Traps in Statistics

Writing $\sigma^2 = \frac{\sum x_i^2}{n}$ instead of $\sigma^2 = \frac{\sum x_i^2}{n} - \bar{x}^2$.

Why: Students remember the first term but forget the crucial subtraction of $\bar{x}^2$.

If each $x_i$ becomes $x_i + k$, variance does NOT change. Only the mean shifts by $k$.

Why: Students apply the shift to everything, including variance, since the mean changes.

Choosing the class with the highest frequency as the median class instead of the class where cumulative frequency first exceeds $N/2$.

Why: Confusing modal class (highest frequency) with median class (cumulative frequency criterion).

Reporting $\sigma$ when asked for $\sigma^2$ or vice versa. Forgetting to take the square root.

Why: Both measure spread, and students rush to write the final answer without checking which one is asked.

In JEE, variance uses $N$ in the denominator (population variance). Using $N-1$ (sample variance) gives wrong answers.

Why: Some textbooks and statistics courses use $N-1$ (Bessel's correction), but JEE uses $N$.

When all observations are multiplied by $c$, new variance = $c^2 \times$ old variance, not $c \times$ old variance.

Why: Students scale variance by $c$ instead of $c^2$, or forget to square the scaling factor.