CHSE Odisha Elective Statistics cheat sheet

Arithmetic mean: (\bar{x} = \frac{1}{n}\sum x_i); for grouped data (\bar{x} = \frac{\sum f_i x_i}{\sum f_i}).

Median: for ungrouped data, middle value after sorting; for grouped data, (L + \frac{(N/2 - F)}{f} \times h), where (L) = lower boundary of median class, (F) = cumulative frequency before median class, (f) = frequency of median class, (h) = class width.

Mode: most frequent value; for grouped data, (L + \frac{(f_m - f_1)}{(2f_m - f_1 - f_2)} \times h), where (f_m) = modal frequency, (f_1) and (f_2) = frequencies of preceding and following classes.

Range = max − min; quartile deviation = ((Q_3 - Q_1)/2); coefficient of quartile deviation = ((Q_3 - Q_1)/(Q_3 + Q_1)).

Mean absolute deviation = (\frac{1}{n}\sum |x_i - A|) where (A) is mean, median, or mode; it is minimum when (A) is the median.

Sample variance (s^2 = \frac{1}{n-1}\sum (x_i - \bar{x})^2); population variance (\sigma^2 = \frac{1}{N}\sum (X_i - \mu)^2).

Coefficient of variation (CV) = (\frac{s}{\bar{x}} \times 100%); used to compare dispersion between data sets with different units or means.

For grouped data, variance (s^2 = \frac{1}{n-1}\left[\sum f_i x_i^2 - \frac{(\sum f_i x_i)^2}{n}\right]); computational formula avoids rounding errors.

Empirical relation for moderately skewed data: mean − mode ≈ 3(mean − median); if mean > median > mode, distribution is positively skewed.

Karl Pearson correlation coefficient: (r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}); (-1 \le r \le 1).

Spearman rank correlation: (r_s = 1 - \frac{6\sum d_i^2}{n(n^2 - 1)}), where (d_i) = difference in ranks; use when data are ordinal or when Pearson assumptions fail.

Regression line of (y) on (x): (\hat{y} = a + bx), with (b = r\frac{s_y}{s_x}) and (a = \bar{y} - b\bar{x}).

Regression line of (x) on (y): (\hat{x} = c + dy), with (d = r\frac{s_x}{s_y}) and (c = \bar{x} - d\bar{y}).

Relationship: (r = \pm\sqrt{b_{yx} \cdot b_{xy}}), where (b_{yx}) and (b_{xy}) are the two regression coefficients; sign of (r) matches sign of both slopes.

Coefficient of determination (r^2) = proportion of variance in (y) explained by (x) in linear regression; (0 \le r^2 \le 1).

Correlation does not imply causation; (r) measures only linear association and is sensitive to outliers.

If (r = 0), the variables are uncorrelated linearly, but they may still have a nonlinear relationship.

Axioms: (P(S) = 1), (P(A) \ge 0), and for mutually exclusive events (P(\cup A_i) = \sum P(A_i)).

Addition rule: (P(A \cup B) = P(A) + P(B) - P(A \cap B)); if mutually exclusive, (P(A \cap B) = 0).

Conditional probability: (P(A|B) = \frac{P(A \cap B)}{P(B)}), provided (P(B) > 0).

Multiplication rule: (P(A \cap B) = P(A|B)P(B) = P(B|A)P(A)); independent events satisfy (P(A \cap B) = P(A)P(B)).

Bayes' theorem: (P(A_i|B) = \frac{P(B|A_i)P(A_i)}{\sum_j P(B|A_j)P(A_j)}), for a partition ({A_j}).

Discrete random variable: probability mass function (p(x) = P(X = x)) with (p(x) \ge 0) and (\sum p(x) = 1).

Continuous random variable: probability density function (f(x) \ge 0), (\int_{-\infty}^{\infty} f(x),dx = 1), and (P(a \le X \le b) = \int_a^b f(x),dx).

Expected value (E(X) = \sum x,p(x)) or (\int x f(x),dx); variance (\operatorname{Var}(X) = E(X^2) - [E(X)]^2).

For constants (a, b): (E(aX + b) = aE(X) + b) and (\operatorname{Var}(aX + b) = a^2 \operatorname{Var}(X)).

Binomial (B(n, p)): (P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}), mean (np), variance (np(1-p)); conditions: fixed (n), independent trials, constant (p).

Poisson (P(\lambda)): (P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}), mean = variance = (\lambda); limit of binomial when (n) large, (p) small, (np = \lambda).

Normal (N(\mu, \sigma^2)): density (f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-(x-\mu)^2/(2\sigma^2)}); standardize (Z = \frac{X - \mu}{\sigma}).

Standard normal table gives (\Phi(z) = P(Z \le z)); by symmetry (P(Z \le -z) = 1 - \Phi(z)).

Uniform (U(a, b)): density (f(x) = \frac{1}{b-a}) on ([a, b]), mean ((a+b)/2), variance ((b-a)^2/12).

