Statistics: Tricky Questions Solved (3)

Question

The mean of $10$ observations is $25$. If each observation is multiplied by $3$ and then $5$ is subtracted from each, what is the new mean? Then, if the standard deviation of the original data was $4$, what is the new standard deviation? This kind of “linear transformation of data” question is a CBSE Class 10/11 staple.

Solution — Step by Step

Why This Works

Mean is a linear statistic — it commutes with linear operations. Add a constant to every data point, mean increases by that constant. Multiply every data point, mean multiplies the same way.

Standard deviation measures spread around the mean. Adding a constant shifts the whole distribution but doesn’t stretch or shrink it — spread unchanged. Multiplying scales every deviation from the mean by $|a|$, so SD scales by $|a|$. The absolute value matters because SD is always non-negative even if $a$ is negative.

Alternative Method

Use the variance formula $\sigma^2 = \tfrac{1}{n}\sum (x_i - \bar{x})^2$ directly. Substitute $y_i = ax_i + b$ and $\bar{y} = a\bar{x} + b$:

$\sigma_y^2 = \tfrac{1}{n}\sum (ax_i + b - a\bar{x} - b)^2 = \tfrac{1}{n}\sum a^2(x_i - \bar{x})^2 = a^2 \sigma_x^2$

Hence $\sigma_y = |a|\sigma_x$. This derivation is the kind of result CBSE asks in 4-mark questions.

Common Mistake

Final answer: New mean $= 70$, new standard deviation $= 12$.