UGC NET MBA Unit-8

Statistics for Management, Operations, and Operations Research

PART 1: STATISTICS FOR MANAGEMENT

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1. Concept and Scope

Statistics is both a science and an art of collecting, classifying, presenting, analyzing, and interpreting numerical data to aid rational decision-making under uncertainty.

It provides quantitative foundations for managerial functions such as planning, control, and forecasting.

Branches of Statistics

1. Descriptive Statistics – summarizing data through tables, charts, and averages.

2. Inferential Statistics – drawing conclusions about populations from samples using probability theory.

Role of Statistics in Management

• Marketing: Market surveys, consumer behaviour analysis.

• Finance: Portfolio risk analysis, stock price movements.

• Production: Quality control, forecasting demand.

• HR: Wage analysis, performance evaluation.

• Operations: Scheduling, process optimization.

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2. Data and Its Types

A. Based on Source

• Primary Data: Collected first-hand for a specific study (surveys, interviews).

• Secondary Data: Collected earlier for another purpose (reports, journals, databases).

B. Based on Nature

• Qualitative (Attribute): Categorical (e.g., gender, brand preference).

• Quantitative (Variable): Numeric (e.g., income, profit).

C. Based on Measurement Scale

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🟩 3. MEASURES OF CENTRAL TENDENCY

Central tendency expresses the “typical” or “representative” value of a dataset.

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A. Arithmetic Mean

$$\bar{X}=\frac{\sum X}{N}$$

For grouped data:

$$\bar{X}=\frac{\sum fX}{\sum f}$$

Merits: Simple, algebraically tractable.
Limitations: Affected by extreme values.

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B. Median

The middle value when data are arranged in order.

$$\text{Median}=L+\frac{(\frac{N}{2}-CF)}{f}\times h$$

• Less affected by outliers.

• Appropriate for skewed data.

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C. Mode

Most frequent value.
For grouped data:

$$\text{Mode}=L+\frac{(f_1-f_0)}{(2f_1-f_0-f_2)}\times h$$

Used for qualitative data like brand or color preference.

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D. Relationship among Mean, Median, and Mode

$$\text{Mode}=3(\text{Median})-2(\text{Mean})$$

Useful for estimating one measure from the other two.

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🟩 4. MEASURES OF DISPERSION

Dispersion measures how values deviate from the average — indicating consistency or risk.

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Example:
Dataset A: Mean = 50, SD = 10 → CV = 20%
Dataset B: Mean = 80, SD = 16 → CV = 20%
→ Both have equal relative variability.

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🟩 5. PROBABILITY AND PROBABILITY DISTRIBUTIONS

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A. Concept of Probability

Probability quantifies the likelihood of occurrence of an event.

$$P(A)=\frac{\text{Favourable outcomes}}{\text{Total outcomes}}$$

Range: 0 ≤ P(A) ≤ 1

• P(A) = 1: Certain event

• P(A) = 0: Impossible event

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B. Rules of Probability

1. Addition Rule:
   If A and B are mutually exclusive,

   $$P(A\cup B)=P(A)+P(B)$$

2. Multiplication Rule:
   For independent events,

   $$P(A\cap B)=P(A)\times P(B)$$

3. Conditional Probability:

   $$P(A\mathrm{\mid }B)=\frac{P(A\cap B)}{P(B)}$$

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C. Probability Distributions

(1) Binomial Distribution

Discrete distribution used for number of successes in n independent trials.

$$P(X=x)=\left(\genfrac{}{}{0px}{}{n}{x}\right)p^x(1-p{)}^{n-x}$$

• Mean = np

• Variance = np(1−p)

Example: Probability of 3 defective bulbs out of 10 when defect rate = 0.1.

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(2) Poisson Distribution

For rare events (e.g., accidents per day).

$$P(X=x)=\frac{e^{-m}{m}^x}{x!}$$

• Mean = Variance = m

Used when n → large, p → small, and np = constant.

