Statistics

1. Introduction to Statistics

Statistics is the science of collecting, organizing, presenting, analyzing and interpreting numerical data for decision making. It is widely used in economics, agriculture, medicine, engineering, business, banking and government planning.

Branches of Statistics

1. Descriptive Statistics

• Organizes and summarizes data.
• Includes tables, charts, averages and dispersion.

2. Inferential Statistics

• Draws conclusions about a population using sample data.
• Includes estimation, hypothesis testing and prediction.

Functions

• Collection of data
• Classification
• Tabulation
• Presentation
• Analysis
• Interpretation
• Decision making

Limitations

• Deals only with numerical data.
• Results depend on data quality.
• Cannot establish absolute truth.
• May be misused.

2. Collection of Data

Data are numerical observations collected for analysis.

Types of Data

Primary Data

Collected first-hand.

Methods:

• Direct interview
• Questionnaire
• Observation
• Experiment
• Telephone survey

Advantages:

• Original
• Reliable
• Suitable for objectives

Disadvantages:

• Costly
• Time-consuming

Secondary Data

Collected by others.

Sources:

• Census
• NSS
• RBI reports
• Economic Survey
• Statistical Handbooks
• Journals

Advantages:

• Cheap
• Easily available

Disadvantages:

• May be outdated
• Accuracy uncertain

3. Classification and Tabulation

Classification means arranging data into groups.

Types

• Qualitative
• Quantitative
• Chronological
• Geographical

Frequency Distribution

Groups observations into class intervals.

Terms:

• Class interval
• Frequency
• Class limit
• Mid value
• Class width

Benefits

• Simplifies large data
• Easy comparison
• Graphical presentation

4. Diagrammatic and Graphical Presentation

Diagrams

• Bar diagram
• Multiple bar
• Pie chart
• Rectangle
• Square

Graphs

• Histogram
• Frequency polygon
• Frequency curve
• Ogive
• Line graph

Applications

• Comparison
• Trend analysis
• Forecasting
• Presentation

5. Measures of Central Tendency

Represents the central value.

Arithmetic Mean

Most commonly used average.

Properties:

• Based on all observations.
• Easy to calculate.
• Algebraically convenient.

Merits:

• Stable
• Scientific
• Useful in further analysis

Demerits:

• Affected by extreme values.

Median is preferred when data contain outliers.

Mode is preferred for qualitative data.

Median

Middle value after arranging observations.

Advantages:

• Not affected by extremes.
• Suitable for skewed data.

Disadvantages:

• Ignores many observations.

Mode

Most frequently occurring value.

Applications:

• Market research
• Fashion industry
• Consumer preference

Relationship

6. Measures of Dispersion

Shows variability.

Range

Largest − Smallest

Advantages:

• Simple

Disadvantages:

• Uses only two observations.

Quartile Deviation

Half of interquartile range.

Useful for skewed distributions.

Mean Deviation

Average of absolute deviations.

Standard Deviation

Most important measure of dispersion.

Characteristics:

• Uses all observations.
• Stable.
• Widely applied.

Applications:

• Finance
• Research
• Risk analysis
• Quality control

Variance is the square of standard deviation.

Coefficient of Variation (CV)

Lower CV indicates greater consistency.

7. Skewness

Measures asymmetry.

Symmetrical Distribution

Mean = Median = Mode

Positive Skew

Mean > Median > Mode

Long right tail.

Negative Skew

Mean < Median < Mode

Long left tail.

8. Kurtosis

Measures peakedness.

9. Probability

Probability measures uncertainty.

Range: 0 ≤ P(A) ≤ 1

• 0 = Impossible
• 1 = Certain

Types

• Classical
• Empirical
• Subjective

Addition Rule

Multiplication Rule

Conditional Probability

Probability of A given B.

Bayes Theorem

Used for updating probabilities.

Applications

• Medical diagnosis
• Machine learning
• Artificial Intelligence
• Banking
• Fraud detection

10. Random Variable

A variable whose value depends on chance.

Types

Discrete

Examples: Number of heads, Defective bulbs

Continuous

Examples: Height, Weight, Rainfall

11. Probability Distributions

Binomial Distribution

Conditions:

• Fixed trials
• Two outcomes
• Independent trials
• Constant probability

Applications:

• Quality inspection
• Coin tossing
• Success-failure experiments

Poisson Distribution

Suitable for rare events.

Applications:

• Road accidents
• Machine failures
• Phone calls

Normal Distribution

Bell-shaped curve.

Characteristics:

• Symmetric
• Mean = Median = Mode
• Total area = 1

Applications:

• IQ
• Examination scores
• Biological measurements
• Manufacturing

12. Sampling

Sampling means selecting part of population.

