Karnataka II PUC Statistics (KSEAB) cheat sheet

Statistics as a science deals with collection, organisation, presentation, analysis and interpretation of numerical data; as a singular noun it means statistical data themselves.

Scope includes descriptive statistics (summarising data) and inferential statistics (generalising from sample to population); it is widely applied in economics, business, demography, medicine and education.

Limitations: statistics handles only quantitative or quantifiable facts, studies aggregates not individuals, and results are true only on average; conclusions can be misused without context.

Primary data are collected first-hand by the investigator; secondary data are already collected by someone else and reused.

Direct personal interview, indirect oral interview, information from correspondents, questionnaires mailed to informants, and schedules filled by enumerators are the main primary-data methods.

A pilot survey (pre-test) on a small group detects ambiguous questions before the main survey; it is a standard step in questionnaire design.

Census method studies every unit of the population; sample method studies a representative part - census is costlier but free of sampling error.

Sampling and non-sampling errors: sampling error arises from studying only a part; non-sampling errors include bias in frame, non-response, measurement and recording mistakes.

A pilot or pre-test also helps estimate variance for fixing sample size; a good questionnaire uses short, unambiguous, non-leading questions and avoids double-barrelled items.

Classification arranges data into homogeneous groups; bases are geographical (place), chronological (time), qualitative (attributes) and quantitative (class intervals).

In exclusive (continuous) classes the upper limit is excluded, e.g. 10–20, 20–30; in inclusive classes both limits are included, e.g. 10–19, 20–29, needing a correction factor.

Class width = upper limit − lower limit; class mark (mid-point) = (L + U)/2; relative frequency = frequency / total frequency.

A frequency distribution may be discrete (ungrouped) or continuous (grouped); cumulative frequency gives 'less than' or 'more than' type ogives.

Tabulation presents data in rows and columns; parts of a table include table number, title, captions (column heads), stubs (row heads), body, and source note.

A good table has a clear title, self-explanatory headings, consistent units, and no unnecessary lines or ditto marks in the body.

Diagrams: bar charts compare categories; pie charts show shares of a whole (angle = component/total × 360°); pictograms use symbols.

Graphs: histogram for continuous frequency distributions, frequency polygon by joining mid-points, and ogive (cumulative curve) for locating median and quartiles.

Histogram area is proportional to frequency; the modal class is the tallest bar, whereas the frequency polygon's peak may differ from the histogram mode.

Arithmetic mean (ungrouped) = Σx/n; for grouped data x̄ = Σfx/Σf where x is the class mark; combined mean of two groups = (n₁x̄₁ + n₂x̄₂)/(n₁ + n₂).

Weighted mean = Σwx/Σw; the mean is affected by every observation and is distorted by extreme values.

Median is the positional average of ordered data; for grouped data median = L + [(N/2 − cf)/f] × h, with L the lower limit of the median class.

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

Empirical relation for moderately skewed distributions: Mode = 3 Median − 2 Mean.

Geometric mean = antilog(Σlog x / n), used for ratios and growth rates; harmonic mean = n/Σ(1/x), used for average speed.

Partition values: quartiles divide data into four parts, deciles into ten, percentiles into hundred; Q₁, D₅ and P₅₀ all equal the median in symmetric data.

Properties: Σ(x − x̄) = 0; the mean of a constant added to each value shifts the mean by that constant; multiplying each value by k multiplies the mean by k.

Choice rule: mean for symmetric quantitative data, median for skewed data or open-end classes, mode for most typical/frequent category.

Range = maximum − minimum; coefficient of range = (max − min)/(max + min). Range is simple but uses only two observations.

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

Mean deviation about A: MD = Σ|x − A|/n (or Σf|x − A|/N); it is least when A is the median.

Variance = Σ(x − x̄)²/n for a population and s² = Σ(x − x̄)²/(n − 1) for a sample; standard deviation is its positive square root.

Short-cut variance formula: σ² = (Σfx²/N) − (Σfx/N)², using class marks for grouped data.

Coefficient of variation CV = (σ/x̄) × 100; it compares variability of two series and is unit-free - larger CV means less consistency.

Combined standard deviation uses σ² = (n₁σ₁² + n₂σ₂²)/(n₁ + n₂) plus a between-group term when means differ.