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(3) Normal Distribution

Continuous, bell-shaped curve.

$$f(x)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}$$

Properties:

• Symmetrical about mean.

• 68.26% within ±1σ, 95.45% within ±2σ, 99.73% within ±3σ.
  Used in hypothesis testing and control charts.

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(4) Exponential Distribution

Used to model time between events (e.g., waiting time).

$$f(x)=\lambda e^{-\lambda x},x\ge 0$$

Mean = 1/λ
Variance = 1/λ²

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🟩 6. DATA COLLECTION AND QUESTIONNAIRE DESIGN

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Data Collection

• Primary: Through direct observation, survey, or experimentation.

• Secondary: Government reports, journals, internet sources.

Questionnaire Design

1. Define objectives

2. Select information to collect

3. Choose question type:

   • Open-ended (qualitative insights)

   • Closed-ended (quantitative analysis)

4. Logical sequencing (easy → complex)

5. Pilot testing and revision

Common Mistakes: Ambiguous wording, double-barreled questions, poor scaling.

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🟩 7. SAMPLING THEORY

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A. Basic Concepts

• Population (Universe): Entire group under study

• Sample: Representative subset

• Sampling Unit: Element from which data is collected

B. Steps in Sampling Process

1. Define population

2. Select sampling frame

3. Decide sample size

4. Choose technique

5. Collect and analyze

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C. Probability Sampling Techniques

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D. Non-Probability Sampling Techniques

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🟩 8. HYPOTHESIS TESTING

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A. Key Definitions

• Parameter: Numerical summary of population (μ, σ²).

• Statistic: Calculated from sample (x̄, s²).

Goal: Use sample data to infer about population.

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B. Hypothesis Types

• Null Hypothesis (H₀): No significant difference.

• Alternative Hypothesis (H₁): Significant difference exists.

Errors:

• Type I (α): Rejecting true H₀

• Type II (β): Accepting false H₀

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C. Testing Steps

1. Formulate H₀ and H₁

2. Choose significance level (α = 0.05 or 0.01)

3. Choose appropriate test statistic (Z, t, F, χ²)

4. Compute test statistic

5. Compare with critical value

6. Draw conclusion

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D. Parametric Tests

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E. Non-Parametric Test

Chi-Square (χ²) Test:

$${\chi }^2=\sum \frac{(O-E{)}^2}{E}$$

Used for testing independence or goodness of fit.

Example: Relationship between gender and brand preference.

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🟩 9. CORRELATION AND REGRESSION

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A. Correlation

Measures strength and direction of linear relationship.

Karl Pearson’s Coefficient (r):

$$r=\frac{\mathrm{\Sigma }(X-\bar{X})(Y-\bar{Y})}{\sqrt{\mathrm{\Sigma }(X-\bar{X}{)}^2\mathrm{\Sigma }(Y-\bar{Y}{)}^2}}$$

Range: -1 to +1
r = +1 → Perfect positive
r = -1 → Perfect negative

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B. Rank Correlation (Spearman’s ρ)

$$\rho =1-\frac{6\mathrm{\Sigma }{D}^2}{n(n^2-1)}$$

Used when data are ordinal (ranks).

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C. Regression Analysis

Used to predict value of dependent variable (Y) from independent variable (X).

Simple Linear Regression:

$$Y=a+bX$$

where
$b=\frac{\mathrm{\Sigma }(X-\bar{X})(Y-\bar{Y})}{\mathrm{\Sigma }(X-\bar{X}{)}^2}$

Multiple Regression:
$Y=a+b_1{X}_1+b_2{X}_2+\mathrm{.}\mathrm{.}\mathrm{.}+b_n{X}_n$

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🟩 10. OPERATIONS MANAGEMENT

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A. Concept

Operations Management deals with conversion of inputs (materials, labour, capital, information) into outputs (goods/services) efficiently.