Census vs Sampling

Census

• Entire population
• Accurate
• Costly

Sampling

• Partial study
• Faster
• Economical

Sampling Methods

Probability Sampling

• Simple Random
• Stratified
• Systematic
• Cluster

Non-probability Sampling

• Convenience
• Judgment
• Quota
• Snowball

Advantages

• Saves time
• Reduces cost
• Suitable for large populations

13. Sampling Errors

Sampling Error

Difference due to sample selection.

Can be reduced by increasing sample size.

Non-sampling Error

Occurs due to:

• Bias
• Wrong recording
• Measurement error
• Processing error

14. Estimation

Estimating unknown population parameters.

Point Estimation

Single value estimate.

Interval Estimation

Range of values with confidence level.

Confidence Levels

• 90%
• 95%
• 99%

15. Hypothesis Testing

Hypothesis is an assumption about population.

Null Hypothesis (H₀)

No difference.

Alternative Hypothesis (H₁)

Difference exists.

Steps

1. Form hypothesis
2. Select significance level
3. Calculate statistic
4. Decision

Errors

Type I Error

Reject true H₀.

Type II Error

Accept false H₀.

Common Tests

• Z-test
• t-test
• Chi-square test
• F-test

16. Correlation

Correlation measures the degree and direction of relationship between two or more variables. It indicates how one variable changes with another but does not prove cause and effect.

Types of Correlation

1. Positive Correlation

Both variables increase or decrease together.

Examples: Income and expenditure, Height and weight, Education and salary

2. Negative Correlation

One variable increases while the other decreases.

Examples: Price and demand, Speed and travel time, Interest rate and investment

3. Zero Correlation

No relationship exists.

Example: Shoe size and intelligence

4. Linear Correlation

Relationship follows a straight line.

5. Non-linear Correlation

Relationship follows a curve.

Karl Pearson's Correlation Coefficient (r)

• Value ranges from -1 to +1
• r = +1 → Perfect positive correlation
• r = -1 → Perfect negative correlation
• r = 0 → No correlation

Spearman's Rank Correlation

Used when observations are ranked instead of measured.

Applications

• Agriculture
• Economics
• Meteorology
• Psychology
• Engineering
• Medical research

Limitations

• Does not imply causation.
• Sensitive to extreme values.
• Measures mainly linear relationships.

17. Regression Analysis

Regression predicts one variable using another.

Types

• Simple Regression
• Multiple Regression

Regression Lines

• Regression of Y on X
• Regression of X on Y

Applications

• Sales forecasting
• Rainfall prediction
• Crop yield estimation
• Demand forecasting
• Financial analysis

Differences between Correlation and Regression

• Correlation: Measures relationship; two-way measure; coefficient lies between -1 and +1; no cause-effect implication.
• Regression: Predicts values; one dependent variable; regression coefficient has no fixed limits; used for prediction.

18. Analysis of Variance (ANOVA)

ANOVA tests whether three or more population means are equal.

Instead of comparing groups individually, it compares overall variation.

Types

One-way ANOVA

One factor influences observations.

Example: Yield of crops under different fertilizers.

Two-way ANOVA

Two factors influence observations.

Example: Crop yield based on fertilizer and irrigation.

Assumptions

• Random samples
• Normal distribution
• Equal variances
• Independent observations

Applications

• Agriculture
• Medicine
• Industry
• Psychology
• Education

Advantages

• Compares multiple means simultaneously.
• Reduces experimental error.
• Widely used in research.

19. Design of Experiments (DOE)

Experimental design helps conduct experiments scientifically.

Basic Principles

Randomization

Treatments are assigned randomly.

Purpose: Eliminates bias.

Replication

Each treatment is repeated.

Purpose: Improves accuracy.

Local Control (Blocking)

Groups similar experimental units.

Purpose: Reduces experimental error.

Common Experimental Designs

Applications

• Agricultural research
• Clinical trials
• Industrial experiments

20. Time Series Analysis

Time series is a sequence of observations arranged over time.

Examples: Daily temperature, Annual rainfall, Population growth, Monthly sales, Stock prices

Components

Trend (T)

Long-term movement.

Example: Population increase.

Seasonal Variation (S)

Occurs within one year.

Example: Festival sales.

Cyclical Variation (C)

Occurs over many years.

Example: Business cycles.

Irregular Variation (I)

Unexpected changes.

Example: Pandemic, Flood, War

Methods of Measuring Trend

• Freehand curve
• Semi-average
• Moving average
• Least squares

Applications

• Budget planning
• Economic forecasting
• Business planning
• Weather forecasting

21. Index Numbers

Index numbers measure relative change over time.

Uses

• Measure inflation
• Cost of living
• Economic growth
• Industrial production

Types

Price Index

Measures price changes.

Examples: Consumer Price Index (CPI), Wholesale Price Index (WPI)

Quantity Index

Measures production changes.

Value Index

Measures total value changes.

Important Formulae

Laspeyres Index

Uses base year quantities.

