1. Meaning, Scope and Limitations

Its scope covers the entire journey of data: collection, classification and tabulation, presentation, analysis, interpretation, and forecasting. In business it is a tool — it does not decide anything by itself, but it removes guesswork from decisions on production, pricing, sales targets and quality control.

2. Importance in Business & Management

• Helps management set realistic targets based on past performance rather than guesswork.
• Supports forecasting of demand, sales and inventory needs.
• Enables comparison of performance across branches, products or time periods.
• Backs quality control and cost control with measurable standards.
• Provides the evidence base for policy-making at the firm and national level.

3. Types and Sources of Data

Primary data is collected first-hand by the investigator for a specific purpose. Secondary data is already collected and published by someone else — census reports, company records, government publications — and re-used for a new study.

4. Methods of Collection

Primary data methods: direct personal interview, indirect oral investigation, mailed or enumerator-filled questionnaires, and telephone/online surveys.

Secondary data sources: published (government reports, journals, newspapers) and unpublished (research theses, internal company records).

5. Precautions & Problems in Data Collection

Common problems include non-response from respondents, ambiguous or leading questions, sampling bias, cost and time constraints, and enumerator bias during personal interviews.

1. Classification of Data

Need & objectives: it condenses bulky data, makes comparison possible, highlights the significant features of the data, and prepares it for further statistical analysis.

Types of classification: Geographical (by area/region), Chronological (by time), Qualitative (by attribute, e.g. gender), and Quantitative (by magnitude, e.g. income groups).

2. Constructing a Frequency Distribution

A frequency distribution groups raw data into class intervals and records how many observations (frequency) fall in each class. Key principles: decide the number of classes (Sturges’ Rule: k = 1 + 3.322 log N), keep class width uniform where possible, and clearly define class limits and class boundaries so there is no overlap.

3. Tabular & Diagrammatic Presentation

A good statistical table has a title, column/row headings, the main body of figures, and a source note. Diagrams compare discrete values visually: a simple bar diagram compares one variable across categories, a multiple bar diagram compares two or more variables side by side, and a pie diagram splits a total into 360° slices proportional to each component’s share.

4. Graphic Presentation

• Histogram — adjoining bars over class boundaries for continuous data; bars touch each other (unlike a bar diagram).
• Frequency Polygon — a line graph joining the mid-points of the tops of histogram bars.
• Frequency Curve — a smoothed, freehand version of the frequency polygon.
• Ogive (cumulative frequency curve) — plots cumulative frequency against class boundaries; the “less than” and “more than” ogives intersect at a point whose X-value gives the median graphically.

1. Arithmetic Mean (AM)

Simple AM = Σx / n. Weighted AM = Σwx / Σw (used when items carry different importance). For grouped data: AM = Σfx / Σf (direct method), or A + (Σfd/Σf) × i using the step-deviation/short-cut method, where A is an assumed mean.

2. Geometric Mean (GM) & Harmonic Mean (HM)

GM = Antilog(Σlog x / n) — best suited for rates of growth, ratios and index numbers. HM = n / Σ(1/x) — best suited for averaging speeds, rates and prices.

3. Median & Partition Values

The Median is the value of the middle item when data is arranged in order — a positional average, unaffected by extreme values. For grouped data: Median = L + [(N/2 − cf) / f] × i. Quartiles (Q₁, Q₂, Q₃), deciles and percentiles are calculated the same way, replacing N/2 with N/4, N/10 or N/100 respectively.

4. Mode

The Mode is the value that occurs most frequently. For grouped data: Mode = L + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × i, where f₁ is the frequency of the modal class, f₀ the preceding class and f₂ the following class.

5. Properties & Choice of an Average

A good average should be rigidly defined, based on all observations, easy to calculate and understand, and capable of further algebraic treatment. The Arithmetic Mean is used most often, but it is distorted by extreme values — in that case Median or Mode is preferred.

1. Absolute Measures

• Range = Largest value − Smallest value.
• Quartile Deviation (QD) = (Q₃ − Q₁) / 2 — based on the middle 50% of data, unaffected by extreme values.
• Standard Deviation (σ) = √[ Σf(x − x̄)² / N ] — the most reliable and widely used measure, since it uses every observation.

2. Relative Measures

Relative measures remove the unit of measurement, which makes them ideal for comparing two different series.

• Coefficient of Range = (L − S) / (L + S)
• Coefficient of Quartile Deviation = (Q₃ − Q₁) / (Q₃ + Q₁)
• Coefficient of Variation (CV) = (σ / x̄) × 100

3. Lorenz Curve

The Lorenz Curve is a graphical method for showing inequality in the distribution of income or wealth. Cumulative % of population is plotted on the X-axis against cumulative % of income on the Y-axis; the further this curve bends away from the 45° “line of equal distribution,” the greater the inequality.

