AlgebraHigh Yield (1-2 Qs in JEE)

Statistics Formulas

Mean, variance, standard deviation, mean deviation, and effect of linear change of origin and scale.

8 Formulas•332+ Questions in Bank

Core Formulas & Mathematical Identities

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

Measures of central tendency and dispersion.

Examiner Tip: Variance = (Mean of squares) - (Square of mean).

y_i = a x_i + b \implies \bar{y} = a\bar{x} + b, \quad \sigma_y^2 = a^2 \sigma_x^2, \quad \sigma_y = |a| \sigma_x

Shift of origin (b) does not affect variance or standard deviation.

Examiner Tip: Adding constant to every observation leaves variance completely unchanged.

\bar{x}_{12} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}, \quad \sigma_{12}^2 = \frac{n_1(\sigma_1^2 + d_1^2) + n_2(\sigma_2^2 + d_2^2)}{n_1 + n_2} \quad (d_i = \bar{x}_i - \bar{x}_{12})

Pooled variance of two combined statistical groups.

Examiner Tip: Standard deviation cannot be simply averaged.

MD(\bar{x}) = \frac{\sum |x_i - \bar{x}|}{n}, \quad MD(M) = \frac{\sum |x_i - M|}{n}

Mean absolute dispersion of observations from central tendency. Mean deviation is strictly MINIMUM when measured about the Median.

Examiner Tip: Coefficient of MD = (Mean Deviation) / (Mean or Median).

\text{C.V.} = \frac{\sigma}{\bar{x}} \times 100\%

Relative measure of dispersion used to compare consistency between two datasets.

Examiner Tip: The series or team with smaller C.V. is considered more consistent and uniform.

\bar{x} = \frac{n + 1}{2}, \quad \sigma^2 = \frac{n^2 - 1}{12}, \quad \sigma = \sqrt{\frac{n^2 - 1}{12}}

Direct closed-form results for arithmetic mean and variance of first n positive integers {1, 2, ..., n}.

Examiner Tip: Frequently asked direct question in JEE Main (e.g. variance of first 50 natural numbers is (2500 - 1)/12 = 208.25).

\text{Mode} = 3\,\text{Median} - 2\,\text{Mean} \iff \text{Mean} - \text{Mode} = 3(\text{Mean} - \text{Median})

Karl Pearson's empirical formula connecting mean, median, and mode for moderately skewed unimodal frequency distributions.

Examiner Tip: For perfectly symmetrical distribution, Mean = Median = Mode.

\text{Even } \{2, 4, \dots, 2n\}: \bar{x} = n + 1, \; \sigma^2 = \frac{n^2 - 1}{3}; \quad \text{Odd } \{1, 3, \dots, 2n-1\}: \bar{x} = n, \; \sigma^2 = \frac{n^2 - 1}{3}

Variance of first n consecutive even and odd positive integers.

Examiner Tip: Notice that variance of first n even numbers and first n odd numbers is identical: (n² - 1)/3.

Frequently tested misconceptions in Statistics

•Variance is affected by SCALE (multiplying by k multiplies variance by k²), but UNCHANGED by change of ORIGIN (adding a constant).
•Standard deviation is √Variance (always non-negative).