Paasche Index

Uses current year quantities.

Fisher's Ideal Index

Geometric mean of Laspeyres and Paasche.

It satisfies:

• Time Reversal Test
• Factor Reversal Test

Characteristics of Good Index Number

• Simple
• Reliable
• Representative
• Accurate
• Easy to compute

22. Vital Statistics

Vital statistics deals with population events.

Includes: Birth, Death, Marriage, Divorce, Migration

Important Rates

Applications

• Health planning
• Population policy
• Medical research
• Government welfare

23. Demography

Study of human population.

Demographic Variables

• Age
• Sex
• Literacy
• Occupation
• Fertility
• Mortality
• Migration

Population Growth

Depends on Births, Deaths, Migration.

Population Pyramid

Represents age-sex distribution.

Types:

• Expanding
• Stable
• Declining

Applications

• Education planning
• Employment
• Housing
• Healthcare

24. Official Statistics in India

India has one of the largest statistical systems.

Major Organizations

Ministry of Statistics and Programme Implementation (MoSPI)

Functions: National statistical system, Surveys, National accounts, SDG monitoring

National Sample Survey (NSS)

Collects socio-economic data.

National Statistical Office (NSO)

Conducts Household surveys, Industrial surveys, Economic census.

Registrar General of India

Conducts Population Census, Civil Registration System.

Reserve Bank of India

Publishes Banking statistics, Monetary statistics, Inflation indicators.

25. Statistical Quality Control (SQC)

Maintains product quality using statistical methods.

Objectives

• Reduce defects
• Improve quality
• Reduce cost
• Increase productivity

Quality Control Tools

• Control Charts – Monitor manufacturing process.
• Acceptance Sampling – Accept or reject production lot.
• Process Control – Checks variation during production.

Types of Control Charts

Applications

• Automobile industry
• Pharmaceutical industry
• Electronics
• Textile industry

26. Operations Research (OR)

Operations Research applies mathematical techniques for decision-making.

Characteristics

• Scientific approach
• Mathematical models
• Optimization
• Resource allocation

Applications

• Defence
• Airlines
• Hospitals
• Manufacturing
• Transportation
• Banking

27. Linear Programming (LPP)

Linear Programming finds the best solution under constraints.

Components

• Objective Function – Maximize profit or minimize cost.
• Decision Variables – Unknown quantities.
• Constraints – Available resources.
• Non-negativity Restriction – Variables cannot be negative.

Applications

• Production planning
• Transportation
• Agriculture
• Marketing
• Finance

28. Transportation Problem

Determines minimum transportation cost.

Objectives

• Reduce transportation cost.
• Efficient distribution.

Methods

• North-West Corner Rule
• Least Cost Method
• Vogel's Approximation Method (VAM)
• MODI Method

Applications

• Supply chain
• Warehousing
• Logistics

29. Assignment Problem

Special case of transportation problem.

One worker performs one job.

Methods

• Hungarian Method

Applications

• Staff allocation
• Machine assignment
• Project scheduling

30. Inventory Control

Inventory means stock of materials.

Objectives

• Avoid shortage
• Reduce storage cost
• Maintain optimum inventory

Inventory Costs

• Ordering Cost
• Holding Cost
• Shortage Cost
• Purchase Cost

EOQ (Economic Order Quantity)

Determines optimum order size that minimizes total inventory cost.

Applications

• Retail stores
• Warehouses
• Manufacturing

31. Queueing Theory

Studies waiting lines.

Components

• Customers
• Server
• Arrival pattern
• Service rate
• Queue discipline

Common Disciplines

• FIFO
• LIFO
• Priority
• Random

Applications

• Banks
• Hospitals
• Railway reservation
• Call centers
• Toll plazas

32. Game Theory

Helps make decisions under competition.

Types

• Two-person games
• Zero-sum games
• Mixed strategy

Applications

• Business
• Military
• Economics
• Marketing

33. Decision Theory

Decision-making under uncertainty.

Decision Environments

• Certainty
• Risk
• Uncertainty

Decision Criteria

• Maximax
• Maximin
• Minimax Regret
• Hurwicz Criterion
• Laplace Criterion

Applications

• Business investment
• Production
• Government planning

34. Reliability Theory

Reliability is the probability that a system performs without failure for a specified period.

Applications

• Electronics
• Aerospace
• Mechanical engineering
• Defence
• Medical equipment

35. Sampling Surveys in India

Important Large-scale Surveys

• Population Census
• National Family Health Survey (NFHS)
• Periodic Labour Force Survey (PLFS)
• Annual Survey of Industries (ASI)
• National Sample Surveys (NSS)
• Economic Census

36. Computer Applications in Statistics

Common Statistical Software

Applications

• Data analysis
• Regression
• Forecasting
• Machine learning
• Data visualization

37. TNPSC CTS Important One-Liners

TNPSC CTS Examination Focus