69  Statistics for Management

69.1 What is Statistics?

Statistics is the science of collecting, organising, analysing, interpreting and presenting data. In management, statistics is the toolkit that turns data into insight for decisions — from forecasting demand to evaluating product quality to inferring customer preferences. The Indian standard text by S.P. Gupta defines statistics as “the science which deals with classification, tabulation, analysis and interpretation of numerical data” (gupta2019?).

TipTwo Branches of Statistics
Branch What it does Example
Descriptive statistics Summarise and present data Mean, median, charts
Inferential statistics Draw conclusions about populations from samples Hypothesis tests, confidence intervals

69.2 Types of Data and Levels of Measurement

TipTypes of Data
Type Description Examples
Qualitative / Categorical Non-numerical Gender, brand preference
Quantitative — discrete Whole-number counts Number of customers
Quantitative — continuous Any value within a range Height, weight, time
TipStevens’s Four Levels of Measurement (1946)
Level What it captures Examples Permitted statistics
Nominal Categories, no order Gender, religion Mode, frequency
Ordinal Order, no equal intervals Likert scale, rank Median, percentile
Interval Equal intervals, no true zero Celsius, Fahrenheit Mean, std dev
Ratio Equal intervals + true zero Income, weight, age All operations including ratios

69.3 Measures of Central Tendency

TipThree Classical Measures of Central Tendency
Measure Definition Best for
Mean (arithmetic) Σx ÷ n Symmetric, no extreme outliers
Median Middle value when data is sorted Skewed data
Mode Most frequent value Categorical or unimodal data

Karl Pearson’s empirical relationship (for moderately skewed distributions): Mode = 3 × Median − 2 × Mean.

69.4 Measures of Dispersion

TipFive Common Measures of Dispersion
Measure Formula What it captures
Range Max − Min Crude spread
Quartile deviation (Q3 − Q1) / 2 Middle 50% spread
Mean deviation Σ|x − x̄| ÷ n Average absolute deviation
Variance Σ(x − x̄)² ÷ n Average squared deviation
Standard deviation √Variance Same units as data; most-used
Coefficient of Variation (σ ÷ x̄) × 100 Relative dispersion (compare different units)

69.5 Measures of Skewness and Kurtosis

TipSkewness and Kurtosis
Concept What it captures Symmetric value
Skewness Asymmetry of distribution 0 (symmetric)
Kurtosis Peakedness / tail heaviness 3 (normal); excess kurtosis = 0

Pearson’s coefficient of skewness: Skp = (Mean − Mode) / σ.

69.6 Probability Foundations

TipThree Approaches to Probability
Approach Definition
Classical / a-priori Favourable outcomes ÷ Total outcomes (assumes equally likely)
Empirical / Frequentist Long-run relative frequency
Subjective Personal degree of belief
TipThree Probability Rules
Rule Statement
Addition P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Multiplication P(A ∩ B) = P(A) × P(B
Conditional P(A
Bayes P(A

69.7 Probability Distributions

TipCommon Discrete and Continuous Distributions
Distribution Type Use
Bernoulli Discrete Single yes/no trial
Binomial Discrete n independent yes/no trials
Poisson Discrete Rare events; mean = variance = λ
Geometric Discrete Trials until first success
Normal (Gaussian) Continuous Bell curve; many natural phenomena
Standard Normal Continuous Mean 0, SD 1; z-scores
t-distribution Continuous Small samples; replaces Normal
Chi-square Continuous Goodness-of-fit, independence
F-distribution Continuous Variance ratio; ANOVA
Exponential Continuous Time between events
Uniform Continuous Equal probability across range 

flowchart LR
  D[Probability<br/>Distributions] --> DI[Discrete]
  D --> CO[Continuous]
  DI --> B[Binomial]
  DI --> P[Poisson]
  DI --> G[Geometric]
  CO --> N[Normal]
  CO --> T[t]
  CO --> CS[Chi-square]
  CO --> F[F]
  style D fill:#FCE4EC,stroke:#AD1457
  style N fill:#E8F5E9,stroke:#1B5E20

