NIMCET 2025 — Mathematics PYQ

The correct option (the statement that is NOT correct) is (B).

Here is the step-by-step breakdown and verification of each statement:

Analysis of Option (B) - Incorrect Statement

• Statement: "The variance is expressed in the same units as the units of observation."

• Explanation: Variance is calculated by squaring the deviations of individual data points from the mean. Mathematically, if the observations $x_i$ have a unit of measurement (e.g., meters, $\text{m}$), their variance is given by:

  $${\sigma }^2=\frac{1}{N}\sum _{i=1}^N(x_i-\bar{x}{)}^2$$

• Because the terms are squared, the units of variance become the square of the units of observation (e.g., square meters, ${\text{m}}^2$). Therefore, this statement is false, making it the right choice for this question.

Verification of Other Options (Correct Statements)

Option (A)

• Statement: "The standard deviation is greater than or equal to the mean deviation (about mean)"

• Explanation: By mathematical inequality (specifically the Cauchy-Schwarz inequality), Root Mean Square Deviation (Standard Deviation) is always greater than or equal to Mean Absolute Deviation.

  $$\text{Standard Deviation }(\sigma )\ge \text{Mean Deviation}$$

• Thus, this statement is mathematically correct.

Option (C)

• Statement: "The value of standard deviation changes by a change of scale."

• Explanation: Measures of dispersion like standard deviation are independent of a change of origin (adding/subtracting a constant) but are dependent on a change of scale (multiplying/dividing by a constant). If every observation is multiplied by a constant $k$, the new standard deviation becomes $|k|\sigma$.

• Thus, this statement is correct.

Option (D)

• Statement: "The sum of squares of deviations is minimum when taken from the mean"

• Explanation: This is a fundamental algebraic property of the arithmetic mean. If we take deviations from any arbitrary value $A$, the sum of squares $\sum (x_i-A{)}^2$ reaches its absolute minimum value precisely when $A=\bar{x}$ (the arithmetic mean).

• Thus, this statement is correct.

Correct Answer:

The statement in Option (B) is NOT correct.