Assam Higher Secondary Statistics cheat sheet

Primary data are collected first-hand by the investigator for the enquiry at hand; secondary data were collected earlier by someone else and are reused.

Primary collection methods: direct personal interview, indirect oral interview, information from local correspondents, questionnaires sent by post, and schedules filled by enumerators.

A census covers every unit of the population; a sample survey covers a representative part, costs less but carries sampling error.

Classification is chronological (time), spatial (place), qualitative (attributes) or quantitative (variables).

Exclusive classes exclude the upper limit (10–20, 20–30); inclusive classes include it (10–19, 20–29) and need a 0.5 adjustment at both limits before graphing.

Class mark = (lower limit + upper limit)/2; class width = upper limit − lower limit.

Relative frequency = f/Σf; percentage frequency = 100f/Σf; the last cumulative frequency equals n.

Sturges' rule gives the number of classes as k ≈ 1 + 3.322 log₁₀ n.

Histogram for continuous frequency data, frequency polygon to compare two distributions, ogive for cumulative frequencies and the median, pie chart for components of a total.

Mean = Σx/n for raw data and Σfx/Σf for a frequency distribution.

Combined mean = (n₁x̄₁ + n₂x̄₂)/(n₁ + n₂); weighted mean = Σwx/Σw.

The mean uses every observation, so it is pulled by extreme values, and it cannot be computed for open-end classes.

Median = L + [(N/2 − cf)/f] × h, where L is the lower boundary of the median class, cf the cumulative frequency before it, f its frequency and h the class width.

Q₁ = L + [(N/4 − cf)/f] × h and Q₃ = L + [(3N/4 − cf)/f] × h, using the corresponding quartile classes.

Mode = L + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h for grouped data; for raw data it is the most frequently occurring value.

Mode ≈ 3 Median − 2 Mean holds approximately for a moderately skewed unimodal distribution.

Geometric mean = antilog(Σ log x / n); harmonic mean = n/Σ(1/x); for positive values HM ≤ GM ≤ AM.

Σ(x − x̄) = 0, and Σ(x − A)² is smallest when A equals the mean.

Range = H − L; coefficient of range = (H − L)/(H + L).

Quartile deviation = (Q₃ − Q₁)/2; coefficient of quartile deviation = (Q₃ − Q₁)/(Q₃ + Q₁).

Mean deviation about A = Σ|x − A|/n, and it is least when A is the median.

σ² = Σ(x − x̄)²/n for a population and Σ(x − x̄)²/(n − 1) for a sample; σ = √σ².

Step-deviation form: σ = h√[Σfy²/N − (Σfy/N)²], where y = (x − A)/h and f are the class frequencies.

Combined variance = [n₁σ₁² + n₂σ₂² + n₁d₁² + n₂d₂²]/(n₁ + n₂), with dᵢ = x̄ᵢ − combined mean.

Coefficient of variation = (σ/x̄) × 100; of two series the one with the smaller CV is more consistent.

Variance is unaffected by a change of origin but multiplied by a² under a scale change: Var(aX + b) = a²Var(X).

r-th moment about A: μ′ᵣ = Σ(x − A)ʳ/n; central moments set A = x̄, giving μᵣ = Σ(x − x̄)ʳ/n.

μ₁ = 0 and μ₂ = σ² = μ′₂ − (μ′₁)².

μ₃ = μ′₃ − 3μ′₂μ′₁ + 2(μ′₁)³ and μ₄ = μ′₄ − 4μ′₃μ′₁ + 6μ′₂(μ′₁)² − 3(μ′₁)⁴.

β₁ = μ₃²/μ₂³ with γ₁ = √β₁ measures skewness; β₂ = μ₄/μ₂² with γ₂ = β₂ − 3 measures kurtosis.

Karl Pearson's coefficient of skewness = (Mean − Mode)/σ = 3(Mean − Median)/σ.

Bowley's coefficient = (Q₃ + Q₁ − 2Q₂)/(Q₃ − Q₁), which always lies between −1 and +1.

A symmetric distribution has μ₃ = 0 and β₁ = 0; positive skewness gives mean > median > mode, negative skewness reverses the order.

β₂ = 3 mesokurtic, β₂ > 3 leptokurtic (more peaked, heavier tails), β₂ < 3 platykurtic (flatter).

Moments about the mean are unchanged by a shift of origin and are multiplied by aʳ under the transformation X → aX.

r = Σ(x − x̄)(y − ȳ)/√[Σ(x − x̄)² · Σ(y − ȳ)²], computed in practice as [nΣxy − ΣxΣy]/√{[nΣx² − (Σx)²][nΣy² − (Σy)²]}.

r carries the sign of the covariance and lies between −1 and +1; r = 0 means no linear relation, not necessarily independence.

r is independent of any change of origin or scale in x or y, except that a negative scale factor reverses its sign.

Spearman's ρ = 1 − 6Σd²/[n(n² − 1)]; with tied ranks add (t³ − t)/12 to Σd² once for each group of t tied ranks.

