Statistics: Mean, Variance & Standard Deviation

Statistics in JEE Main focuses on measures of central tendency (mean, median, mode) and measures of dispersion (range, mean deviation, variance, standard deviation). The emphasis is heavily on variance and standard deviation problems requiring algebraic manipulation.

Measures of Central Tendency:

Arithmetic Mean (AM): For raw data x1, x2, ..., xn: Mean (x-bar) = (sum of xi)/n. For grouped frequency data: x-bar = (sum of fixi)/(sum of fi), where fi is frequency and xi is class mark. Three methods: direct, assumed mean (x-bar = A + sum(fidi)/N where di = xi - A), and step deviation (x-bar = A + h * sum(fi*ui)/N where ui = (xi - A)/h).

Median: The middle value when data is arranged in ascending order. For n observations: if n is odd, median = ((n+1)/2)th value; if n is even, median = average of (n/2)th and (n/2 + 1)th values. For grouped data: Median = l + [(N/2 - cf)/f] * h, where l = lower limit of median class, cf = cumulative frequency before median class, f = frequency of median class, h = class width.

Mode: The most frequently occurring value. For grouped data: Mode = l + [(f1 - f0)/(2f1 - f0 - f2)] * h, where f1 = frequency of modal class, f0 = preceding class frequency, f2 = succeeding class frequency.

Measures of Dispersion:

Range: Maximum value - Minimum value. Simplest but least reliable measure.

Mean Deviation: MD about mean = (sum of |xi - x-bar|)/n. MD about median = (sum of |xi - M|)/n. Mean deviation about median is minimum among all MD values.

Variance ($\sigma$²): The average of squared deviations from the mean.

• For raw data: $\sigma$² = (sum of (xi - x-bar)²)/n = (sum of $xi^{2}$)/n - (x-bar)²
• For frequency data: $\sigma$² = (sum of fi*(xi - x-bar)²)/N = (sum of fi*$xi^{2}$)/N - (x-bar)²

The second formula (sum of squares/n minus square of mean) is the computational shorthand: Var(X) = E($X^{2}$) - [E(X)]².

Standard Deviation ($\sigma$): $\sigma$ = $\sqrt{variance}$. It has the same unit as the data.

Properties of Variance Under Transformations:

• If yi = xi + a (shift), then Var(Y) = Var(X), $\sigma_{Y}$ = $\sigma_{X}$. Adding a constant does not change spread.
• If yi = b*xi (scale), then Var(Y) = $b^{2}$ * Var(X), $\sigma_{Y}$ = |b| * $\sigma_{X}$.
• If yi = a + bxi, then Var(Y) = $b^{2}$ * Var(X), Mean(Y) = a + bMean(X).

Combined Mean and Variance: For two groups of sizes n1, n2 with means x-bar1, x-bar2 and variances $s1^{2}$, $s2^{2}$:

• Combined mean: x-bar = (n1x-bar1 + n2x-bar2)/(n1 + n2)
• Combined variance: $s^{2}$ = [n1($s1^{2}$ + $d1^{2}$) + n2($s2^{2}$ + $d2^{2}$)]/(n1+n2) where d1 = x-bar1 - x-bar, d2 = x-bar2 - x-bar.

Coefficient of Variation (CV): CV = ($\sigma$/x-bar) * 100%. Lower CV means more consistent data. Used to compare variability of datasets with different units or means.

The key problem-solving concept is mastering the computational formula for variance: $\sigma$² = (sum $xi^{2}$)/n - (mean)², and understanding how linear transformations affect mean and variance. JEE problems frequently provide partial information (like sum of xi, sum of $xi^{2}$, and one value is changed) and ask for the new variance.