69.8 The Normal Distribution

The most important continuous distribution. Properties:

TipProperties of the Normal Distribution
Property What it says
Symmetry Mean = median = mode
Bell-shaped Single peak at the mean
Asymptotic Tails approach but never touch the X-axis
68-95-99.7 rule ~68% within ±1σ, 95% within ±2σ, 99.7% within ±3σ
Standardisation Any normal converts to standard normal via z = (x − μ) / σ
Central Limit Theorem Sample means converge to normal as n grows

69.9 Practice Questions

Q 01 Branches Easy

Drawing conclusions about a population based on a sample is the domain of:

  • ADescriptive statistics
  • BInferential statistics
  • CTabulation
  • DGraphing
View solution
Correct Option: B
Inferential statistics generalise from sample to population using probability.
Q 02 Stevens Medium

Temperature in Celsius is measured on which level?

  • ANominal
  • BOrdinal
  • CInterval
  • DRatio
View solution
Correct Option: C
Celsius has equal intervals but no true zero (0°C is not the absence of temperature) — interval. Kelvin would be ratio.
Q 03 Pearson Empirical Medium

For a moderately skewed distribution with Mean = 50 and Median = 45, the Mode is approximately:

  • A35
  • B40
  • C47
  • D55
View solution
Correct Option: A
Pearson's empirical: Mode ≈ 3 × Median − 2 × Mean = 3 × 45 − 2 × 50 = 135 − 100 = 35.
Q 04 CV Medium

The Coefficient of Variation is most useful for:

  • ACalculating mean
  • BComparing relative variability across data sets with different units or means
  • CComputing standard deviation
  • DGraphing data
View solution
Correct Option: B
CV = (σ ÷ x̄) × 100 — a unit-less measure of relative dispersion, ideal for comparing variability across different scales.
Q 05 Bayes Medium

Bayes's theorem expresses:

  • AP(A|B) = P(B|A) × P(A) ÷ P(B)
  • BP(A) + P(B)
  • CP(A) × P(B)
  • DP(A) − P(B)
View solution
Correct Option: A
P(A|B) = P(B|A) × P(A) ÷ P(B) — the foundation of Bayesian inference.
Q 06 Poisson Medium

In a Poisson distribution, mean and variance are:

  • AEqual
  • BUnequal
  • CNegative of each other
  • DReciprocals
View solution
Correct Option: A
A defining feature of the Poisson — mean = variance = λ.
Q 07 Normal Medium

In a normal distribution, approximately what % of values lie within ±2σ of the mean?

  • A50%
  • B68%
  • C95%
  • D99.7%
View solution
Correct Option: C
68-95-99.7 rule: 95% within ±2σ.
Q 08 CLT Medium

The Central Limit Theorem says that:

  • APopulation is always normal
  • BSample size doesn't matter
  • CDistribution of sample means tends to normal as sample size grows, regardless of the population's distribution
  • DVariance equals mean
View solution
Correct Option: C
CLT — a foundation of inferential statistics. Sample means become approximately normal as n grows even if the population is not normal.
ImportantQuick recall
  • Statistics = science of data. Two branches: descriptive + inferential.
  • Stevens’s four levels: Nominal · Ordinal · Interval · Ratio.
  • Measures of central tendency: Mean · Median · Mode. Pearson empirical: Mode ≈ 3M − 2X̄.
  • Dispersion: Range · QD · MD · Variance · SD · CV (relative).
  • Probability rules: addition, multiplication, conditional, Bayes.
  • Distributions: Discrete (Bernoulli, Binomial, Poisson, Geometric) and Continuous (Normal, t, Chi-square, F, Exponential, Uniform).
  • Normal: 68-95-99.7 rule; CLT says sample means converge to normal.