Advanced Statistical Techniques CBMBT4T05

Syllabus topics (RTMNU)

• Introduction to Advanced Statistics in Business Analytics
• Role of Statistics in Business Decision-Making
• Descriptive vs. Inferential Statistics
• Applications of Advanced Statistics in Business
• Data Types, Measurement Scales, and Data Collection
• Qualitative vs. Quantitative Data
• Levels of Measurement (Nominal, Ordinal, Interval, Ratio)
• Data Collection Techniques in Business Analytics
• Exploratory Data Analysis (EDA)
• Measures of Central Tendency (Mean, Median, Mode)
• Measures of Dispersion (Variance, Standard Deviation, IQR)
• Detecting Outliers using Boxplots and Z-scores
• Skewness and Kurtosis
• Graphical Representation of Data
• Histograms, Bar Charts, and Pie Charts
• Scatter Plots for Correlation Analysis
• Heatmaps, Treemaps, and Bubble Charts
• Box-and-Whisker Plots for Distribution Analysis
• Introduction to SPSS for EDA
• IBM SPSS Interface and Data Import
• Data Cleaning and Handling Missing Values
• Generating Summary Statistics and Charts in SPSS

Introduction to Advanced Statistics in Business Analytics – Role of Statistics in Business Decision-Making

Statistics is the science of collecting, organising, analysing and interpreting numerical data to take decisions under uncertainty. Advanced statistics goes beyond averages to probability, testing, regression, multivariate and non-parametric methods.

Role in business decisions:

1. Planning and forecasting – estimating next year's sales or demand.
2. Quality control – checking whether a product meets standards (control charts).
3. Market research – understanding customer likes through surveys.
4. Finance – risk, return and credit analysis in banks and mutual funds.
5. HR – salary structure, performance and attrition analysis.
6. Testing ideas – checking if a new advertisement really increased sales (hypothesis testing).

Descriptive vs. Inferential Statistics

Applications of Advanced Statistics in Business

Demand forecasting (time series), customer segmentation (cluster analysis), credit scoring (logistic regression), A/B testing of websites (t-test), quality improvement (Six Sigma), product preference studies (chi-square), and risk modelling (Monte Carlo simulation) – used by banks, FMCG companies like HUL, e-commerce firms and insurers.

Data Types, Measurement Scales and Data Collection – Qualitative vs. Quantitative Data

• Qualitative (categorical) data – describe qualities or categories: gender, brand used, city, satisfaction level.
• Quantitative (numerical) data – measured in numbers: income, sales, age. They are discrete (countable: number of customers) or continuous (any value: weight, time).

Levels of Measurement (Nominal, Ordinal, Interval, Ratio)

Data Collection Techniques in Business Analytics

1. Primary data – collected first-hand: surveys/questionnaires (Google Forms), interviews, observation, experiments (A/B tests), focus groups.
2. Secondary data – already collected by others: company records, RBI and SEBI publications, Census, NSO reports, websites.
3. Digital/automatic – POS billing systems, website and app logs, social media, sensors (IoT), web scraping.

Exploratory Data Analysis (EDA) – Measures of Central Tendency and Dispersion

EDA means first examining data with summaries and graphs to find patterns, errors and outliers before formal analysis (idea of J.W. Tukey).

• Mean = Σx/n; Median = middle value; Mode = most frequent value.
• Variance σ² = Σ(x − x̄)²/n (sample: ÷ n − 1); SD = √variance; IQR = Q₃ − Q₁.

Detecting Outliers using Boxplots and Z-scores

An outlier is a value far away from the rest. Boxplot rule: outlier if below Q₁ − 1.5×IQR or above Q₃ + 1.5×IQR. Z-score rule: z = (x − x̄)/σ; |z| > 3 is usually an outlier (for small samples |z| > 2 is treated as unusual).

Illustration 1: Monthly sales (₹ thousand) of 10 salesmen: 22, 25, 27, 28, 30, 30, 31, 33, 35, 60. Find mean, median, mode, variance, SD, IQR; check outliers and skewness.

• Mean = 321/10 = 32.1; Median = (30 + 30)/2 = 30; Mode = 30.
• Variance = 992.90/10 = 99.29; SD = √99.29 = 9.96.
• Q₁ (2.75th value) = 25 + 0.75(27 − 25) = 26.5; Q₃ (8.25th value) = 33 + 0.25(35 − 33) = 33.5; IQR = 7.
• Fences: 26.5 − 10.5 = 16 and 33.5 + 10.5 = 44. 60 > 44, so 60 is an outlier. Its z-score is 2.80 (above 2, unusual for a sample of 10).
• Karl Pearson's skewness = 3(Mean − Median)/SD = 3(32.1 − 30)/9.96 = +0.63 → positively skewed (long right tail, caused by 60).

