Quantitative Techniques gives managers the numeric tools to describe data, handle uncertainty and optimise decisions. It splits into two halves: statistics (describing and inferring from data) and operations research (choosing the best action under constraints). Each idea below has a small worked number.
1. Describing data
| Measure | Meaning | Note |
|---|---|---|
| Mean | Average | Sensitive to extreme values |
| Median | Middle value | Robust; use for income |
| Mode | Most frequent | Categories |
| Range, IQR | Spread | IQR ignores extremes |
| Variance and standard deviation (SD) | Average squared and typical deviation from the mean | SD in the same unit as the data |
| Coefficient of variation (CV) | SD ÷ mean × 100% | Compare variability across different units |
| Skewness and kurtosis | Asymmetry and tail weight | Shape of distribution |
Example: delivery times of 2, 4, 4, 4, 5, 5, 7, 9 days. Mean = 40 ÷ 8 = 5, median = 4.5, mode = 4. Charts: histograms, box plots, scatter plots.
In the real world: average salaries are pulled up by a few high earners, so reports quote the median. A service level agreement that quotes only the mean response time hides the slow tail, which is why engineers track the 95th and 99th percentiles.
2. Probability and distributions
Probability measures likelihood from 0 to 1. Rules: addition P(A or B) = P(A) + P(B) − P(A and B); multiplication P(A and B) = P(A) × P(B | A) (independent events: P(A) × P(B)). Bayes’ theorem updates a probability with evidence: P(A | B) = P(B | A) P(A) ÷ P(B).
Example: a test catches 90% of faulty items (sensitivity) and falsely flags 5% of good ones; 2% of items are faulty. P(faulty | flagged) = (0.9 × 0.02) ÷ (0.9 × 0.02 + 0.05 × 0.98) = 0.018 ÷ 0.067 ≈ 27%. Even with a good test, most flags are false alarms because faults are rare.
| Distribution | Use | Parameters |
|---|---|---|
| Binomial | Number of successes in n independent trials | n, p; mean np |
| Poisson | Count of events in a fixed interval (calls per hour) | λ; mean = variance = λ |
| Normal | Measurement errors, many natural variables | μ, σ; about 68%, 95%, 99.7% within 1, 2, 3 SD |
| Exponential | Time between Poisson events | Rate λ |
Example: a call centre receives 4 calls a minute on average (Poisson, λ = 4). The chance of exactly 2 calls in a minute is e⁻⁴ × 4² ÷ 2! = 0.0183 × 8 ≈ 0.147. Expected value E(X) = Σ x·p(x) guides decisions under risk.
In the real world: the Bayes example is why screening a whole population for a rare condition produces many false positives, and why spam and fraud filters weigh evidence rather than flag any single clue. Poisson models underlie queue and staffing plans.
3. Sampling and estimation
The central limit theorem says the average of a large sample is approximately normal, whatever the population shape, with standard error = σ ÷ √n. A confidence interval for the mean is x̄ ± z × SE. Example: a sample of 100 customers has a mean spend of ₹800 with SD ₹200; SE = 20, so the 95% interval is 800 ± 1.96 × 20 ≈ ₹761 to ₹839. Larger samples shrink the interval; quadrupling n halves it. (Hypothesis tests and sampling methods are covered in Research Methods.)
4. Correlation, regression and time series
Correlation coefficient (r) measures linear association from −1 to +1. Simple regression fits Y = a + bX by least squares, where b = Σ(x − x̄)(y − ȳ) ÷ Σ(x − x̄)².
Example: advertising (₹ lakh) 1, 2, 3, 4, 5 and sales (₹ lakh) 10, 14, 15, 19, 22. x̄ = 3, ȳ = 16; b = 29 ÷ 10 = 2.9 and a = 16 − 2.9 × 3 = 7.3. Forecast for ad spend 6: 7.3 + 2.9 × 6 = ₹24.7 lakh. Multiple regression uses several predictors.
Time series splits data into trend, seasonal, cyclical and irregular components, with forecasting by moving averages, exponential smoothing and trend equations. Index numbers compare prices or quantities with a base period: the Laspeyres index uses base-period quantities, the Paasche index uses current ones, and the consumer price index (CPI) measures inflation.