Adding a constant to every observation leaves σ unchanged; multiplying every observation by k multiplies σ by |k|.

Dispersion measures are absolute (range, QD, MD, σ) or relative (coefficients); relative measures allow comparison across different units.

Skewness measures asymmetry of a distribution; kurtosis measures peakedness or tail weight.

Karl Pearson's coefficient: SK = (Mean − Mode)/σ, or SK = 3(Mean − Median)/σ when mode is ill-defined.

Bowley's coefficient: SK = (Q₃ + Q₁ − 2Q₂)/(Q₃ − Q₁), based on quartiles.

SK = 0 means symmetric; SK > 0 means positively skewed (right tail longer, Mean > Median > Mode); SK < 0 means negatively skewed.

Moment-based skewness: β₁ = μ₃²/μ₂³ and γ₁ = √β₁; μ₂, μ₃ are second and third central moments.

Kurtosis: β₂ = μ₄/μ₂²; γ₂ = β₂ − 3. β₂ = 3 (γ₂ = 0) is mesokurtic, β₂ > 3 leptokurtic (sharper peak, heavier tails), β₂ < 3 platykurtic (flatter).

Moments: μ_r = Σf(x − x̄)^r / N; μ₁ = 0, μ₂ = σ², and μ₃ > 0 for positive skewness.

For a symmetric distribution all odd central moments vanish; skewness is zero but kurtosis can still be lepto- or platykurtic.

Coefficient of skewness is unit-free, so distributions measured in different units can be compared directly.

Correlation measures the degree and direction of linear association between two variables; it does not imply causation.

Karl Pearson's coefficient r = Σ(x − x̄)(y − ȳ) / √[Σ(x − x̄)² Σ(y − ȳ)²]; it lies between −1 and +1.

Short-cut form: r = [NΣxy − ΣxΣy] / √{[NΣx² − (Σx)²][NΣy² − (Σy)²]}.

r = +1 perfect positive, r = −1 perfect negative, r = 0 no linear relation; r is independent of change of origin and scale (sign changes if a variable is reflected).

Spearman's rank correlation ρ = 1 − 6Σd²/[n(n² − 1)], with d the rank difference; apply the correction Σd² + (t³ − t)/12 for tied ranks.

Regression predicts one variable from another: y on x gives ŷ = a + bx with b_yx = r(σ_y/σ_x); x on y gives b_xy = r(σ_x/σ_y).

Regression coefficients satisfy b_yx × b_xy = r²; both lines pass through (x̄, ȳ).

The two regression lines coincide when r = ±1 and are perpendicular when r = 0.

Coefficient of determination r² is the proportion of variation in y explained by the linear regression on x.

An index number measures relative change in a variable or group of variables over time, place or category; the base year is set to 100.

Simple price relative = (p₁/p₀) × 100; simple aggregative index = (Σp₁/Σp₀) × 100, which ignores quantity weights.

Laspeyres index = (Σp₁q₀/Σp₀q₀) × 100 (base-year quantities); Paasche index = (Σp₁q₁/Σp₀q₁) × 100 (current-year quantities).

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

Marshall–Edgeworth index uses average quantities (q₀ + q₁)/2; Dorbish–Bowley uses the arithmetic mean of Laspeyres and Paasche.

Time-reversal test: P₀₁ × P₁₀ = 1; factor-reversal test: P₀₁ × Q₀₁ = value ratio Σp₁q₁/Σp₀q₀.

Cost of living index (CPI) = Σ(p₁/p₀)w / Σw × 100, where w are base-period consumption weights.

Chain index numbers link relatives year by year; splicing joins an old and a new series by rescaling to a common base.

Deflating converts nominal to real values: real value = (nominal value / price index) × 100.

Components: secular trend (T), seasonal variation (S), cyclical variation (C) and irregular/random variation (I).

Multiplicative model Y = T × S × C × I; additive model Y = T + S + C + I is used when components are independent of trend.

Trend by semi-averages: split data into two halves, average each half, and join the two plotted points.

Moving averages smooth out short-term fluctuations; a 4-year moving average is centred with a 2-period average to align with the original years.