1. Skewness

Skewness formulas reuse the Mean, Median and Mode from Unit 3, and the Standard Deviation from Unit 4 — worth revising both before this chapter.

Karl Pearson’s coefficient: Sk = (Mean − Mode) / σ, or 3(Mean − Median) / σ when the mode is ill-defined.
Bowley’s coefficient (quartile-based): Sk = (Q₃ + Q₁ − 2 × Median) / (Q₃ − Q₁).

2. Five-Number Summary & Box-Whisker Plot

The five-number summary is: Minimum, Q₁, Median, Q₃, Maximum. Plotted as a box-whisker diagram, it shows the spread, the middle 50% of data as a box, and any skewness or outliers (points beyond 1.5 × IQR from the box) at a glance.

3. Kurtosis

Kurtosis measures the peakedness of a distribution relative to the normal curve: Leptokurtic (sharply peaked), Mesokurtic (normal), Platykurtic (flat-topped). By the percentile method, Kurtosis coefficient = QD / (P₉₀ − P₁₀).

4. Moments

Moments describe a distribution’s shape mathematically. Central moments (about the mean) are μᵣ = Σf(x − x̄)ʳ / N. Note μ₁ is always 0, and μ₂ equals the variance. Skewness by moments = β₁ = μ₃² / μ₂³; Kurtosis by moments = β₂ = μ₄ / μ₂².

1. Karl Pearson’s Correlation Coefficient

r = Σ(x − x̄)(y − ȳ) / √[Σ(x − x̄)² × Σ(y − ȳ)²], and it always lies between −1 and +1. A value near +1 means strong positive correlation, near −1 strong negative correlation, and near 0 little to no linear relationship. The same formula extends to a bivariate frequency distribution by weighting with cell frequencies.

Coefficient of Determination = r² — the proportion of variation in Y that is explained by X.
Probable Error = 0.6745 × (1 − r²) / √n — used to judge whether r is statistically significant.

2. Spearman’s Rank Correlation

Used when data is ranked/qualitative rather than measured: R = 1 − [6Σd² / (n(n² − 1))], where d is the difference between the ranks of each pair.

3. Regression

Two regression lines exist: Y on X (used to estimate Y from X) and X on Y (used to estimate X from Y). The regression coefficient byx = r × (σy/σx), and bxy = r × (σx/σy).

1. Components of a Time Series

• Secular Trend — the long-term, general direction (upward, downward or stable) over many years.
• Seasonal Variation — regular, short-term fluctuations tied to a fixed calendar period (a quarter, month or season).
• Cyclical Variation — longer, wave-like ups and downs tied to business cycles, usually spanning more than a year.
• Irregular/Random Variation — unpredictable movements from one-off events (disasters, strikes, pandemics).

2. Measuring Trend

Semi-average method: split the series into two equal halves, find the average of each half, and join the two points. Moving average method: smooths out fluctuations by replacing each value with the average of itself and its neighbours over a fixed period. Method of Least Squares: fits a trend line Yc = a + bX that minimises the sum of squared deviations — the most mathematically precise method and the one TU expects for numerical problems.

3. Measuring Seasonal Variation

Simple average method: averages the seasonal values for each period as a percentage of the overall average. Ratio-to-moving-average method: expresses each actual value as a percentage of the corresponding moving average (trend) value — the standard method used for quarterly seasonal indices.

1. Types & Construction Problems

Main types: Price Index, Quantity Index, and Value Index. Key problems in construction: selecting the base period, selecting representative items, deciding the weights to assign, and choosing the right average and formula.

2. Methods of Construction

3. Cost of Living Index

Aggregative Expenditure Method = Σp₁q₀ / Σp₀q₀ × 100 (same structure as Laspeyre’s). Family Budget Method = Σ(I × W) / ΣW, where I is the price relative of each item and W is its assigned weight.

4. Base Shifting & Deflating

Base shifting recalculates a whole index series against a new base year without re-collecting original data. Deflating converts a nominal (current price) value into a real value: Real Value = Nominal Value / Price Index × 100 — this is exactly how “real GDP” or “real wages” are calculated.

1. Definition

2. Addition Theorem

For any two events: P(A∪B) = P(A) + P(B) − P(A∩B). If A and B are mutually exclusive (cannot occur together), P(A∩B) = 0, so P(A∪B) = P(A) + P(B).

3. Multiplication Theorem

For independent events: P(A∩B) = P(A) × P(B). For dependent events: P(A∩B) = P(A) × P(B|A), where P(B|A) is the conditional probability of B given A has occurred.