Regression of y on x: ŷ = a + bx with b_yx = Σ(x − x̄)(y − ȳ)/Σ(x − x̄)² = r·σy/σx and a = ȳ − b·x̄.

b_xy = r·σx/σy, hence b_yx · b_xy = r², and r = ±√(b_yx × b_xy) with the sign taken from the coefficients.

The two regression lines intersect at (x̄, ȳ) and coincide only when |r| = 1.

Coefficient of determination r² = explained variation / total variation = 1 − residual variation / total variation.

Classical probability: P(A) = number of favourable outcomes / total number of equally likely outcomes.

P(A ∪ B) = P(A) + P(B) − P(A ∩ B); the last term vanishes for mutually exclusive events.

P(A|B) = P(A ∩ B)/P(B) and P(A ∩ B) = P(A)P(B|A); for independent events P(A ∩ B) = P(A)P(B).

Bayes' theorem: P(Aᵢ|B) = P(Aᵢ)P(B|Aᵢ) / Σⱼ P(Aⱼ)P(B|Aⱼ) over mutually exclusive, exhaustive events Aⱼ.

Binomial: P(X = r) = C(n, r)pʳqⁿ⁻ʳ with mean np and variance npq; symmetric when p = q = 0.5.

Poisson: P(X = r) = e⁻ᵐmʳ/r! with mean = variance = m; it is the limiting form of the binomial for large n and small p.

Standard normal z = (x − μ)/σ; the curve is symmetric about μ with total area 1 and mean = median = mode = μ.

Normal areas: 68.27% within μ ± σ, 95.45% within μ ± 2σ and 99.73% within μ ± 3σ.

For a discrete random variable E(X) = Σx·P(x) and Var(X) = E(X²) − [E(X)]².

Population = the complete set of units under study; sample = a subset used to estimate population parameters.

Probability designs: simple random, systematic, stratified, cluster and multistage; non-probability designs include quota and judgement sampling.

Simple random sampling gives every unit an equal chance of selection and can be drawn by lottery or random number tables, with or without replacement.

Stratified sampling splits the population into internally homogeneous strata; proportional allocation takes n_i = n·N_i/N and generally reduces sampling variance.

Systematic sampling picks every k-th unit after a random start, with k = N/n; it is unsafe when the frame has a periodic pattern.

Sampling error decreases as n increases and can be estimated; non-sampling errors (non-response, faulty measurement, wrong frame) do not shrink with n.

Standard error of the mean = σ/√n; for sampling without replacement from a finite population multiply by √[(N − n)/(N − 1)].

The sampling frame is the list from which the sample is actually drawn; a mismatch with the target population produces coverage error.

H₀ is the null hypothesis tested and H₁ the alternative; a two-tailed test splits α between the two tails, a one-tailed test places it in one tail.

Type I error = rejecting a true H₀, with probability α; Type II error = accepting a false H₀, with probability β; power = 1 − β.

Large-sample test for a mean: z = (x̄ − μ)/(σ/√n), referred to the standard normal table.

Small-sample test for a mean: t = (x̄ − μ)/(s/√n) with n − 1 degrees of freedom, used when σ is unknown.

Difference of two means: t = (x̄₁ − x̄₂)/[s√(1/n₁ + 1/n₂)], where s² is the pooled variance with n₁ + n₂ − 2 df.

χ² goodness of fit: χ² = Σ(O − E)²/E with df = k − 1, reduced by one for each parameter estimated from the data.

F = s₁²/s₂² tests equality of two variances with df (n₁ − 1, n₂ − 1); put the larger variance in the numerator for a one-tailed test.

Confidence interval for a mean = x̄ ± z·σ/√n, with z = 1.96 at 95% and z = 2.58 at 99%.

Decision rule: reject H₀ when the computed statistic exceeds the critical value or when the p-value is less than α.

A time series has four components: trend T, seasonal variation S, cyclical movement C and irregular fluctuation I.

Additive model Y = T + S + C + I; multiplicative model Y = T × S × C × I, the usual choice for economic data.

Moving averages smooth a series; the period must equal the length of the seasonal cycle, and an even period requires centring of the averages.

Least squares trend Y = a + bX with b = Σ(X − X̄)(Y − Ȳ)/Σ(X − X̄)²; coding X as deviations from the middle period gives a = Ȳ.

Seasonal index = (seasonal average / grand average) × 100, and the indices average 100 over the year.

Laspeyres price index = (Σp₁q₀/Σp₀q₀) × 100, using base-year quantities as weights; it tends to overstate the price rise.

Paasche price index = (Σp₁q₁/Σp₀q₁) × 100, using current-year quantities as weights; it tends to understate the rise.

Fisher's ideal index = √(Laspeyres × Paasche); it satisfies both the time reversal and factor reversal tests.

Base shifting: new index = (old index ÷ index of the new base period) × 100; splicing joins an old and a new series at a common overlapping period.