Illustration 2 (comparing with z-scores): A student scored 72 in Accounts (class mean 60, SD 8) and 65 in Statistics (mean 50, SD 5). z(Accounts) = (72 − 60)/8 = 1.5; z(Statistics) = (65 − 50)/5 = 3.0. Relative performance is better in Statistics.

Skewness and Kurtosis

Skewness measures lack of symmetry. Symmetric: mean = median = mode. Positive skew: mean > median > mode (e.g. income). Negative skew: mean < median < mode. Bowley's coefficient = (Q₃ + Q₁ − 2Median)/(Q₃ − Q₁).

Kurtosis measures peakedness/tail heaviness compared to the normal curve: mesokurtic (normal, β₂ = 3), leptokurtic (tall peak, heavy tails, β₂ > 3), platykurtic (flat, β₂ < 3). β₂ = μ₄/μ₂². SPSS reports "excess kurtosis" = β₂ − 3.

Graphical Representation of Data – Histograms, Bar Charts and Pie Charts

• Histogram – adjoining bars for continuous class intervals; shows shape of distribution.
• Bar chart – separate bars for categories (sales by region).
• Pie chart – circle divided into sectors showing shares of a total (market share); angle = (value/total) × 360°.

Scatter Plots for Correlation Analysis

Points plotted for pairs (x, y). Upward pattern = positive correlation, downward = negative, no pattern = no correlation. Also shows outliers and non-linear patterns.

Heatmaps, Treemaps and Bubble Charts

• Heatmap – a grid where colour intensity shows value (sales by store × month, correlation matrix).
• Treemap – nested rectangles whose area shows size (sales by category and sub-category).
• Bubble chart – scatter plot where bubble size shows a third variable (x = price, y = sales, size = profit).

Box-and-Whisker Plots for Distribution Analysis

Shows the five-number summary: minimum, Q₁, median, Q₃, maximum. The box covers the IQR, whiskers extend to the last non-outlier values, and outliers appear as dots. Useful to compare branches or products side by side.

Introduction to SPSS for EDA – IBM SPSS Interface and Data Import

SPSS (Statistical Package for the Social Sciences, now IBM SPSS Statistics) is menu-driven software for analysis. Windows: Data View (rows = cases, columns = variables), Variable View (name, type, label, values, measure scale), Output Viewer (results) and Syntax Editor. Import: File → Open → Data (Excel, CSV, text).

Data Cleaning and Handling Missing Values

Check wrong codes and duplicates (Data → Identify Duplicate Cases), define missing values in Variable View, and treat them by deleting cases (listwise/pairwise) or replacing with mean/median (Transform → Replace Missing Values).

Generating Summary Statistics and Charts in SPSS

Analyze → Descriptive Statistics → Frequencies (tables, mode, percentiles, charts), Descriptives (mean, SD, z-scores via "Save standardized values"), Explore (boxplots, outliers, skewness, kurtosis). Graphs → Chart Builder for histograms, bar, pie and scatter charts.

Likely exam questions

1. (Long, 8/16 marks) Explain the role of statistics in business decision-making. Distinguish between descriptive and inferential statistics, and explain the four levels of measurement with examples.
2. (Long, 8/16 marks) What is Exploratory Data Analysis? Monthly sales (₹ thousand) of 10 salesmen are 22, 25, 27, 28, 30, 30, 31, 33, 35, 60. Calculate mean, median, standard deviation and IQR, detect outliers using the boxplot rule and z-scores, and comment on skewness.
3. (Short, 4 marks) Distinguish between qualitative and quantitative data.
4. (Short, 4 marks) Explain skewness and kurtosis with diagrams in words.
5. (Short, 4 marks) Explain heatmaps, treemaps and bubble charts.

