Mean, Variance & SDJEE Main

Spread of a data set — interactive Mathematics simulation for IIT-JEE.

Concept

Once you have a data set, two questions matter: where is its centre and how spread out is it? The mean locates the centre; the variance and standard deviation measure dispersion about that centre.

Key formula

\bar x = \frac{1}{n}\sum x_i, \qquad \sigma^2 = \frac{1}{n}\sum (x_i-\bar x)^2, \qquad \sigma = \sqrt{\sigma^2}

Derivation

The mean $\bar x$ is the balance point of the data. Each deviation $x_i-\bar x$ tells how far a point sits from it.

Squaring removes the sign (deviations would otherwise sum to zero) and averaging gives the variance $\sigma^2$ . Its square root, the standard deviation $\sigma$ , returns to the original units.

Scenarios to explore

Mean, Variance & SD — Centre and spread of a data set.

Real-world applications

Comparing the consistency of two data sets with equal means.
Standardizing scores (z-scores) in statistics.
Risk and error analysis in physics experiments.

JEE exam tips

Shortcut: σ² = (Σx²)/n − x̄².
Adding a constant to every value leaves σ unchanged; scaling by k scales σ by |k|.
Variance is never negative; it is zero only if all values are equal.

Common mistakes

Forgetting to square the deviations before averaging.
Mixing up variance (squared units) and SD (original units).
Using ÷(n−1) when the population formula ÷n is wanted (or vice-versa).

Exam traps to avoid

Mean deviation uses absolute values, not squares — different from SD.
JEE usually means the population variance (÷n) unless 'sample' is stated.