70  Statistics for Management

70.1 What is Statistics?

Statistics = the science of collecting, organising, analysing, interpreting and presenting data — Croxton & Cowden. Modern statistics has two branches: Descriptive Statistics (summarising data) and Inferential Statistics (drawing conclusions about populations from samples). Foundational figures: Karl Pearson (correlation), R.A. Fisher (ANOVA, MLE), W.S. Gosset (t-test), Jerzy Neyman (hypothesis testing), C.R. Rao and P.C. Mahalanobis (Indian Statistical Institute, 1931).

70.2 Types of Data

TipData types
  • Qualitative / Categorical — Nominal (gender, brand) · Ordinal (ranking).
  • Quantitative — Discrete (count) · Continuous (height, weight).
  • Time-series vs Cross-sectional vs Panel data.
  • Primary vs Secondary data.

70.3 Levels of Measurement — Stevens (1946)

TipStevens’ 4 scales
  • Nominal — names, categories (Yes/No).
  • Ordinal — ordered ranks (Likert).
  • Interval — equal intervals, arbitrary zero (temperature °C).
  • Ratio — true zero, ratios meaningful (income, height).

70.4 Measures of Central Tendency

TipCentral tendency
  • Mean — arithmetic average; sum/n.
  • Median — middle value (50th percentile).
  • Mode — most frequent value.
  • Geometric Mean — nth root of product; for ratios.
  • Harmonic Mean — reciprocal of avg of reciprocals; for rates.
  • Quartiles · Deciles · Percentiles — positional measures.

70.5 Measures of Dispersion

TipDispersion measures
  • Range = Max − Min.
  • Interquartile Range (IQR) = Q3 − Q1.
  • Mean Deviation.
  • Variance (σ²) = Σ(X − X̄)² / n.
  • Standard Deviation (σ) = √Variance.
  • Coefficient of Variation (CV) = σ / X̄ × 100.
  • Quartile Deviation = (Q3 − Q1)/2.

70.6 Measures of Shape

TipSkewness and Kurtosis
  • Skewness — asymmetry.
    • Positive (right-skewed) — tail on right; Mean > Median > Mode.
    • Negative (left-skewed) — tail on left.
    • Karl Pearson’s coefficient = (Mean − Mode) / σ; or 3(Mean − Median)/σ.
  • Kurtosis — peakedness.
    • Mesokurtic = normal (k = 3).
    • Leptokurtic = peaked (k > 3).
    • Platykurtic = flat (k < 3).

70.7 Probability

TipProbability concepts
  • Classical = favourable/total outcomes.
  • Empirical = relative frequency.
  • Subjective = personal belief.
  • Axiomatic — Kolmogorov (1933).
  • Conditional probability = P(A|B) = P(A∩B)/P(B).
  • Bayes’ Theorem — Reverend Bayes (1763) — P(A|B) = P(B|A)·P(A)/P(B).
  • Independent events — P(A∩B) = P(A)·P(B).

70.8 Probability Distributions

TipKey probability distributions
Distribution Use case
Binomial Bernoulli trials; success/failure
Poisson Rare events per time period
Hypergeometric Without-replacement sampling
Normal (Gauss) Bell curve; many natural phenomena
Standard Normal (Z) μ=0, σ=1
Student’s t Small-sample inference; Gosset 1908
Chi-square (χ²) Goodness of fit, independence
F-distribution ANOVA, variance ratios; Fisher-Snedecor
Exponential Time between events
Uniform Equal probability over range

70.9 Central Limit Theorem

CLT (Laplace 1810; refined by Lyapunov, Lindeberg-Lévy) — distribution of sample means approaches Normal as sample size n increases, regardless of population distribution shape. n ≥ 30 is rule of thumb.

70.10 Index Numbers

TipMajor index numbers
  • Laspeyres’ Index — base-year weights.
  • Paasche’s Index — current-year weights.
  • Fisher’s Ideal Index — √(Laspeyres × Paasche).
  • Marshall-Edgeworth — average of base and current weights.
  • CPI (Consumer Price Index) — India: CPI-IW, CPI-AL, CPI-RL.
  • WPI (Wholesale Price Index).
  • IIP (Index of Industrial Production).
  • Tests: Time Reversal · Factor Reversal · Circular.

70.11 Time-Series Analysis

TipTime-series components
  • Trend (T) — long-term direction.
  • Cyclical (C) — business-cycle fluctuations.
  • Seasonal (S) — repeating pattern.
  • Irregular (I) — random.

Multiplicative model: Y = T × C × S × I. Additive model: Y = T + C + S + I.

Methods: Moving averages · Exponential smoothing · ARIMA (Box-Jenkins 1970) · Holt-Winters.

70.12 Indian Statistical System

TipIndian statistical agencies
  • ISI — Indian Statistical Institute (P.C. Mahalanobis, 1931, Kolkata).
  • CSO / NSO — Central Statistical Office, now National Statistical Office (under MoSPI).
  • NSSO / NSO Field Operations — surveys.
  • MoSPI — Ministry of Statistics and Programme Implementation.
  • RBI — financial statistics.
  • Census of India — Registrar General.
  • Mahalanobis — pioneer of Indian sample surveys; Mahalanobis Distance (1936).

70.14 Practice Questions

Q 01ISIMedium

Indian Statistical Institute was founded in 1931 by:

  • AP.C. Mahalanobis
  • BC.R. Rao
  • CR.A. Fisher
  • DKarl Pearson
View solution
Correct Option: A
P.C. Mahalanobis, Kolkata.
Q 02StevensMedium

Stevens (1946) identified how many measurement scales?

