Learning Objectives

By the end of this chapter, you will be able to:

  • Distinguish between descriptive and inferential statistics
  • Define population, sample, parameter, and statistic
  • Identify when to use descriptive vs inferential methods
  • Understand the relationship between samples and populations

The Two Branches of Statistics

Statistics as a science can be divided into two main branches:

flowchart TB
    A[STATISTICS] --> B[Descriptive Statistics]
    A --> C[Inferential Statistics]

    B --> B1["Summarizes data"]
    B --> B2["Describes characteristics"]
    B --> B3["Uses tables, charts, measures"]
    B --> B4["No generalizations beyond data"]

    C --> C1["Makes inferences"]
    C --> C2["Generalizes to population"]
    C --> C3["Uses probability theory"]
    C --> C4["Tests hypotheses"]

    style B fill:#e1f5fe
    style C fill:#fff3e0

Descriptive Statistics

Descriptive statistics consists of methods for organizing, displaying, and summarizing data using tables, graphs, and numerical measures.

Purpose of Descriptive Statistics

  • Organize raw data into meaningful form
  • Summarize large datasets using key measures
  • Present data in tables and charts
  • Describe the basic features of the data

Key Components

mindmap
  root((Descriptive Statistics))
    Organizing Data
      Frequency Tables
      Cross-tabulations
    Graphical Display
      Bar Charts
      Histograms
      Pie Charts
      Line Graphs
    Numerical Measures
      Central Tendency
        Mean
        Median
        Mode
      Dispersion
        Range
        Variance
        Standard Deviation

Example: Descriptive Statistics in Public Administration

Scenario: A district administration collects data on 500 households regarding their access to public services.

Service Households with Access Percentage
Drinking Water 425 85%
Electricity 475 95%
Health Post 350 70%
Primary School 460 92%
Road Access 380 76%

Descriptive Summary:

  • Average access across services: 83.6%
  • Most accessible service: Electricity (95%)
  • Least accessible service: Health Post (70%)

Key Point: Descriptive statistics only describes THIS dataset. It does not make claims about other districts or the entire country.

Inferential Statistics

Inferential statistics consists of methods that use sample data to make generalizations, predictions, or decisions about a larger population.

Purpose of Inferential Statistics

  • Estimate population characteristics from samples
  • Test hypotheses about populations
  • Make predictions based on data
  • Quantify uncertainty in conclusions

The Logic of Inference

flowchart LR
    A[Population<br/>N = Large/Unknown] -->|Select Sample| B[Sample<br/>n = Small/Known]
    B -->|Compute| C[Sample Statistics<br/>x̄, s, p]
    C -->|Estimate| D[Population Parameters<br/>μ, σ, P]
    D -->|Decision Making| E[Conclusions about<br/>Population]

    style A fill:#ffcdd2
    style B fill:#c8e6c9
    style E fill:#b3e5fc

Example: Inferential Statistics in Public Administration

Scenario: A government wants to know the average satisfaction level of all public service users in Nepal (population = 30 million citizens).

Process:

  1. Population: All citizens using public services (too large to survey all)
  2. Sample: Randomly select 2,000 citizens from different regions
  3. Measure: Average satisfaction score in sample = 6.8 out of 10
  4. Inference: “We estimate that the average satisfaction level of ALL citizens is between 6.5 and 7.1 (with 95% confidence)”

Key Point: Inferential statistics allows us to make statements about populations based on samples, with a known degree of uncertainty.

Key Terminology

Understanding these terms is essential for the entire course:

Population vs Sample

Concept Definition Symbol Example
Population Complete set of all items of interest $N$ All government employees in Nepal
Sample Subset of the population selected for study $n$ 500 randomly selected employees

Parameter vs Statistic

Concept Definition Notation Example
Parameter Numerical measure describing a population Greek letters ($\mu$, $\sigma$, $P$) True average salary of ALL employees
Statistic Numerical measure computed from a sample Roman letters ($\bar{x}$, $s$, $p$) Average salary of 500 sampled employees 
flowchart TB
    subgraph "Population"
        A[Parameter μ = Unknown<br/>True population mean]
    end

    subgraph "Sample"
        B[Statistic x̄ = Known<br/>Calculated sample mean]
    end

    A -.->|"Sampling"| B
    B -->|"Inference"| A

    style A fill:#ffecb3
    style B fill:#c8e6c9

Common Notation

Measure Population Parameter Sample Statistic
Mean $\mu$ (mu) $\bar{x}$ (x-bar)
Standard Deviation $\sigma$ (sigma) $s$
Proportion $P$ $p$
Size $N$ $n$
Variance $\sigma^2$ $s^2$