Syllabus topics (RTMNU)

• Probability Concepts
• Rules of Probability (Addition and Multiplication)
• Conditional Probability and Bayes’ Theorem
• Random Variables and Probability Distributions
• Discrete and Continuous Distributions
• Binomial, Poisson, and Hypergeometric Distributions
• Normal, Exponential, and Uniform Distributions
• Central Limit Theorem and Its Business Applications
• Sampling and Estimation
• Probability Sampling Methods (Simple Random, Stratified, Systematic, Cluster)
• Non-Probability Sampling (Convenience, Judgment, Snowball)
• Confidence Intervals for Means and Proportions
• Hypothesis Testing
• Null and Alternative Hypotheses
• Type I and Type II Errors
• Z-Test, T-Test, Paired Sample T-Test, and Independent Sample T-Test
• ANOVA for Multiple Group Comparisons
• SPSS Applications in Probability and Hypothesis Testing
• Running Probability Simulations in SPSS
• Conducting Hypothesis Testing in SPSS
• Interpreting SPSS Outputs for Business Decision-Making

Probability Concepts – Rules of Probability (Addition and Multiplication)

Probability is a number between 0 and 1 that measures the chance of an event. P(A) = favourable outcomes ÷ total equally likely outcomes.

• Addition rule: P(A or B) = P(A) + P(B) − P(A and B). For mutually exclusive events (cannot happen together), P(A or B) = P(A) + P(B).
• Multiplication rule: P(A and B) = P(A) × P(B|A). For independent events, P(A and B) = P(A) × P(B).

Example: 30% of customers buy tea, 20% buy coffee, 10% buy both. P(tea or coffee) = 0.30 + 0.20 − 0.10 = 0.40.

Conditional Probability and Bayes' Theorem

Conditional probability P(A|B) = P(A and B)/P(B) – chance of A when B is known to have happened. Bayes' theorem revises (updates) prior probabilities using new evidence: P(A_i|B) = P(A_i)P(B|A_i) / Σ P(A_j)P(B|A_j).

Illustration 1: Machine A makes 60% of output with 2% defectives; Machine B makes 40% with 5% defectives. A unit is found defective. Find the chance it came from A.

Random Variables and Probability Distributions

A random variable takes numerical values decided by chance (number of defectives, daily demand). Discrete – countable values; continuous – any value in a range. A probability distribution lists every value with its probability (total = 1). Expected value E(X) = Σ x·p(x).

Discrete Distributions – Binomial, Poisson and Hypergeometric

• Binomial – n independent trials, each success/failure with constant p. P(X = r) = ⁿC_r p^r q^n−r; mean = np, variance = npq.
• Poisson – number of rare events in a fixed time/space, average λ. P(X = r) = e^−λλ^r/r!; mean = variance = λ.
• Hypergeometric – like binomial but sampling without replacement from a small finite lot: P(X = r) = [^KC_r × ^N−KC_n−r] / ^NC_n.

Illustration 2: (a) 10% of bulbs are defective. In a pack of 5, P(none defective) = (0.9)⁵ = 0.5905; P(at least one) = 1 − 0.5905 = 0.4095. (b) A bank branch gets on average 3 cheque-bounce cases a day (Poisson). P(no case) = e⁻³ = 0.0498; P(exactly 2) = e⁻³×9/2 = 0.2240.

Continuous Distributions – Normal, Exponential and Uniform

• Normal – bell-shaped, symmetric about mean μ; mean = median = mode. Standardise with Z = (X − μ)/σ. About 68%, 95% and 99.7% of values lie within 1, 2 and 3 SD of the mean. Example: pack weight μ = 500 g, σ = 50 g; P(X > 600) = P(Z > 2) = 0.0228.
• Exponential – time between events (time between customer arrivals); mean = 1/λ.
• Uniform – all values in [a, b] equally likely; mean = (a + b)/2.

Central Limit Theorem and Its Business Applications

CLT: for a large sample (n ≥ 30), the sample mean is approximately normally distributed with mean μ and standard error σ/√n, whatever the shape of the population. Use: it allows z-tests and confidence intervals for average sales, average bill amount, quality checks, even when the data are skewed.

Sampling and Estimation – Probability Sampling Methods

1. Simple random – every unit has an equal chance (lottery, random numbers).
2. Stratified – population divided into groups (strata, e.g. urban/rural) and a random sample taken from each.
3. Systematic – every k-th unit from a list (every 10th invoice).
4. Cluster – whole groups (e.g. a few branches) selected randomly and all units in them studied.

Non-Probability Sampling (Convenience, Judgment, Snowball)

• Convenience – whoever is easy to reach (shoppers at a mall).
• Judgment (purposive) – the researcher chooses units he thinks are typical (expert dealers).
• Snowball – existing respondents refer others (useful for rare groups like startup founders).