Exponential with mean (\theta): (f(x) = \frac{1}{\theta} e^{-x/\theta}) for (x > 0), variance (\theta^2), memoryless property (P(X > s+t | X > s) = P(X > t)).

Negative binomial counts number of trials to get (r) successes; geometric distribution is the special case (r = 1).

Hypergeometric models sampling without replacement from a finite population: (P(X = k) = \frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}).

Simple random sampling (SRS): every sample of size (n) has equal probability (1/\binom{N}{n}); sampling may be with or without replacement.

Sampling distribution of (\bar{X}): mean (\mu_{\bar{X}} = \mu); standard error (\sigma_{\bar{X}} = \sigma/\sqrt{n}) for infinite population or with replacement.

Finite population correction (fpc): (\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} \sqrt{\frac{N-n}{N-1}}) for sampling without replacement.

Central limit theorem: for large (n) (typically (n \ge 30)), (\bar{X}) is approximately normal regardless of population shape.

If population is normal, (\bar{X} \sim N(\mu, \sigma^2/n)) exactly for any (n).

Sample proportion (\hat{p}): mean (p), standard error (\sqrt{p(1-p)/n}); normal approximation valid when (np \ge 5) and (n(1-p) \ge 5).

Sample variance (S^2) from a normal population: (\frac{(n-1)S^2}{\sigma^2} \sim \chi^2_{n-1}).

Stratified sampling: divide population into homogeneous strata, sample within each; reduces variance compared with SRS when strata are internally homogeneous.

Systematic sampling selects every (k)-th unit after a random start; (k = N/n) is the sampling interval.

Null hypothesis (H_0) states no effect or no difference; alternative (H_1) is one-sided or two-sided.

Type I error: rejecting (H_0) when true, probability (\alpha); Type II error: failing to reject (H_0) when false, probability (\beta); power = (1 - \beta).

Test statistic for mean, (\sigma) known or large (n): (z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}).

Test statistic for mean, (\sigma) unknown and small (n): (t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}) with (df = n-1).

Two-sample (t)-test (pooled): (t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{1/n_1 + 1/n_2}}), (s_p^2 = \frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1 + n_2 - 2}), (df = n_1 + n_2 - 2).

Paired (t)-test: compute differences (d_i), then (t = \frac{\bar{d}}{s_d/\sqrt{n}}) with (df = n-1).

Test for proportion: (z = \frac{\hat{p} - p_0}{\sqrt{p_0(1-p_0)/n}}) under (H_0).

Chi-square goodness-of-fit: (\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}) with (df = k - 1 - m), where (m) = number of estimated parameters.

Confidence interval for mean: (\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}) (large sample) or (\bar{x} \pm t_{\alpha/2, n-1} \frac{s}{\sqrt{n}}) (small sample).

Principles of experimental design: replication, randomization, and local control (blocking).

Completely randomized design (CRD): treatments assigned randomly to all experimental units; one-way ANOVA tests equality of (k) treatment means.

One-way ANOVA: total SS = between-group SS + within-group SS; (F = \frac{MS_{\text{between}}}{MS_{\text{within}}}) with (df_1 = k-1), (df_2 = N-k).

Between-group SS = (\sum n_i (\bar{x}_i - \bar{x})^2); within-group SS = (\sum (n_i - 1)s_i^2).

Randomized block design (RBD): each treatment appears once in each block; two-way ANOVA without interaction, with (df_{\text{treat}} = k-1), (df_{\text{block}} = b-1), (df_{\text{error}} = (k-1)(b-1)).

Latin square design: controls two nuisance factors; requires number of treatments = number of rows = number of columns.

Factorial experiment: studies effects of two or more factors simultaneously; (2^2) factorial has 4 treatment combinations and can estimate main effects and interaction.

ANOVA assumptions: independent observations, normal populations, and equal variances (homoscedasticity).

If (F) is significant in ANOVA, post-hoc tests (e.g., Tukey's HSD) identify which means differ.

Components of a time series: trend ((T)), seasonal ((S)), cyclical ((C)), and irregular ((I)) variations.

Multiplicative model: (Y = T \times S \times C \times I); additive model: (Y = T + S + C + I).

Moving average method: averages successive (k) observations; for even (k), use centered moving averages to align with time periods.

Seasonal index = (original value / centered moving average) × 100; average seasonal indices over identical periods to remove irregular variation.

Laspeyres price index: (P_{L} = \frac{\sum p_{1} q_{0}}{\sum p_{0} q_{0}} \times 100), using base-year quantities.

Paasche price index: (P_{P} = \frac{\sum p_{1} q_{1}}{\sum p_{0} q_{1}} \times 100), using current-year quantities.

Fisher's ideal index: (P_{F} = \sqrt{P_{L} \times P_{P}}); satisfies both time reversal and factor reversal tests.

Time reversal test: (P_{01} \times P_{10} = 1); factor reversal test: (P_{01} \times Q_{01} = \frac{\sum p_{1} q_{1}}{\sum p_{0} q_{0}}).

Chain index: links successive period indices; chain index for period (n) = product of link relatives divided by 100 raised to appropriate power.