4. Combination Rule in Probability

When the order of selection does not matter (e.g. choosing a committee), the number of ways to choose r items from n is ⁿCᵣ = n! / [r!(n−r)!]. This is combined with the basic probability formula whenever a question asks for the probability of drawing a specific combination of items.

5. Conditional Probability

P(A|B) = P(A∩B) / P(B) — the probability of A occurring, given that B has already occurred.

1. Population, Sample, Census vs. Sampling

The population is the entire group under study; a sample is a subset drawn from it. A census studies every single unit of the population, while sampling studies only a representative part and uses it to infer conclusions about the whole population.

2. Sampling Techniques

Probability sampling (every unit has a known chance of selection): simple random, stratified, systematic, and cluster sampling. Non-probability sampling (selection based on judgment/convenience, no fixed probability): convenience, judgment, and quota sampling.

3. Sampling Distribution & Standard Error

Standard Error (SE) = σ / √n — it measures how much the sample mean is expected to fluctuate from the true population mean, and it shrinks as the sample size n increases.

4. Estimation

Estimation is the process of inferring an unknown population parameter from a known sample statistic (the estimator). A point estimate is a single value (e.g. x̄ estimates μ); an interval estimate gives a range with an attached confidence level (e.g. x̄ ± Z × SE at 95% confidence).

1. Decision-Making Under Uncertainty

Used when the probabilities of future states of nature are not known at all.

• Maximax (optimism): pick the alternative with the highest of the maximum possible payoffs.
• Maximin (pessimism / Wald criterion): pick the alternative with the highest of the minimum possible payoffs — the safest, most cautious choice.
• Minimax Regret (Savage criterion): pick the alternative that minimises the maximum possible regret (opportunity loss) across all states.

2. Decision-Making Under Risk

Used when probabilities of each state of nature are known or can be estimated.

• EMV (Expected Monetary Value) = Σ (payoff × probability) for each alternative; choose the one with the highest EMV.
• EPPI (Expected Profit with Perfect Information) = Σ (best possible payoff under each state × its probability).
• EVPI (Expected Value of Perfect Information) = EPPI − Best EMV — the maximum amount worth paying to obtain perfect information before deciding.

3. Linear Programming Problem (LPP)

LPP is used to maximise or minimise an objective function (e.g. profit or cost) subject to a set of linear constraints (e.g. limited labour, raw material or budget). Formulation requires: decision variables, an objective function, the constraints, and non-negativity restrictions. With exactly two decision variables, the graphical method plots each constraint, marks the feasible region, and evaluates the objective function at every corner point of that region to find the optimum.

1. Definition & Calculation

For a 2×2 determinant |a b; c d|, the value = ad − bc. For a 3×3 determinant, the value is found by expansion along any row or column using minors and cofactors, with alternating signs (+, −, +…).

2. Properties of Determinants

• The value is unchanged if rows and columns are interchanged (transposed).
• The sign changes if any two rows (or two columns) are interchanged.
• The value is zero if any two rows or columns are identical.
• If every element of a row/column is multiplied by k, the determinant’s value is also multiplied by k.
• The value is unchanged if a multiple of one row/column is added to another.

3. Cramer’s Rule

For a system of linear equations, Cramer’s Rule finds each unknown as a ratio of determinants: x = Dx/D, y = Dy/D, z = Dz/D — where D is the determinant of the coefficient matrix, and Dx, Dy, Dz are formed by replacing the respective column with the column of constants. Works cleanly for up to three variables.

1. Definition & Types

A matrix is a rectangular array of numbers arranged in rows and columns. Common types: row matrix, column matrix, square matrix, diagonal matrix, identity (unit) matrix, null (zero) matrix, and symmetric matrix.

2. Matrix Operations

Addition/Subtraction: possible only when both matrices have the same order; corresponding elements are simply added or subtracted. Multiplication: possible only when the number of columns in the first matrix equals the number of rows in the second; each entry is the dot product of a row from the first matrix and a column from the second.

3. Cofactor, Transpose, Adjoint & Inverse

The cofactor of an element is its signed minor. The transpose (Aᵀ) flips rows into columns. The adjoint is the transpose of the matrix of cofactors. The inverse A⁻¹ = Adjoint(A) / |A|, and it exists only when the determinant |A| is non-zero (a “non-singular” matrix).

4. Matrix Method for Simultaneous Equations

Writing a system of equations as AX = B, the solution is found as X = A⁻¹B. This works for up to three unknown variables and is the matrix-algebra alternative to Cramer’s Rule covered in Unit 12.