Business Statistics and Operations Management

Statistics for management

Data must first be described before it can be used. Measures of central tendency locate the centre: the mean is the arithmetic average, sensitive to extreme values; the median is the middle value when data are ordered, robust to extremes; and the mode is the most frequent value, useful for categorical data. A manager who reports only the mean for skewed salary or sales data misleads, because a few very high values pull the mean above what a typical employee or store experiences; median and mode reveal that skew. Measures of dispersion describe spread: range is the difference between maximum and minimum, simple but driven by extremes; quartile deviation uses the middle half of data; mean deviation averages absolute deviations from a central value; variance averages squared deviations, and standard deviation, its square root, returns spread to original units and is the most widely used measure. Coefficient of variation expresses standard deviation as a percentage of the mean, allowing comparison of variability across data sets with different scales. Skewness describes asymmetry: positive skew has a long right tail and mean above median, negative skew the reverse, and a symmetric distribution has mean, median and mode approximately equal.

Probability distributions model uncertainty. The binomial distribution models the number of successes in a fixed number of independent trials with constant probability of success, such as defective items in a sample. The Poisson distribution models the number of events in a fixed interval when events occur independently at a constant average rate, such as customer arrivals or defects per unit. The normal distribution is the bell-shaped, symmetric distribution defined by its mean and standard deviation, and it is central because many natural and measurement variables approximate it, and because sampling distributions of means approach normality as sample size grows, the central limit theorem. Standardising a normal value by subtracting the mean and dividing by standard deviation gives a standard normal value, allowing probabilities to be read from standard tables. Index numbers measure change over time relative to a base period, such as price indices that track inflation or quantity indices that track output; a Laspeyres index uses base-period weights, a Paasche index uses current-period weights, and a Fisher index, the geometric mean of the two, is often treated as ideal because it balances both. Sampling itself matters: probability sampling, including simple random, stratified and systematic sampling, gives every unit a known chance of selection and supports inference, while non-probability sampling, including convenience and judgement sampling, is cheaper but cannot support formal generalisation. A statistic computed from a sample estimates a population parameter, and the difference between them is sampling error, which shrinks as sample size grows but never disappears entirely.

Hypothesis testing, correlation and regression

Hypothesis testing is a formal procedure for judging whether sample evidence is strong enough to reject a claim about a population. The null hypothesis states no effect or no difference, while the alternative hypothesis states the effect or difference the researcher seeks to show. A test statistic, such as a z or t value, measures how far the sample result lies from the null value in standard error units. The significance level, commonly five percent, is the probability of rejecting a true null hypothesis, a Type I error; a Type II error is failing to reject a false null hypothesis. Power is the probability of correctly rejecting a false null hypothesis, and it rises with sample size, effect size and significance level. The p-value is the probability of obtaining a result at least as extreme as the observed result if the null hypothesis were true; a small p-value, below the significance level, leads to rejection, but it is not the probability that the null hypothesis is true, nor a measure of practical importance. A statistically significant result with a tiny effect may matter little in practice, while a practically important effect may fail significance in a small sample. Tests are chosen by question and data: one-sample and two-sample tests for means, paired tests for matched observations, chi-square tests for categorical association and goodness of fit, and analysis of variance for comparing more than two means.

Correlation measures the strength and direction of linear association between two variables, with the correlation coefficient ranging from minus one to plus one. A value near plus one indicates a strong positive linear relationship, near minus one a strong negative relationship, and near zero little linear relationship — though a zero correlation does not rule out a non-linear relationship. Correlation does not imply causation: two variables may move together because one causes the other, because a third variable causes both, or by coincidence, and only theory and design, not correlation alone, support causal claims. Regression goes further by fitting a line that predicts one variable from another: simple linear regression estimates an intercept and slope, where the slope is the expected change in the dependent variable for a one-unit change in the independent variable, and the coefficient of determination, R-squared, is the proportion of variation in the dependent variable explained by the model. Multiple regression extends this to several predictors, but it requires care with multicollinearity, when predictors are highly correlated and individual effects become unstable, and with extrapolation, predicting beyond the range of observed data where the relationship may not hold. Residual analysis, checking whether errors are random, constant in variance and approximately normal, tests whether the model fits; a regression used without these checks is a line drawn on hope. Managers use regression for forecasting sales from advertising, price or income, for cost estimation, and for identifying drivers of performance, always remembering that prediction is only as good as the model, the data and the stability of the relationship into the future.

Operations management

Operations management is the design, operation and improvement of systems that create and deliver goods and services. Its decisions are strategic, tactical and operational: strategic decisions concern process choice, facility location and capacity; tactical decisions concern aggregate planning, inventory policy and workforce levels; operational decisions concern scheduling, sequencing and day-to-day control. Process types match volume and variety: project processes handle unique, complex outputs; job shops handle low volume and high variety with flexible equipment; batch processes handle moderate volume and variety in lots; mass or line processes handle high volume and low variety with standardised flow; and continuous processes handle very high volume, undifferentiated flow such as chemicals or refining. The product-process matrix predicts that mismatch — for example, a job shop trying to mass-produce, or a line process forced into endless variety — creates inefficiency and quality problems. Service processes follow a similar logic, varying by customer contact and customisation: professional services are high contact and customised, service shops moderate, and mass services low contact and standardised.