Confidence Intervals for Means and Proportions

Mean: x̄ ± z × σ/√n (use t and s for small samples). Proportion: p ± z × √[p(1 − p)/n]. z = 1.96 for 95%, 2.576 for 99%.

Illustration 3: (a) n = 64 bills, x̄ = ₹480, s = ₹40. 95% CI = 480 ± 1.96 × 40/8 = 480 ± 9.8 = ₹470.2 to ₹489.8. (b) 120 of 400 customers prefer a new brand; p = 0.30, SE = √(0.3 × 0.7/400) = 0.0229; 95% CI = 0.30 ± 0.0449 = 25.5% to 34.5%.

Hypothesis Testing – Null and Alternative Hypotheses; Type I and Type II Errors

H₀ (null) – statement of no difference/no effect (μ = 500 g). H₁ (alternative) – what we suspect (μ ≠ 500 g). Steps: state H₀ and H₁, choose level of significance α (5%), compute test statistic, compare with table value (or p-value with α), decide.

Z-Test, T-Test, Paired Sample T-Test and Independent Sample T-Test

• Z-test – large sample (n ≥ 30) or σ known: z = (x̄ − μ)/(σ/√n).
• One-sample t-test – small sample, σ unknown: t = (x̄ − μ)/(s/√n), d.f. = n − 1.
• Paired t-test – same units measured twice (before/after training): t = d̄/(s_d/√n).
• Independent t-test – two separate groups (male vs female customers): t = (x̄₁ − x̄₂)/√[s²(1/n₁ + 1/n₂)] with pooled s².

Illustration 4: A company claims packs weigh 500 g. A sample of 9: 492, 498, 495, 490, 497, 494, 499, 491, 496. x̄ = 494.67, s = 3.162, t = (494.67 − 500)/(3.162/3) = −5.06. Table t (8 d.f., 5%, two-tailed) = 2.306. |t| > 2.306 → reject H₀; packs are under-weight.

Illustration 5 (paired): Sales of 5 salesmen before/after training: 50→54, 55→58, 48→50, 60→63, 52→55. d = 4, 3, 2, 3, 3; d̄ = 3, s_d = 0.707; t = 3/(0.707/√5) = 9.49 > 2.776 (4 d.f.) → training increased sales.

ANOVA for Multiple Group Comparisons

ANOVA (Analysis of Variance, R.A. Fisher) tests whether three or more group means are equal. F = MSB/MSW (between-group variance ÷ within-group variance).

Illustration 6: Sales (₹ thousand) under three ad strategies: A: 10, 12, 14; B: 15, 17, 16; C: 20, 18, 22. Means 12, 16, 20; grand mean 16.

Table F (2, 6) at 5% = 5.14. 16 > 5.14 → reject H₀; strategies differ.

SPSS Applications in Probability and Hypothesis Testing

• Running probability simulations: Transform → Compute Variable with functions such as RV.NORMAL(μ, σ), RV.BINOM(n, p), RV.POISSON(λ); CDF.NORMAL gives probabilities.
• Conducting tests: Analyze → Compare Means → One-Sample T Test / Independent-Samples T Test / Paired-Samples T Test / One-Way ANOVA.
• Interpreting output: read the "Sig." (p-value) column. If Sig. < 0.05 reject H₀. For independent t-test first check Levene's test for equal variances.

Likely exam questions

1. (Long, 8/16 marks) State Bayes' theorem. Machine A produces 60% of output with 2% defectives and Machine B 40% with 5% defectives. A unit picked at random is defective; find the probability that it was made by Machine A. Also explain binomial and Poisson distributions.
2. (Long, 8/16 marks) Explain the steps of hypothesis testing and Type I and Type II errors. A sample of 9 packs weighs 492, 498, 495, 490, 497, 494, 499, 491, 496 g. Test at 5% whether the mean weight is 500 g (table t for 8 d.f. = 2.306).
3. (Short, 4 marks) State the Central Limit Theorem and its business use.
4. (Short, 4 marks) Distinguish between probability and non-probability sampling.
5. (Short, 4 marks) 120 of 400 customers prefer a brand. Find a 95% confidence interval for the proportion.

