Solution

Question: Which one of the following is NOT a correct statement?

1. The value of standard deviation changes by a change of scale.
2. The standard deviation is greater than or equal to the mean deviation (about mean).
3. The sum of squares of deviations is minimum when taken from the mean.
4. The variance is expressed in the same units as the units of observation.

Answer: Option 4

Why: Variance has squared units, not the same units as the data (e.g., if data are in cm, variance is in cm²). Standard deviation (the square root of variance) has the same units as the data.

Notes:

• Change of scale: if each value is multiplied by \(k\), then \(\text{SD}\) becomes \(|k|\cdot \text{SD}\) and \(\text{Var}\) becomes \(k^2\cdot \text{Var}\).
• \(\text{SD} \ge \text{Mean Deviation (about mean)}\) is a standard inequality.
• \(\sum (x_i - a)^2\) is minimized at \(a = \bar{x}\) (the mean).