Tamil Nadu HSE First Year Statistics flashcards

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

Statistics as data and methods

Statistics denotes numerical facts (data) and also the methods of collecting, organising, presenting, analysing and interpreting them. Interpretation without analysis is meaningless.

Primary versus secondary data

Primary data are collected first-hand by the investigator for the current enquiry. Secondary data were gathered earlier by someone else and are reused, so their reliability must be checked.

Census versus sample method

The census method enumerates every unit of the population, giving complete accuracy at high cost. The sample method studies only a part, saving time and cost and suiting destructive tests.

Class mark and class width

For a class with lower limit L and upper limit U, the class mark (mid-value) is (L + U)/2 and the class width is U − L. Class marks represent the classes in grouped calculations.

Combined arithmetic mean

When two groups of sizes n₁ and n₂ have means x̄₁ and x̄₂, the combined mean is (n₁x̄₁ + n₂x̄₂)/(n₁ + n₂), a weighted average of the group means.

Median of grouped data

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

Mode of grouped data

Mode = L + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h, where f₁ is the modal class frequency, f₀ and f₂ the frequencies of the preceding and succeeding classes.

Empirical mean-median-mode relation

For a moderately skewed distribution, Mode = 3 Median − 2 Mean. It lets you estimate the mode when the distribution is not badly skewed and the mode is ill-defined.

Geometric and harmonic mean

Geometric mean of n positive values is (x₁x₂…xₙ)^(1/n), used for growth rates and ratios. Harmonic mean is n/Σ(1/x), used for averaging speeds over equal distances.

Coefficient of variation

CV = (standard deviation/mean) × 100. It is a unit-free measure of relative dispersion, so it compares consistency of two series measured in different units.

Quartile deviation

Quartile deviation is (Q₃ − Q₁)/2, half the interquartile range, and its coefficient is (Q₃ − Q₁)/(Q₃ + Q₁). It ignores the extreme 50 per cent of observations.

Variance shortcut formula

Variance of raw data equals (Σx²)/n − x̄²; for a frequency distribution it is (Σfx²)/Σf − x̄². Standard deviation is the positive square root of this quantity.

Combined standard deviation

Combined variance is [n₁(σ₁² + d₁²) + n₂(σ₂² + d₂²)]/(n₁ + n₂), where d₁ and d₂ are the deviations of the group means from the combined mean.

Pearson's coefficient of skewness

SK = (Mean − Mode)/σ, or 3(Mean − Median)/σ when the mode is ill-defined. A positive value means a right tail, a negative value a left tail.

Bowley's coefficient of skewness

Bowley's measure is (Q₃ + Q₁ − 2Q₂)/(Q₃ − Q₁). It depends only on quartiles, always lies between −1 and +1, and needs no mean or standard deviation.

Moments and beta coefficients

The rth moment about the mean is μᵣ = Σf(x − x̄)ʳ/Σf. Then β₁ = μ₃²/μ₂³ measures skewness and β₂ = μ₄/μ₂² measures kurtosis.

Mesokurtic, leptokurtic, platykurtic

A normal curve is mesokurtic with β₂ = 3. β₂ greater than 3 is leptokurtic (more peaked with heavier tails); β₂ less than 3 is platykurtic (flatter).

Karl Pearson's correlation coefficient

r = Σ(x − x̄)(y − ȳ)/√[Σ(x − x̄)²Σ(y − ȳ)²]. It lies between −1 and +1 and is unchanged by shifting origin or rescaling either variable.

Spearman's rank correlation

ρ = 1 − 6Σd²/[n(n² − 1)], where d is the difference between ranks of the same item. Repeated ranks require adding Σ(t³ − t)/12 to Σd².

Regression coefficients and r

b_yx = r(σ_y/σ_x) and b_xy = r(σ_x/σ_y). Their geometric mean with the sign of r gives r, and their product never exceeds one.

Addition theorem of probability

For any two events, P(A ∪ B) = P(A) + P(B) − P(A ∩ B). When A and B are mutually exclusive the intersection term vanishes.

Conditional probability

P(A|B) = P(A ∩ B)/P(B), defined when P(B) > 0. It gives the chance of A given that B is known to have occurred, restricting the sample space to B.

Independent versus mutually exclusive events

Mutually exclusive events cannot occur together, so P(A ∩ B) = 0. Independent events do not influence each other, so P(A ∩ B) = P(A)P(B). The two conditions differ.

Bayes' theorem

For a partition E₁, …, Eₙ, P(Eᵢ|A) = P(Eᵢ)P(A|Eᵢ)/Σ P(Eⱼ)P(A|Eⱼ). It reverses conditional probabilities using prior probabilities and the total probability rule.

Expectation of a random variable

E(X) = Σx·p(x) for a discrete variable and ∫x·f(x)dx for a continuous one. Expectation is linear: E(aX + b) = aE(X) + b.

Variance in terms of expectation

Var(X) = E(X²) − [E(X)]². Multiplying X by a gives variance a²Var(X); adding a constant leaves the variance unchanged.

Distribution function properties

F(x) = P(X ≤ x) is non-decreasing and satisfies 0 ≤ F(x) ≤ 1, with F(−∞) = 0 and F(∞) = 1. For continuous variables it is the integral of the density.

Binomial mean and variance

A binomial variable counts successes in n independent Bernoulli trials with success probability p. Its mean is np and its variance is npq, where q = 1 − p.

Stratified sampling

The population is divided into internally homogeneous strata and a sample drawn from each. Under proportional allocation, stratum i contributes nᵢ = n(Nᵢ/N) units.

Standard error of the mean

The standard error of x̄ is σ/√n, estimated by s/√n when σ is unknown. It measures the spread of the sampling distribution and shrinks as sample size grows.

Sampling versus non-sampling error

Sampling error arises because only part of the population is observed and falls with larger samples. Non-sampling errors come from faulty frames, non-response, or recording mistakes.

Laspeyres and Paasche indices

Laspeyres uses base-year quantities, Σp₁q₀/Σp₀q₀ × 100, and tends to overstate price rises. Paasche uses current quantities, Σp₁q₁/Σp₀q₁ × 100, and tends to understate them.

Fisher's ideal index

Fisher's index is the geometric mean of Laspeyres and Paasche: √[(Σp₁q₀/Σp₀q₀)(Σp₁q₁/Σp₀q₁)] × 100. It satisfies both the time-reversal and factor-reversal tests.