Syllabus topics (RTMNU)

• Correlation and Regression Analysis
• Pearson and Spearman Correlation
• Simple and Multiple Linear Regression
• Model Assumptions and Diagnostics (Multicollinearity, Heteroscedasticity)
• Time Series Analysis and Forecasting
• Components of Time Series Data (Trend, Seasonality, Cyclical, Irregular)
• Moving Averages and Exponential Smoothing
• ARIMA Models for Business Forecasting
• Multivariate Data Analysis
• Principal Component Analysis (PCA) for Dimensionality Reduction
• Factor Analysis and Variable Reduction Techniques
• Cluster Analysis for Market Segmentation
• Decision Trees and Machine Learning Basics
• Classification and Regression Trees (CART)
• CHAID Algorithms
• Logistic Regression for Classification Problems
• SPSS Applications in Regression and Forecasting
• Performing Regression and Forecasting in SPSS
• Conducting Factor and Cluster Analysis in SPSS
• Case Studies in Business Analytics

Correlation and Regression Analysis – Pearson and Spearman Correlation

Correlation measures the strength and direction of the relationship between two variables (−1 to +1).

• Karl Pearson's r = Σdx·dy / √(Σdx² × Σdy²) – for interval/ratio data and linear relationships.
• Spearman's rank correlation R = 1 − 6ΣD² / [n(n² − 1)] – for ranks (ordinal data) or when data are not normal. D = difference in ranks.

Illustration 1 (Spearman): Two judges rank 6 products:

R = 1 − (6 × 6)/(6 × 35) = 1 − 36/210 = 0.829 → high agreement between the judges.

Simple and Multiple Linear Regression

Regression estimates the value of a dependent variable (Y) from one or more independent variables (X). Simple: Ŷ = a + bX, where b = Σdx·dy/Σdx² and a = ȳ − b x̄. Multiple: Ŷ = a + b₁X₁ + b₂X₂ + … (e.g. sales on advertising, price and number of outlets). R² (coefficient of determination) shows the share of variation in Y explained by the model.

Illustration 2: Advertising (X) 2, 4, 5, 6, 8 and Sales (Y) 10, 14, 15, 19, 22 (₹ lakh). x̄ = 5, ȳ = 16, Σdx·dy = 41, Σdx² = 20, Σdy² = 86.

• b = 41/20 = 2.05; a = 16 − 2.05 × 5 = 5.75. Regression line: Ŷ = 5.75 + 2.05X.
• If advertising = ₹7 lakh, estimated sales = 5.75 + 14.35 = ₹20.10 lakh.
• r = 41/√(20 × 86) = 0.989; R² = 0.977 → 97.7% of variation in sales is explained by advertising.

Model Assumptions and Diagnostics (Multicollinearity, Heteroscedasticity)

Assumptions: linear relationship, independent errors, errors normally distributed, constant error variance, and no exact relationship among X variables.

• Multicollinearity – independent variables are highly correlated with each other (e.g. income and spending power). Coefficients become unstable. Check with VIF (Variance Inflation Factor); VIF above 10 (some use 5) signals a problem. Remedy: drop or combine variables.
• Heteroscedasticity – the spread of errors is not constant (errors grow with firm size). Check with a residual plot (funnel shape) or Breusch–Pagan test. Remedy: log transformation, weighted least squares.

Time Series Analysis and Forecasting – Components of Time Series Data

1. Trend (T) – long-term upward or downward movement (rising mobile data use).
2. Seasonality (S) – regular pattern within a year (higher sales in Diwali season).
3. Cyclical (C) – up-and-down movement over several years (business cycles).
4. Irregular (I) – random, unpredictable changes (floods, strikes, pandemic).

Models: additive Y = T + S + C + I; multiplicative Y = T × S × C × I.

Moving Averages and Exponential Smoothing

Moving average – average of a fixed number of consecutive periods, moved one period at a time; smooths short-term ups and downs. Exponential smoothing – F_t+1 = F_t + α(A_t − F_t), where α (0 to 1) is the smoothing constant; recent data get more weight.

Illustration 3: Sales of a company (₹ crore), 2019–2025: 20, 24, 22, 26, 30, 28, 32. Find 3-yearly moving averages and exponential smoothing forecasts with α = 0.3 (take forecast for 2020 = actual 2019 = 20).

Working: F₂₀₂₁ = 20 + 0.3(24 − 20) = 21.20; F₂₀₂₂ = 21.20 + 0.3(22 − 21.20) = 21.44; and so on. Forecast for 2026 = 25.88 + 0.3(32 − 25.88) = ₹27.71 crore. The moving averages (22 → 30) show a clear rising trend.