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B. Functions

• Product design & process selection

• Plant layout and facility location

• Capacity planning

• Scheduling and inventory control

• Maintenance and quality management

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C. Objectives

1. Improve productivity

2. Optimize resources

3. Ensure quality and timely delivery

4. Minimize cost

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🟩 11. FACILITY LOCATION AND LAYOUT

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Facility Location

The strategic decision of choosing where to situate production or service facilities.

Quantitative Methods:

• Centre of Gravity Method: minimizes transport cost

• Break-even Analysis: compares fixed and variable cost by site

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Plant Layout

Arrangement of machines, departments, or work areas.

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🟩 12. ENTERPRISE RESOURCE PLANNING (ERP)

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A. Concept

ERP integrates core business functions through a central database.

ERP Modules:

1. Finance

2. HR

3. Production

4. SCM

5. CRM

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B. ERP Implementation Steps

1. Project planning

2. Requirement analysis

3. System design & customization

4. Data migration

5. Training

6. Testing & Go-live

Benefits: Integration, transparency, faster reporting.
Challenges: High cost, change resistance, data migration errors.

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🟩 13. PRODUCTION SCHEDULING AND CONTROL

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A. Loading: Assigning jobs to machines.

B. Scheduling: Determining when and in what sequence jobs are processed.

C. Sequencing: Prioritizing jobs (rules: FCFS, SPT, EDD).

D. Monitoring: Tracking progress, revising schedules.

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🟩 14. QUALITY MANAGEMENT

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A. Quality Concepts

Quality = fitness for purpose.
Quality management ensures that output meets customer expectations.

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B. Statistical Quality Control (SQC)

Uses control charts (mean, range, p-chart, c-chart) to monitor process variation.

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C. Total Quality Management (TQM)

An organization-wide philosophy emphasizing continuous improvement and customer satisfaction.

Principles:

1. Customer orientation

2. Continuous improvement (Kaizen)

3. Employee involvement

4. Scientific decision-making

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D. Quality Tools

• Control Charts

• Fishbone Diagram (Ishikawa)

• Pareto Analysis (80/20 Rule)

• Check Sheets, Histograms, Scatter Diagrams

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E. Kaizen

Continuous small improvements involving all employees.

F. Benchmarking

Comparing performance with best-in-class organizations.

G. Six Sigma

A disciplined methodology targeting defect reduction to 3.4 defects per million opportunities (DPMO).
Focuses on DMAIC cycle – Define, Measure, Analyze, Improve, Control.

H. ISO 9000 Series

Global quality management system standards (documentation, process control, auditing).

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🟩 15. OPERATIONS RESEARCH (OR)

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Definition

Operations Research applies scientific and mathematical models to managerial decision-making for optimization of limited resources.

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Applications:

• Production scheduling

• Inventory control

• Transportation and distribution

• Network planning

• Queuing and service design

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A. Transportation Problem

Objective: Minimize total cost of shipping goods.

$$Z=\sum _{i=1}^m\sum _{j=1}^n{C}_{ij}{X}_{ij}$$

Methods:

1. Initial solution → North-West Corner, Least Cost, Vogel’s Approximation (VAM).

2. Optimality test → MODI method.

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B. Queuing Theory

Studies waiting line systems.

Parameters:
λ = Arrival rate, μ = Service rate
System utilization: ρ = λ / μ
Objective → minimize waiting cost + service cost.

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C. Decision Theory

Used when decisions must be made under risk or uncertainty.

Criteria:

• Maximax (optimistic)

• Maximin (pessimistic)

• Minimax regret (Savage)

• Expected monetary value (probabilistic)

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D. PERT / CPM (Project Scheduling)

PERT Expected Time:

$$t_e=\frac{a+4m+b}{6}$$

Variance $={\left(\frac{b-a}{6}\right)}^2$

Critical Path: Longest path through network; determines project duration.
Slack Time: $LST-EST$ → available delay time.