In the real world: the Sensex and Nifty are market-capitalisation weighted index numbers, the CPI drives wage and policy decisions, and companies fit seasonal models to plan festival-season stock.
5. Linear programming (LP)
LP finds the best value of a linear objective under linear constraints. Steps: define decision variables, write the objective and constraints, solve by the graphical method (two variables: the optimum is at a corner of the feasible region) or the simplex method (larger problems).
Worked example. A firm makes tables (profit ₹300) and chairs (profit ₹200). Each table needs 3 hours of carpentry and each chair 2; 120 hours are available. Each table needs 1 hour of finishing and each chair 2 hours; 80 hours are available. Maximise 300T + 200C subject to 3T + 2C ≤ 120, T + 2C ≤ 80. Corner points: (0, 0) profit 0; (40, 0) profit 12,000; (0, 40) profit 8,000; intersection: subtract the equations, 2T = 40, so T = 20 and C = 30, profit 6,000 + 6,000 = 12,000. Both (40, 0) and (20, 30) give ₹12,000, so any mix along that edge is equally good. Sensitivity analysis and shadow prices show the value of one more hour of a scarce resource.
In the real world: airlines, refiners, logistics firms and food processors solve very large LPs for crew rosters, blending and routing. Cloud cost tools use similar optimisation to pack workloads onto machines.
6. Transportation and assignment
| Problem | Goal | Method |
|---|---|---|
| Transportation | Ship goods from supply points to demand points at least cost | Starting solution (North-West corner, least cost, Vogel’s approximation), then improve with MODI |
| Assignment | Match n jobs to n people at least cost | Hungarian method (reduce rows and columns, cover zeros) |
| Transshipment | Allow intermediate points | Extension of transportation |
Checks: the problem must be balanced (supply = demand; add a dummy if not), and a basic solution has m + n − 1 filled cells. Assignment example: reduce each row by its minimum and each column by its minimum, then assign where zeros allow.
7. Decisions, queues, networks and games
Decision theory: payoff tables with criteria under uncertainty (maximax optimistic, maximin pessimistic, minimax regret, Laplace equal chance) and under risk (expected monetary value). Decision trees lay out sequences of choices and chance events.
Example: launch (60% success giving ₹50 lakh, 40% failure losing ₹20 lakh) has EMV = 0.6 × 50 − 0.4 × 20 = ₹22 lakh, better than doing nothing (₹0).
Queueing (M/M/1): with arrival rate λ and service rate μ, utilisation ρ = λ/μ, average number in system = ρ/(1 − ρ), average time = 1/(μ − λ). If λ = 8 and μ = 10 per hour, ρ = 0.8, average number 4, average time 0.5 hour. At ρ = 0.95 the average number jumps to 19, which shows why waits explode near full utilisation.
PERT and CPM: activity networks with expected time = (a + 4m + b) ÷ 6; the critical path is the longest path and has zero float. Game theory covers two-person zero-sum games, saddle points and the prisoner’s dilemma. Simulation (Monte Carlo) uses random sampling to model uncertain systems.
One-minute recap
| Topic | Key idea | Where you meet it |
|---|---|---|
| Descriptive statistics | Mean, median, SD, percentiles | Reports, SLAs |
| Probability | Rules, Bayes, binomial, Poisson, normal | Risk, screening, staffing |
| Estimation | Central limit theorem, confidence interval | Surveys, sampling |
| Regression and time series | Least squares, trend and seasonality, indices | Forecasts, CPI, Sensex |
| Linear programming | Objective, constraints, corner solutions | Product mix, blending, routing |
| Transportation and assignment | Least cost; Hungarian method | Logistics, rostering |
| Decisions and networks | EMV, M/M/1 queue, PERT and CPM | Investment, capacity, projects |
Next in MBA Revisited: Operations Management II.
Written by Claude, Anthropic’s AI assistant, as part of MBA Revisited, a quick refresher of the PGDM subjects one at a time.