Facility location decisions weigh proximity to customers and suppliers, labour availability and cost, transport infrastructure, taxes and incentives, and quality of life, using methods such as factor rating, which scores locations on weighted criteria, and centre-of-gravity analysis, which finds a location minimising weighted transport distance. Layout then arranges physical resources: process layout groups similar machines together, suiting job shops and variety; product layout arranges in sequence along a line, suiting mass production; fixed-position layout keeps the product still and brings resources to it, suiting large projects such as ships or buildings; and cellular layout groups machines into cells for families of similar products, combining flexibility with flow efficiency. Capacity planning asks how much output capability to build, when, and in what increments, balancing the cost of idle capacity against the cost of lost sales and overtime; strategies include leading demand, building ahead; lagging demand, building after demand appears; and tracking demand, adjusting incrementally. Aggregate planning translates demand forecasts into production, workforce and inventory plans over a medium horizon, using strategies of chasing demand through hiring and overtime, levelling production and using inventory to absorb fluctuation, or mixing both, with costs of hiring, firing, overtime, inventory holding and stockouts traded off explicitly.

Production planning and control then schedules work in detail: master production scheduling states what finished products are needed and when; material requirements planning explodes that schedule through bills of material into component needs, netting against inventory and scheduled receipts to time orders; and shop-floor scheduling sequences jobs through machines, using rules such as first-come-first-served, shortest processing time, earliest due date, and critical ratio, each favouring different objectives — flow time, lateness or utilisation — so no rule dominates on all measures. Inventory control balances ordering and holding costs through economic order quantity under stable demand, reorder points that trigger orders when stock falls to a level covering lead-time demand plus safety stock, and ABC classification, which concentrates control on high-value A items, moderate control on B items, and simple control on low-value C items. Just-in-time systems invert the logic by pulling material only when needed, reducing inventory and exposing problems, but requiring reliable suppliers, quality and transport. Scheduling, planning and inventory together determine whether operations deliver on time, at quality, and at cost, or whether promises made by marketing and sales collapse on the shop floor.

Quality management and operations research

Quality is fitness for purpose as judged by the customer, and quality management builds it into processes rather than inspecting defects out afterwards. Total quality management is a philosophy of customer focus, continuous improvement, employee involvement, process orientation and data-based decisions, supported by leadership commitment. Its tools include cause-and-effect diagrams that trace a problem to categories such as man, machine, method, material, measurement and environment; Pareto analysis, which shows that a small number of causes often account for most defects, so effort should focus there; check sheets for data collection; histograms for distribution; scatter diagrams for relationships; and control charts for monitoring process stability. Statistical process control distinguishes common-cause variation, inherent random variation in a stable process, from special-cause variation, signals of an assignable problem such as a broken tool or new operator. Control charts plot sample statistics against centre line and upper and lower control limits, typically three standard deviations from the centre; points outside limits, or non-random patterns such as runs and trends, signal special causes requiring investigation, while points within limits indicate a stable process that should be improved by system change, not by blaming operators for random variation.

Six Sigma extends this discipline with a target of near-perfect quality, no more than a few defects per million opportunities, and a structured improvement method, DMAIC: define the problem and customer requirements, measure current performance, analyse causes, improve by removing causes and piloting solutions, and control by standardising and monitoring the improved process. Quality awards and standards, including ISO quality management standards, provide frameworks for documented processes, audits and continual improvement, but certification alone does not guarantee quality if documentation replaces genuine process discipline. Quality costs clarify the economics: prevention costs such as training and design, appraisal costs such as inspection and testing, internal failure costs such as scrap and rework, and external failure costs such as warranties, returns and lost reputation. External failure is typically the costliest, which is why investment in prevention usually pays: a defect caught in design costs far less than one caught by a customer. Service quality adds the gap model: gaps between customer expectations and management perceptions, between perceptions and service specifications, between specifications and delivery, and between delivery and communication, together produce the final gap between expected and perceived service.

Operations research applies mathematical models to decision problems with scarce resources. Linear programming optimises a linear objective, such as maximising profit or minimising cost, subject to linear constraints on resources, with a feasible region of solutions satisfying all constraints and an optimal solution at a corner point of that region. The graphical method solves two-variable problems by plotting constraints and objective lines; the simplex method handles larger problems algorithmically. Assumptions are linearity, divisibility, certainty and non-negativity, and when they fail, other models apply: transportation models minimise cost of moving goods from sources to destinations, assignment models allocate one task to each agent at minimum cost, queuing models balance waiting costs against service capacity, and network models find shortest paths or critical paths in projects. Project management uses critical path method to identify the longest path of dependent activities determining project duration, with float or slack showing how much an activity can slip without delaying the project, and programme evaluation and review technique adds probabilistic time estimates for uncertain activities. Operations research does not replace judgement: a model is only as good as its assumptions, data and objective, and a mathematically optimal solution to the wrong problem is still the wrong decision. Its value lies in making trade-offs explicit, testing alternatives systematically, and revealing the cost of constraints.

Measures of centre and spread: when to use each

Process types matched to volume and variety