ARIMA Models for Business Forecasting

ARIMA (p, d, q) – AutoRegressive Integrated Moving Average (Box–Jenkins method). AR (p): uses past values; I (d): differencing to make the series stationary (constant mean and variance); MA (q): uses past forecast errors. Steps: identify (ACF/PACF plots), estimate, check residuals, forecast. Used for sales, stock prices, electricity demand.

Multivariate Data Analysis – Principal Component Analysis (PCA) for Dimensionality Reduction

PCA converts many correlated variables into a few new uncorrelated variables called principal components, which keep most of the information (variance). The first component explains the most variance. Components with eigenvalue > 1 are usually kept. Example: 15 financial ratios reduced to 3 components.

Factor Analysis and Variable Reduction Techniques

Factor analysis finds hidden (latent) factors that explain the correlations among many observed variables. Example: 20 survey questions on a bank grouped into "service quality", "convenience" and "trust". Terms: factor loading (correlation of a variable with a factor), KMO test (sampling adequacy, > 0.6 acceptable), Bartlett's test, and rotation (Varimax) for easy interpretation.

Cluster Analysis for Market Segmentation

Groups customers so that those within a cluster are similar and clusters differ. Methods: hierarchical (dendrogram) and k-means (choose k, assign to nearest centre, recompute centres, repeat). Example: a telecom company forms segments "heavy data users", "voice users", "low users" for different plans.

Decision Trees and Machine Learning Basics – CART and CHAID Algorithms

A decision tree splits data step by step into branches using "if–then" rules until groups are pure.

• CART (Classification and Regression Trees) – binary (two-way) splits; uses Gini index for classification; gives classification or regression trees.
• CHAID (Chi-squared Automatic Interaction Detection) – uses chi-square tests; can make multi-way splits; popular in market research.

Logistic Regression for Classification Problems

Used when Y is binary (buy/not buy, default/no default). log[p/(1 − p)] = a + b₁X₁ + …; p = 1/(1 + e^{−(a + bX)}). Results are read as odds ratios; a case with p ≥ 0.5 is classed "yes". Banks use it for credit scoring.

SPSS Applications in Regression and Forecasting

• Regression: Analyze → Regression → Linear (read R², ANOVA F, coefficients, Sig., VIF under Statistics → Collinearity diagnostics). Binary Logistic for logistic regression.
• Forecasting: Analyze → Forecasting → Create Traditional Models (Expert Modeler, Exponential Smoothing, ARIMA).
• Factor and cluster analysis: Analyze → Dimension Reduction → Factor (KMO and Bartlett, Varimax); Analyze → Classify → K-Means Cluster / Hierarchical Cluster; Analyze → Classify → Tree (CHAID, CRT).
• Case studies in business analytics: e.g. a retail chain regresses monthly sales on advertising and price, then forecasts next quarter; a bank uses factor analysis on a customer-satisfaction survey and clusters customers for targeted offers.

Likely exam questions

1. (Long, 8/16 marks) Explain simple and multiple linear regression and the problems of multicollinearity and heteroscedasticity. From Advertising (₹ lakh) 2, 4, 5, 6, 8 and Sales (₹ lakh) 10, 14, 15, 19, 22, fit the regression line of sales on advertising and estimate sales when advertising is ₹7 lakh.
2. (Long, 8/16 marks) Explain the components of a time series. Sales (₹ crore) for 2019–2025 are 20, 24, 22, 26, 30, 28, 32. Calculate 3-yearly moving averages and explain exponential smoothing and ARIMA models.
3. (Short, 4 marks) Calculate Spearman's rank correlation when ΣD² = 6 and n = 6.
4. (Short, 4 marks) Distinguish between PCA and factor analysis.
5. (Short, 4 marks) Distinguish between CART and CHAID.

Syllabus topics (RTMNU)

• Non-Parametric Statistical Methods
• Chi-Square Tests for Categorical Data
• Wilcoxon Signed-Rank Test
• Kruskal-Wallis Test for Multiple Groups
• Statistical Quality Control and Process Improvement
• Control Charts (X-Bar, R-Chart, p-Chart, c-Chart)
• Six Sigma and Lean Analytics
• Business Process Optimization with Statistical Tools
• Structural Equation Modeling (SEM) and Path Analysis
• Basics of SEM and Its Business Applications
• Model Identification and Goodness-of-Fit Tests
• Path Analysis for Causal Relationships
• Bayesian Statistics and Decision Making
• Introduction to Bayesian Probability
• Bayesian Inference in Business Analytics
• Monte Carlo Simulations
• SPSS Applications in Advanced Analytics
• Running Non-Parametric Tests in SPSS
• Quality Control and Decision Trees in SPSS
• Hands-on Case Studies in Business Performance Analysis

Non-Parametric Statistical Methods – Chi-Square Tests for Categorical Data

Non-parametric tests do not assume that the population is normal; they work with counts, signs or ranks. Useful for small samples and nominal/ordinal data. They are simpler but less powerful than parametric tests when parametric assumptions hold.

