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 $0.01$ .

\text{Var}(\text{height in m}) = (0.01)^2 \times 25 = 0.0001 \times 25 = 0.0025

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

Watch out

Do not say “variance gets multiplied by $k$ ”. It’s $k^2$ . …

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Standard deviation is independent of any change in origin (shifting all values by a constant) but dependent on the change of scale (multiplying all values by a constant). The blanks are filled with "independent" and "dependent" respectively.

Understanding How Transformations Affect Standard Deviation

Standard deviation measures the spread of data around the mean. To understand how it responds to transformations, we need to think about what happens when we shift or stretch the entire dataset.

When we add a constant to every observation (change of origin), we're simply sliding the entire distribution along the number line. The distances between points don't change, so the spread remains identical. But when we multiply every observation by a constant (change of scale), we're stretching or compressing the distribution, which directly affects how far points lie from their mean.

Let's work through this systematically.

Change of Origin (Adding a Constant)

Start with the original data: Suppose we have observations $x_1, x_2, \ldots, x_n$ with mean $\bar{x}$ and standard deviation $\sigma$ .
Apply the transformation: Create new data $y_i = x_i + a$ for some constant $a$ . The new mean becomes:

\bar{y} = \frac{1}{n}\sum_{i=1}^{n}(x_i + a) = \bar{x} + a

Calculate the new variance:

\text{Var}(y) = \frac{1}{n}\sum_{i=1}^{n}(y_i - \bar{y})^2 = \frac{1}{n}\sum_{i=1}^{n}[(x_i + a) - (\bar{x} + a)]^2

The $a$ terms cancel:

= \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2 = \text{Var}(x)

Conclude for standard deviation: Since $\text{SD}(y) = \sqrt{\text{Var}(y)} = \sqrt{\text{Var}(x)} = \sigma$ , the standard deviation is unchanged.

Note

Adding the same amount to every data point shifts the mean but preserves all the distances between points and the mean, leaving the spread untouched.

Change of Scale (Multiplying by a Constant)

Apply the scaling transformation: Create new data $z_i = b \cdot x_i$ for some constant $b \neq 0$ . The new mean is: $\bar{z} = \frac{1}{n}\sum_{i=1}^{n}(b \cdot x_i) = b\bar{x}$ …

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Q.The standard deviation of a data is ___________ of any change in orgin, but is _____ on the change of scale.

Understanding How Transformations Affect Standard Deviation

Change of Origin (Adding a Constant)

Change of Scale (Multiplying by a Constant)

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Q.The standard deviation of a data is _______ of any change in orgin, but is _ on the change of scale.

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