Comparison: Descriptive vs Inferential Statistics

Aspect Descriptive Statistics Inferential Statistics
Purpose Describe and summarize data Make generalizations about population
Scope Limited to collected data Extends beyond collected data
Methods Tables, charts, measures Estimation, hypothesis testing
Uncertainty No uncertainty (describes what IS) Involves probability of error
Population/Sample Can be used for either Specifically uses sample to infer about population
Output Exact values Estimates with confidence levels

Visual Comparison

flowchart TB
    subgraph Descriptive["Descriptive Statistics"]
        D1[Collected Data] --> D2[Summary Measures]
        D2 --> D3["Result: 'The average<br/>in this sample is 75'"]
    end

    subgraph Inferential["Inferential Statistics"]
        I1[Sample Data] --> I2[Statistical Analysis]
        I2 --> I3["Result: 'The population<br/>average is likely<br/>between 72 and 78'"]
    end

    style D3 fill:#e8f5e9
    style I3 fill:#fff3e0

When to Use Each Type

Use Descriptive Statistics When:

✅ You have data from an entire population (census) ✅ You only need to summarize the data at hand ✅ You’re creating reports or dashboards ✅ You’re doing exploratory data analysis

Example: Annual report showing department-wise budget expenditure for the fiscal year.

Use Inferential Statistics When:

✅ You cannot collect data from the entire population ✅ You need to make predictions about a population ✅ You want to test a hypothesis ✅ You need to generalize findings

Example: Using a sample survey to estimate citizen satisfaction across all municipalities.

Practical Application in Public Administration

Case Study: Employee Performance Analysis

Situation: The Public Service Commission wants to understand employee performance patterns.

Descriptive Approach:

  • Collect performance scores of ALL 450,000 government employees
  • Calculate: Mean = 72.5, Median = 74, Mode = 75
  • Create: Distribution chart showing performance categories
  • Conclusion: “The average performance score of government employees is 72.5”

Inferential Approach:

  • Randomly sample 5,000 employees
  • Calculate: Sample mean = 71.8, Standard deviation = 12.3
  • Estimate: Population mean is between 71.4 and 72.2 (95% confidence)
  • Conclusion: “We are 95% confident that the true average performance score of ALL government employees is between 71.4 and 72.2”

Key Insight

Descriptive Inferential
Tells us what the data shows Tells us what the data suggests about the larger picture
Certain but limited Uncertain but generalizable

The Statistical Process

Understanding how descriptive and inferential statistics fit into the overall research process:

flowchart TB
    A[Research Question] --> B[Define Population]
    B --> C{Can we study<br/>entire population?}

    C -->|Yes| D[Census/Complete Data]
    C -->|No| E[Select Sample]

    D --> F[Descriptive Statistics]
    E --> F
    F --> G{Need to generalize?}

    G -->|No| H[Report Descriptive<br/>Summary]
    G -->|Yes| I[Inferential Statistics]

    I --> J[Draw Conclusions<br/>about Population]

    style F fill:#e1f5fe
    style I fill:#fff3e0

Practice Questions

Short Answer Questions

  1. Define descriptive and inferential statistics with examples from public administration.

  2. Explain the relationship between:
    • a) Population and Sample
    • b) Parameter and Statistic

  3. A researcher surveys 1,000 households and reports: “The average household size is 4.2 members.” Is this descriptive or inferential statistics? Explain.

  4. If a sample mean is $\bar{x} = 85$ and we estimate the population mean to be between 82 and 88, which statistics (descriptive or inferential) are being used?

Multiple Choice Questions

Q1. Which of the following is a population parameter?

  • a) Sample mean $\bar{x}$
  • b) Population mean $\mu$ ✓
  • c) Sample size $n$
  • d) Sample standard deviation $s$

Q2. Descriptive statistics is used to:

  • a) Make predictions about populations
  • b) Test hypotheses
  • c) Summarize and describe data ✓
  • d) Estimate population parameters

Q3. Using sample data to make generalizations about a population is called:

  • a) Descriptive statistics
  • b) Inferential statistics ✓
  • c) Probability theory
  • d) Data collection

Q4. A census is best analyzed using:

  • a) Only inferential statistics
  • b) Only descriptive statistics ✓
  • c) Neither type
  • d) Must use both types

Summary Table

Concept Definition Key Feature
Descriptive Statistics Summarizes data No generalization beyond data
Inferential Statistics Makes inferences about population Uses probability theory
Population Complete set of interest Often too large to study entirely
Sample Subset of population Used when population is inaccessible
Parameter Population measure Usually unknown
Statistic Sample measure Calculated from sample data

Next Chapter

In the next chapter, we will study Measures of Central Tendency - methods to find the “center” or typical value in a dataset, including mean, weighted mean, median, and mode.