1.5 Measures of Dispersion¶
Measures of dispersion provide insights into the variability or spread of a dataset.
Common Measures¶
Range¶
- Difference between the largest and smallest values.
- Formula: Range = Max - Min
- Example: For data [1, 1, 2, 2, 4], range = 4 - 1 = 3.
- Limitations: Sensitive to extreme values (outliers).
Interquartile Range (IQR)¶
- Range of the middle 50% of data.
- Formula: IQR = Q3 - Q1 (75th percentile - 25th percentile).
- Example: IQR for salary data = 27 lakhs (Q3) - 24 lakhs (Q1) = 3 lakhs.
- Advantage: Not influenced by outliers.
Variance¶
- Definition: Measures the average squared deviation from the mean.
- Formula: Variance = Σ(xi - x̄)² / (n - 1) for sample data.
- Example: For data [1, 1, 2, 2, 4] with mean = 2:
- Deviations: [-1, -1, 0, 0, +2].
- Squared Deviations: [1, 1, 0, 0, 4].
- Variance = (1 + 1 + 0 + 0 + 4) / (5 - 1) = 1.5.
- In Excel:
VAR.S()function.
Standard Deviation (SD)¶
- Square root of variance.
- Easier to interpret as it is in the same units as the data.
- Example: For variance = 1.5, SD = √1.5 ≈ 1.22.
- In Excel:
STDEV.S()function.
Coefficient of Variation (CV)¶
- Relative measure of dispersion, expressed as a percentage.
- Formula: CV = (SD / Mean) × 100.
- Example: For SD = 3.66 and mean = 25.92, CV = (3.66 / 25.92) × 100 ≈ 14%.
Examples of Application¶
Salary Data (Jyothi Hegde, Dharwad)¶
- Average Salary: 25.9 lakhs.
- SD: 3.66 lakhs, CV: 14%.
- IQR: 27 lakhs - 24 lakhs = 3 lakhs.
Customer Satisfaction Score (Basavaraja, Bellary)¶
- Range: 66 - 42 = 24.
- SD: 6.38, CV: ~11%.
- IQR: 61.75 - 53 = 8.75.
Weekly Sales (Umesh Kamath, Nandini Sweets)¶
- Range: 432 - 34 = 398.
- SD: 88.36, CV: 65%.
- IQR: 136.5 - 82.25 = 54.25.
Property Prices (Vijay Gowda, Mysuru)¶
- Range: 411 - 55 = 356.
- SD: 93, CV: 46%.
- IQR: 243.75 - 149.5 = 94.25.
Z-Scores¶
- Definition: Indicates how many standard deviations a data point is from the mean.
- Formula: Z = (xi - x̄) / SD.
Interpretation¶
- Positive Z: Above mean.
- Negative Z: Below mean.
- Dimensionless: Allows comparison across different datasets.
Properties¶
- Mean of Z-scores: 0.
- SD of Z-scores: 1.
Outlier Detection¶
- Common Threshold: Z < -3 or Z > +3.
Example (Salary Data):¶
- Z-scores computed for each salary point.
- Outliers identified with Z > 3 (e.g., 50 lakhs with Z = +6.58).
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