Unit–I: Measures of Central Tendency, Dispersion, Skewness & Kurtosis
1. Data Processing: Classification, Tabulation, & Frequency Distribution
Data processing transforms raw, unorganized quantitative facts into structured information suitable for statistical analysis.
DATA PROCESSING PIPELINE
|
[ Raw Data Collection ] —> [ Classification ] —> [ Tabulation ] —> [ Visual Graphs ]
| Stage | Operational Definition | Primary Objective |
|---|---|---|
| Classification | Grouping raw data into homogenous classes based on common attributes (geographical, chronological, qualitative, quantitative). | Eliminates complexity and highlights underlying data relationships. |
| Tabulation | Systematic logical arrangement of classified data into rows and columns. | Simplifies data representation for comparison and statistical computation. |
| Frequency Distribution | Tabular arrangement dividing continuous data into class intervals alongside frequency counts (f). | Displays the spread, concentration, and density of continuous observation values. |
Visual Representation of Data
- Diagrams (One & Two Dimensional): Bar charts, sub-divided bar diagrams, and pie charts used for non-continuous or categorical variables.
- Graphs (Frequency Graphs):
- Histogram: Continuous adjacent rectangles representing class frequency counts.
- Frequency Polygon & Curve: Line graphs drawn by connecting the midpoints of histogram bars.
- Ogive Curves (Cumulative Frequency): Less-than and more-than cumulative curves used to determine partition values like the median and quartiles graphically.
2. Measures of Central Tendency (Averages)
A measure of central tendency is a single summary value that represents the center point of a data distribution.
MEASURES OF CENTRAL TENDENCY
|
+————————————+————————————+
| |
MATHEMATICAL AVERAGES POSITIONS & FREQUENCY AVERAGES
• Arithmetic Mean (X̄) • Median (M)
• Weighted Arithmetic Mean (X̄w) • Mode (Mo)
• Geometric Mean (GM) [Theory] • Partition Values (Quartiles, Deciles, Percentiles)
• Harmonic Mean (HM) [Theory]
Summary of Formulae & Theoretical Concepts
| Measure | Formula (Continuous Series) | Key Characteristics & Use Cases |
|---|---|---|
| Arithmetic Mean (X̄) | X̄ = ΣfX / N | Uses all data points; affected by extreme outliers. |
| Weighted Arithmetic Mean (X̄w) | X̄w = ΣwX / Σw | Applied when individual items carry varying relative importance/weights (w). |
| Median (M) | M = L + [(N/2 − c.f.) / f] × h | Positional average dividing ordered data into two equal halves (50/50); unaffected by outliers. |
| Mode (Mo) | Mo = L + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h | Value with the highest frequency density in a dataset. |
| Geometric Mean (GM) (Theory Only) |
GM = Antilog(Σ log X / N) | nth root of product of N values; used for rates, ratios, and growth percentages. |
| Harmonic Mean (HM) (Theory Only) |
HM = N / Σ(1/X) | Reciprocal of the arithmetic mean of reciprocals; used for rates and speeds. |
Empirical Relationship (for moderately skewed distributions): Mode = 3(Median) − 2(Mean)
3. Partition Values: Quartiles, Deciles, and Percentiles
Partition values divide an ordered dataset into equal parts.
PARTITION VALUES
|
+———————————-+———————————-+
| | |
QUARTILES (Qk) DECILES (Dk) PERCENTILES (Pk)
Divides data into 4 equal Divides data into 10 equal Divides data into 100 equal
parts (Q1, Q2, Q3). parts (D1 … D9). parts (P1 … P99).
General Interpolation Formula for Continuous Series: Value = L + [(Rank Position − c.f.) / f] × h
- First Quartile (Q₁): Lower quartile boundary covering the first 25% of data; Rank Position = N/4.
- Second Quartile (Q₂): Median value covering 50% of data; Q₂ = D₅ = P₅₀ = Median.
- Third Quartile (Q₃): Upper quartile boundary covering 75% of data; Rank Position = 3N/4.
4. Measures of Dispersion
Dispersion measures the spread, scatter, or variability of observations around a central value.
