What Happens to the Mean and Standard Deviation If You Multiply Every Data Point by a Constant?

If every data point is multiplied by a constant $c$, the mean is multiplied by $c$, while the standard deviation is multiplied by (|c|). Therefore, if the original mean and standard deviation are $\mu$ and $\sigma$, the transformed values are $c\mu$ and $|c|\sigma$, respectively.

Why This Happens

Let the original data values be $x_1,x_2,\ldots,x_n$, and define each transformed value by $$. The transformed mean is

$$\bar{y}=\frac{\sum_{i=1}^{n}cx_i}{n}=c\frac{\sum_{i=1}^{n}x_i}{n}=c\bar{x}.$$

The constant factors out of the sum, so the centre of the data is scaled by $c$. Each deviation from the transformed mean satisfies

$$y_i-\bar{y}=cx_i-c\bar{x}=c(x_i-\bar{x}).$$

The deviations are therefore multiplied by $c$. Because variance uses squared deviations, it is multiplied by $c^2$:

$$\operatorname{Var}(Y)=c^2\operatorname{Var}(X)=c^2\sigma^2.$$

Taking the positive square root gives

$$\sigma_{\mathrm{new}}=\sqrt{c^2\sigma^2}=|c|\sigma.$$

The absolute value is essential because a standard deviation measures spread and cannot be negative.

Worked Example

The data $2,4,6$ have mean $4$ and population standard deviation $\sqrt{\frac{8}{3}}$. Multiplying every value by gives . The new mean is

$$-3(4)=-12,$$

and the new standard deviation is

$$|-3|\sqrt{\frac{8}{3}}=3\sqrt{\frac{8}{3}}=2\sqrt{6}.$$

The negative multiplier reverses the order and signs of the values, but the spread is multiplied by the positive factor $3$.

More generally, a linear transformation can also include addition: $Y=cX+d$. Adding $d$ shifts every value equally, so it changes the mean by $d$ but leaves all deviations, the variance, and the standard deviation unchanged. Consequently, $E(Y)=cE(X)+d$, while the variance and standard deviation still depend only on the multiplier . This distinguishes a shift from a scale change in exam questions.

Exam Technique

This result is assessed in IB Mathematics AA, SL 4.3, and may appear on Paper 1 or Paper 2. Show the transformation rule and distinguish between variance and standard deviation: variance uses $c^2$, whereas standard deviation uses (|c|). The common misconception is writing $c\sigma$ when $c<0$. At SL, treat the data set as the population unless the question states otherwise.