  • A3
  • B4
  • C5
  • D6
View solution
Correct Option: B
Nominal · Ordinal · Interval · Ratio.
Q 03CLTMedium

Central Limit Theorem rule-of-thumb sample size:

  • A5
  • B10
  • C30
  • D100
View solution
Correct Option: C
n ≥ 30.
Q 04Fisher indexHard

Fisher's Ideal Index is the geometric mean of:

  • ALaspeyres and Paasche
  • BMean and Median
  • CWPI and CPI
  • DRange and SD
View solution
Correct Option: A
√(L × P).
Q 05BayesMedium

Bayes' Theorem (1763) computes:

  • APosterior probability
  • BMarginal probability
  • CJoint probability
  • DVariance
View solution
Correct Option: A
P(A|B) — posterior.
Q 06SkewMedium

When Mean > Median > Mode, distribution is:

  • ASymmetric
  • BPositively skewed
  • CNegatively skewed
  • DBimodal
View solution
Correct Option: B
Right-tail; positive skew.
Q 07CVMedium

CV (Coefficient of Variation) is:

  • Aσ × Mean
  • Bσ / Mean × 100
  • CMean / σ
  • Dσ²
View solution
Correct Option: B
Relative dispersion.
Q 08GossetHard

Student's t-distribution (1908) is by:

  • AW.S. Gosset
  • BFisher
  • CPearson
  • DNeyman
View solution
Correct Option: A
W.S. Gosset, pseudonym "Student" (Guinness).
Q 09MahalanobisHard

"Mahalanobis Distance" was developed in:

  • A1928
  • B1936
  • C1945
  • D1955
View solution
Correct Option: B
P.C. Mahalanobis (1936).
Q 10TS componentsMedium

Time-series components do NOT include:

  • ATrend
  • BCyclical
  • CSeasonal
  • DLinear
View solution
Correct Option: D
T · C · S · I (Irregular).
Q 11KolmogorovHard

Axiomatic probability (1933) is by:

  • AAndrey Kolmogorov
  • BBayes
  • CLaplace
  • DFisher
View solution
Correct Option: A
Russian mathematician (1933).
Q 12ARIMAHard

ARIMA models (1970) are by:

  • ABox & Jenkins
  • BMahalanobis
  • CHolt-Winters
  • DFisher
View solution
Correct Option: A
George Box & Gwilym Jenkins (1970).
Q 13MedianEasy

Median is the:

  • AAverage
  • BMost frequent
  • CMiddle / 50th percentile
  • DRange midpoint
View solution
Correct Option: C
Middle value when sorted.
Q 14CPI IndiaMedium

India's headline retail inflation is measured by:

  • AWPI
  • BCPI (Combined)
  • CIIP
  • DGDP deflator
View solution
Correct Option: B
CPI(C) released by NSO; RBI's primary inflation gauge.
Q 15MatchHard

Match:

(i) t-distribution (a) Mahalanobis
(ii) ANOVA (b) Gosset
(iii) Distance (c) Kolmogorov
(iv) Axiomatic probability (d) R.A. Fisher
  • A(i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)
  • B(i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
  • C(i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
  • D(i)-(d), (ii)-(a), (iii)-(c), (iv)-(b)
View solution
Correct Option: A
t — Gosset; ANOVA — Fisher; Distance — Mahalanobis; Axiomatic — Kolmogorov.

70.14.1 Advanced Format Questions

AR 1Assertion-ReasonHard

A: Median is preferred over mean for skewed data.
R: Mean is affected by extreme values.

  • ABoth true; R explains A
  • BBoth true; R does not explain A
  • CA true, R false
  • DA false, R true
View solution
Correct Option: A
S 1Statement-basedMedium

Dispersion measures: (i) Range. (ii) Variance. (iii) SD. (iv) CV.

  • AAll four
  • B(i) and (ii) only
  • C(iii) and (iv) only
  • D(iv) only
View solution
Correct Option: A
N 1NumericalMedium

Data: 2, 4, 4, 4, 5, 5, 7, 9. Mean is:

  • A5
  • B4
  • C6
  • D4.5
View solution
Correct Option: A
Σ = 40; n = 8; mean = 5.
N 2NumericalHard

Mean = 50, SD = 5. CV is:

  • A10 %
  • B5 %
  • C25 %
  • D50 %
View solution
Correct Option: A
5/50 × 100 = 10%.

70.15 Quick Recall

ImportantQuick recall
  • Statistics — descriptive vs inferential. Pioneers: Pearson · Fisher · Gosset · Neyman · Mahalanobis · C.R. Rao.
  • Stevens (1946) 4 scales: Nominal · Ordinal · Interval · Ratio.
  • Central tendency: Mean · Median · Mode · GM · HM.
  • Dispersion: Range · IQR · MD · Variance · SD · CV = σ/X̄ × 100.
  • Skew: positive (Mean>Median>Mode) · negative · Pearson coeff.
  • Kurtosis: Meso (3) · Lepto (>3) · Platy (<3).
  • Probability: Classical · Empirical · Subjective · Axiomatic (Kolmogorov 1933) · Conditional · Bayes (1763) · Independence.
  • Distributions: Binomial · Poisson · Hypergeometric · Normal · t (Gosset 1908) · χ² · F · Exponential · Uniform.
  • CLT (Laplace 1810) — n ≥ 30.
  • Index: Laspeyres (base) · Paasche (current) · Fisher Ideal (√LP) · Marshall-Edgeworth · CPI · WPI · IIP. Tests: Time/Factor/Circular reversal.
  • Time-series: T · C · S · I; ARIMA (Box-Jenkins 1970) · Holt-Winters · Exponential smoothing.
  • India: ISI (Mahalanobis 1931) · CSO/NSO/MoSPI · Mahalanobis Distance (1936).
  • Modern: Big data · Bayesian revival · ML · Causal inference (Pearl, Imbens Nobel 2021) · streaming · R/Python · Tableau.