MEASURES OF DISPERSION
|
+————————————+————————————+
| |
ABSOLUTE MEASURES RELATIVE MEASURES (Coefficients)
Expressed in original units of data Unitless relative ratios used for comparing distributions
• Range (R) • Coefficient of Range
• Quartile Deviation (QD) • Coefficient of Quartile Deviation
• Mean Deviation (MD) • Coefficient of Mean Deviation
• Standard Deviation (σ) • Coefficient of Variation (CV)
Formula Summary Table
| Measure | Absolute Measure Formula | Relative Coefficient Formula |
|---|---|---|
| Range | Range = L − S (Largest − Smallest value) | Coeff. = (L − S) / (L + S) |
| Quartile Deviation (Semi-Interquartile) | QD = (Q₃ − Q₁) / 2 | Coeff. of QD = (Q₃ − Q₁) / (Q₃ + Q₁) |
| Mean Deviation (from Mean/Median) | MD_X̄ = Σf|X − X̄| / N MD_M = Σf|X − M| / N |
Coeff. of MD = MD_X̄ / X̄ or MD_M / M |
| Standard Deviation (σ) | σ = √[Σf(X − X̄)² / N] | Coeff. of σ = σ / X̄ |
Coefficient of Variation (CV)
The Coefficient of Variation expresses the standard deviation as a percentage of the arithmetic mean. It is used to compare consistency, stability, and variability across two or more distributions.
CV = (σ / X̄) × 100
- Higher CV: Indicates greater variability and lower stability or consistency.
- Lower CV: Indicates higher consistency, uniformity, and stability.
5. Measures of Skewness
Skewness measures the asymmetry or lack of symmetry in a data distribution.
TYPES OF SKEWNESS
|
+———————————-+———————————-+
| | |
SYMMETRICAL (Zero) POSITIVE SKEW NEGATIVE SKEW
Tail spreads equally Tail extends to the right Tail extends to the left
Mean = Median = Mode Mean > Median > Mode Mean < Median < Mode
Dispersion vs. Skewness
| Feature | Dispersion | Skewness |
|---|---|---|
| Core Focus | Measures the degree of variation or spread of data points. | Measures the directional asymmetry of observations around the mean. |
| Analytical Scope | Shows how far items deviate from an average. | Shows whether observations gather more to the left or right of the center. |
Karl Pearson’s & Bowley’s Coefficients of Skewness
Karl Pearson’s Coefficient (Skp): Based on the distance between the mean and mode, normalized by standard deviation.
Skp = (X̄ − Mo) / σ
If Mode is ill-defined: Skp = 3(X̄ − M) / σ
If Mode is ill-defined: Skp = 3(X̄ − M) / σ
Bowley’s Coefficient (Skb): Based on the relative positions of quartiles relative to the median.
Skb = [(Q₃ + Q₁) − 2M] / (Q₃ − Q₁)
6. Concept & Types of Kurtosis
Kurtosis measures the peakedness or flatness of a distribution curve relative to a normal distribution curve.
TYPES OF KURTOSIS
|
+———————————-+———————————-+
| | |
LEPTOKURTIC MESOKURTIC PLATYKURTIC
Highly peaked curve with Normal bell-shaped curve Flat-topped curve with
heavy tails (β₂ > 3 / γ₂ > 0) (β₂ = 3 / γ₂ = 0). dispersed values (β₂ < 3 / γ₂ < 0)
Mathematical Coefficients of Kurtosis
β₂ = μ₄ / μ₂² and γ₂ = β₂ − 3
- Mesokurtic: β₂ = 3, γ₂ = 0
- Leptokurtic: β₂ > 3, γ₂ > 0
- Platykurtic: β₂ < 3, γ₂ < 0
Practical Importance of Skewness and Kurtosis
- Risk Assessment: Helps evaluate financial return distributions, price swings, and market risk.
- Quality & Process Control: Detects systemic manufacturing biases or machine drift.
- Model Selection: Verifies whether sample data meets normality assumptions required for parametric statistical tests (e.g., t-tests, ANOVA).
Key Learning Summary
| Section | Primary Operational Takeaway |
|---|---|
| Central Tendency | Summarizes a dataset using a single representative center point (X̄, M, Mo). |
| Dispersion & CV | Quantifies variability; lower CV indicates greater consistency. |
| Skewness | Evaluates directional asymmetry; compared using Karl Pearson’s or Bowley’s coefficients. |
| Kurtosis | Evaluates peak height and tail thickness (Leptokurtic, Mesokurtic, Platykurtic). |