Assam Higher Secondary Statistics flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Primary data

Data collected firsthand by the investigator for the current enquiry through interviews, questionnaires or observation; it is original but costly and time-consuming.

Exclusive and inclusive classes

Exclusive classes exclude the upper limit (10–20, 20–30); inclusive classes include both limits (10–19, 20–29) and are converted by subtracting 0.5 from the lower and adding 0.5 to the upper limit.

Sturges' rule

Number of classes k ≈ 1 + 3.322 log₁₀ n, where n is the number of observations; used to decide the class width of a frequency distribution.

Ogive

A cumulative frequency curve plotted against class boundaries; the less-than ogive rises and the more-than ogive falls, and their intersection gives the median.

Combined arithmetic mean

For two groups of sizes n₁ and n₂ with means x̄₁ and x̄₂, the combined mean is (n₁x̄₁ + n₂x̄₂)/(n₁ + n₂); the formula extends to any number of groups.

Median of grouped data

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.

Empirical mode relation

For a moderately skewed unimodal distribution, Mode ≈ 3 Median − 2 Mean; the relation is used when the mode is ill-defined or the classes are open-ended.

Geometric mean

GM = antilog(Σ log x / n); it suits ratios, growth rates and index numbers, and for positive data the inequality HM ≤ GM ≤ AM always holds.

Coefficient of variation

CV = (standard deviation / mean) × 100; a unit-free measure used to compare variability between series, where a smaller CV indicates greater consistency.

Combined standard deviation

σ = √[(n₁σ₁² + n₂σ₂² + n₁d₁² + n₂d₂²)/(n₁ + n₂)], where d₁ = x̄₁ − x̄c and d₂ = x̄₂ − x̄c are deviations from the combined mean.

Quartile deviation

QD = (Q₃ − Q₁)/2 and its coefficient is (Q₃ − Q₁)/(Q₃ + Q₁); it ignores the extreme quarter of observations at each end of the distribution.

Variance under change of scale

Var(aX + b) = a²Var(X): adding the constant b shifts the mean but leaves the variance unchanged, while multiplying by a scales the variance by a².

Beta and gamma coefficients

β₁ = μ₃²/μ₂³ with γ₁ = √β₁ measures skewness; β₂ = μ₄/μ₂² with γ₂ = β₂ − 3 measures kurtosis, and γ₂ = 0 for the normal curve.

Leptokurtic distribution

A distribution with β₂ > 3, that is γ₂ > 0; it is more peaked with heavier tails than the normal curve, for which β₂ equals 3.

Bowley's skewness coefficient

SK = (Q₃ + Q₁ − 2Q₂)/(Q₃ − Q₁); it lies between −1 and +1 and depends only on quartiles, so it suits open-end distributions.

Product moment correlation

r = Σ(x − x̄)(y − ȳ)/√[Σ(x − x̄)²·Σ(y − ȳ)²]; it ranges from −1 to +1 and measures only the strength of linear association, not causation.

Spearman's rank correlation

ρ = 1 − 6Σd²/[n(n² − 1)], where d is the difference between the ranks of paired values; used for ranked or qualitative data.

Regression coefficients property

b_yx × b_xy = r²; both regression coefficients carry the sign of r, so r = ±√(b_yx × b_xy) with the sign taken from them.

Two lines of regression

The line of y on x is y − ȳ = b_yx(x − x̄) and the line of x on y is x − x̄ = b_xy(y − ȳ); both pass through the point (x̄, ȳ).

Bayes' theorem

P(Aᵢ|B) = P(Aᵢ)P(B|Aᵢ)/Σ P(Aⱼ)P(B|Aⱼ); it revises prior probabilities of mutually exclusive, exhaustive events once the event B is observed.

Binomial mean and variance

For X ~ B(n, p): mean = np and variance = npq with q = 1 − p; the distribution is symmetric when p equals 0.5.

Poisson distribution

P(X = r) = e⁻ᵐmʳ/r! for r = 0, 1, 2, …; mean and variance both equal m, and it models rare events in a fixed interval.

Normal curve areas

About 68.27% of a normal distribution lies within μ ± σ, 95.45% within μ ± 2σ and 99.73% within μ ± 3σ; the curve is symmetric with total area one.

Stratified sampling

The population is split into non-overlapping homogeneous strata and a sample drawn from each; proportional allocation uses n_i = n·N_i/N and reduces sampling variance.

Standard error of mean

SE(x̄) = σ/√n for large samples; for a finite population sampled without replacement multiply by the finite population correction √[(N − n)/(N − 1)].

Systematic sampling

Units are chosen at every k-th position after a random start, with k = N/n; easy to operate, but risky when the list has a periodic pattern.

Type I and Type II errors

A Type I error rejects a true null hypothesis and has probability α; a Type II error accepts a false null with probability β, so power equals 1 − β.

t statistic for small samples

t = (x̄ − μ)/(s/√n) with n − 1 degrees of freedom, where s is the sample standard deviation; used when σ is unknown and the sample is small.

Chi-square goodness of fit

χ² = Σ(O − E)²/E, compared with table values at degrees of freedom k − 1 minus the number of parameters estimated from the observed data.

Fisher's ideal index

Fisher's index = √(Laspeyres × Paasche); it satisfies both the time reversal and factor reversal tests but requires quantities for both periods.

Multiplicative time series model

Y = T × S × C × I, where trend, seasonal, cyclical and irregular components multiply; seasonal indices are percentages averaging 100 within a year.

Paasche price index

P = (Σp₁q₁/Σp₀q₁) × 100; it uses current-year quantities as weights and usually gives a lower value than the Laspeyres price index.