Least squares trend: fit Y = a + bX with b = [NΣXY − ΣXΣY]/[NΣX² − (ΣX)²]; code X so that ΣX = 0 when n is odd.

Seasonal index = (original value / centred moving average) × 100; average these relatives for each season and adjust so they sum to 400 (quarterly) or 1200 (monthly).

Seasonal variation is measured by the method of simple averages or ratio-to-moving-average; cyclical swings last longer than a year and are irregular.

Growth rate between years = [(Y₁ − Y₀)/Y₀] × 100; the trend line's slope b gives the average annual absolute change.

A moving average cannot be computed for the first and last few periods, and the choice of period length affects the smoothness of the trend.

Classical probability P(A) = number of favourable outcomes / total equally likely outcomes; axioms: 0 ≤ P(A) ≤ 1, P(S) = 1.

Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for mutually exclusive events the last term is 0.

Conditional probability P(A|B) = P(A ∩ B)/P(B); A and B are independent if P(A ∩ B) = P(A)P(B).

Multiplication rule: P(A ∩ B) = P(A)P(B|A); Bayes' theorem reverses conditional probabilities using prior probabilities.

Complement rule: P(A′) = 1 − P(A); P(A ∪ B) = 1 − P(A′ ∩ B′) for complementary events.

A random variable assigns a real number to each outcome; discrete variables take countable values, continuous variables any value in an interval.

Probability distribution of a discrete variable satisfies Σp(x) = 1 and p(x) ≥ 0; mean μ = Σx p(x), variance σ² = Σx² p(x) − μ².

For a continuous variable, probabilities are areas under the density curve: P(a ≤ X ≤ b) = ∫ f(x)dx over [a, b].

Counting aids: permutations nPr = n!/(n − r)!; combinations nCr = n!/[r!(n − r)!]; binomial expansion gives probabilities of r successes.

Binomial: n independent Bernoulli trials with constant p; P(X = r) = nCr p^r q^(n−r), q = 1 − p; mean np, variance npq, σ = √(npq).

Binomial is symmetric when p = 0.5, positively skewed when p < 0.5 and negatively skewed when p > 0.5.

Poisson: P(X = r) = e^(−m) m^r / r!, where m = mean = variance; used for rare events in time or space.

Poisson as a limiting case of binomial when n is large, p is small and np = m is moderate (rule of thumb n ≥ 50, p ≤ 0.05).

Normal: f(x) = [1/(σ√2π)] exp[−(x − μ)²/(2σ²)], with mean μ and variance σ²; the curve is bell-shaped and symmetric about x = μ.

Standard normal Z = (X − μ)/σ; areas: P(−1 ≤ Z ≤ 1) ≈ 0.6826, P(−2 ≤ Z ≤ 2) ≈ 0.9544, P(−3 ≤ Z ≤ 3) ≈ 0.9974.

Normal curve properties: mean = median = mode = μ, total area 1, inflection points at μ ± σ, and skewness zero.

Binomial tends to normal for large n with p not near 0 or 1; a continuity correction of ±0.5 is applied when using normal tables.

Poisson tends to normal when m is large; the normal distribution is used to approximate binomial and Poisson probabilities.

Population: the complete set of units under study; sample: a representative subset drawn from it; sampling frame is the list of units.

Probability sampling methods: simple random (lottery or random numbers), systematic, stratified, cluster and multistage sampling.

Non-probability methods: judgement, quota and convenience sampling; they do not give a known selection probability.

Simple random sampling with replacement has variance σ²/n; without replacement it is (σ²/n) × (N − n)/(N − 1), where (N − n)/(N − 1) is the finite population correction.

Standard error (SE) of the sample mean is σ/√n (or s/√n when σ is unknown); it decreases as n increases.

Sampling distribution of x̄ is approximately normal for large n by the central limit theorem, regardless of the population shape.

Standard error of sample proportion p̂ is √(pq/n); for difference of means it is √(σ₁²/n₁ + σ₂²/n₂).

An estimator is unbiased if E(θ̂) = θ; x̄ and s² (with divisor n − 1) are unbiased for μ and σ², while the sample range is biased.

Larger sample size reduces sampling error but not non-sampling error; a random sample is the basis for valid inference.