Chi-square (χ²) test compares observed frequencies (O) with expected frequencies (E): χ² = Σ (O − E)²/E. Uses: (1) test of independence/association between two attributes, (2) goodness of fit. d.f. = (r − 1)(c − 1). Condition: each expected frequency should be at least 5.

Illustration 1: 140 customers – Male: Brand A 40, Brand B 20; Female: Brand A 30, Brand B 50. Is brand preference independent of gender? (5% level)

H₀: preference is independent of gender. Row totals 60, 80; column totals 70, 70.

d.f. = (2 − 1)(2 − 1) = 1; table χ² at 5% = 3.841. Since 11.667 > 3.841, reject H₀: brand preference depends on gender (women prefer Brand B).

Wilcoxon Signed-Rank Test

Non-parametric alternative to the paired t-test. Steps: find differences for each pair, drop zero differences, rank the absolute differences, attach signs, add positive ranks (T⁺) and negative ranks (T⁻); test statistic T = smaller of the two, compared with the table value (reject H₀ if T ≤ table value). Example: customer ratings of a restaurant before and after renovation.

Kruskal-Wallis Test for Multiple Groups

Non-parametric alternative to one-way ANOVA for three or more independent groups. Rank all observations together, find the rank sum R_i for each group. H = [12/(N(N + 1))] Σ(R_i²/n_i) − 3(N + 1); compared with χ² with (k − 1) d.f. Example: comparing satisfaction scores of three bank branches.

Statistical Quality Control and Process Improvement – Control Charts (X-Bar, R-Chart, p-Chart, c-Chart)

SQC uses statistics to monitor a process. A control chart has a centre line (CL), an upper control limit (UCL) and a lower control limit (LCL), usually at ±3σ. Points within limits = only chance causes (process in control); a point outside = assignable cause to find and remove.

• X̄ chart (variables) – monitors process average: CL = X̿, limits X̿ ± A₂R̄.
• R chart – monitors variability: CL = R̄, UCL = D₄R̄, LCL = D₃R̄.
• p chart (attributes) – proportion defective: p̄ ± 3√[p̄(1 − p̄)/n].
• c chart – number of defects per unit: c̄ ± 3√c̄.

Illustration 2: Five samples of 4 bolts (diameter, mm). Constants for n = 4: A₂ = 0.729, D₃ = 0, D₄ = 2.282.

• X̿ = 56.5/5 = 11.3; R̄ = 11/5 = 2.2.
• X̄ chart: UCL = 11.3 + 0.729 × 2.2 = 12.90; LCL = 11.3 − 1.60 = 9.70.
• R chart: UCL = 2.282 × 2.2 = 5.02; LCL = 0.
• All sample means (10.0 to 12.0) and ranges (2 to 3) are within limits → process is in control.

Illustration 3: (a) 10 samples of 100 items had 80 defectives in total. p̄ = 80/1000 = 0.08; σ_p = √(0.08 × 0.92/100) = 0.0271. UCL = 0.08 + 0.0814 = 0.161; LCL = 0.08 − 0.0814 is negative, so LCL = 0. (b) Average 4 defects per cloth piece: c chart UCL = 4 + 3√4 = 10, LCL = 4 − 6 < 0, so 0.

Six Sigma and Lean Analytics

Six Sigma (started at Motorola, made famous by GE) aims at near-perfect quality – no more than 3.4 defects per million opportunities (DPMO). Method: DMAIC – Define, Measure, Analyze, Improve, Control. Roles: Champions, Black Belts, Green Belts. Lean (from the Toyota Production System) removes waste (excess inventory, waiting, transport, over-processing, defects, motion, over-production). Lean analytics uses data and KPIs (cycle time, lead time, defect rate) to track waste reduction. Lean Six Sigma combines both. Indian example: the Mumbai dabbawalas are often cited for very low error rates.

Business Process Optimization with Statistical Tools

Tools: process maps/flowcharts, Pareto chart (80/20 rule – few causes create most problems), cause-and-effect (fishbone) diagram, histograms, scatter diagrams, control charts, check sheets, regression and design of experiments. Process capability index C_p = (USL − LSL)/6σ; C_p ≥ 1.33 is generally considered capable.

Structural Equation Modeling (SEM) and Path Analysis – Basics of SEM and Its Business Applications

SEM is a multivariate technique that combines factor analysis and multiple regression to test a theory of relationships among latent variables (concepts not measured directly, like satisfaction, trust, loyalty) measured through observed indicators (survey items). It has a measurement model (indicators → latent variables) and a structural model (latent → latent). Software: AMOS, SmartPLS, LISREL. Applications: service quality → satisfaction → loyalty in banks; technology acceptance of UPI; employee engagement studies.

Model Identification and Goodness-of-Fit Tests

Identification: a model can be estimated only if the number of known values (variances and covariances) is at least the number of parameters. Just-identified (equal), over-identified (more knowns – preferred), under-identified (cannot be estimated). Fit indices (common rules of thumb): χ²/df < 3, CFI and TLI ≥ 0.90 (better ≥ 0.95), RMSEA ≤ 0.08 (good ≤ 0.05), GFI ≥ 0.90, SRMR ≤ 0.08.

Path Analysis for Causal Relationships

Path analysis (developed by Sewall Wright) studies direct and indirect effects among observed variables using a path diagram with arrows. Indirect effect = product of path coefficients along the route. Example: Advertising → Brand awareness (0.6) → Purchase intention (0.5): indirect effect = 0.30.

Bayesian Statistics and Decision Making – Introduction to Bayesian Probability

Bayesian probability treats probability as a degree of belief that is updated with evidence. Posterior ∝ Likelihood × Prior. Prior = belief before data; likelihood = how probable the data are under each belief; posterior = revised belief.

Bayesian Inference in Business Analytics

Example: a manager believes a new product has a 40% chance of success (prior). A test market gives a positive result; such results occur in 80% of successful and 30% of failed products. P(success | positive) = (0.4 × 0.8)/(0.4 × 0.8 + 0.6 × 0.3) = 0.32/0.50 = 0.64. Uses: spam filters, fraud detection, A/B testing, demand forecasting with expert opinion.

Monte Carlo Simulations

Uses random numbers to imitate an uncertain system many times. Steps: (1) define the probability distribution of each uncertain variable, (2) find cumulative probabilities and allot random-number ranges, (3) draw random numbers and read values, (4) compute the outcome, (5) repeat many times and analyse. Example: daily demand 10/20/30 units with probabilities 0.3/0.5/0.2 → random numbers 00–29, 30–79, 80–99. Used for project risk, inventory, portfolio value-at-risk.

SPSS Applications in Advanced Analytics

• Running non-parametric tests: Analyze → Descriptive Statistics → Crosstabs → Statistics → Chi-square; Analyze → Nonparametric Tests → Legacy Dialogs → 2 Related Samples (Wilcoxon) / K Independent Samples (Kruskal-Wallis H). Reject H₀ if Sig. < 0.05.
• Quality control and decision trees: Analyze → Quality Control → Control Charts (X-bar/R, p, c); Analyze → Classify → Tree (CHAID, CRT).
• Hands-on case studies in business performance analysis: e.g. a mobile showroom tests whether brand preference depends on age group (chi-square), monitors service time with control charts, and compares branch ratings with Kruskal-Wallis.

Likely exam questions

1. (Long, 8/16 marks) What are non-parametric tests? Explain the chi-square test, Wilcoxon signed-rank test and Kruskal-Wallis test. Of 140 customers, males prefer Brand A 40 and Brand B 20; females Brand A 30 and Brand B 50. Test at 5% whether brand preference is independent of gender (table χ² for 1 d.f. = 3.841).
2. (Long, 8/16 marks) What is statistical quality control? Explain X̄, R, p and c charts. Five samples of size 4 have means 11.5, 11.0, 12.0, 10.0, 12.0 and ranges 3, 2, 2, 2, 2. Compute control limits for the X̄ and R charts (A₂ = 0.729, D₃ = 0, D₄ = 2.282) and comment.
3. (Short, 4 marks) Explain Six Sigma and the DMAIC method.
4. (Short, 4 marks) What is Structural Equation Modeling? State any four goodness-of-fit indices.
5. (Short, 4 marks) Explain Bayesian inference with a simple business example.