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Complete Guide to Descriptive Statistics: Mean, Median, Mode & Dispersion
Descriptive statistics summarize and characterize the central tendencies and dispersion of numerical data points. In data analytics, market research, quality control, and scientific reporting, choosing between the mean, median, or mode determines whether insights accurately represent the whole population or get distorted by outliers.
Measures of Central Tendency
- 1. Arithmetic Mean (x̄):
x̄ = (Σx) / n. Sum of values divided by count. Highly sensitive to extreme outliers. - 2. Median (Q2): The exact 50th percentile midpoint of sorted data. Impervious to skewed high/low values.
- 3. Mode: The most frequent recurring element. Useful for categorical and discrete survey data.
Measures of Statistical Dispersion
Dispersion metrics quantify how spread out values are around the mean:
Range: R = Maximum Value - Minimum Value
Variance (s²): s² = Σ(x - x̄)² / (n - 1)
Standard Deviation (σ): σ = √Variance (Spread in original units)
Comparison of Central Tendency Metrics & Practical Scenarios
| Metric | Mathematical Definition | Outlier Sensitivity | Best Real-World Application |
|---|---|---|---|
| Arithmetic Mean | Sum of values / Total count | Very High (Easily skewed) | Class test averages, daily temperatures |
| Median | Middle value of ordered dataset | Robust / Resistant | Real estate prices, employee salary surveys |
| Mode | Highest frequency value | None | Shoe inventory sizing, voting ballots |
| Standard Deviation | Root mean square deviation | Moderate to High | Stock market volatility, manufacturing tolerances |
Frequently Asked Questions (FAQ)
Why is Median often preferred over Mean for salary reports?
In income statistics, a tiny handful of ultra-wealthy individuals (billionaires) drastically drags the arithmetic mean upward, creating an artificially inflated number. The median represents the exact compensation of a typical middle-income earner.
Can a dataset have more than one mode or no mode at all?
Yes! If every number occurs with the same frequency, there is no mode. If two numbers tie for the highest frequency, the dataset is bimodal. If three or more tie, it is multimodal.
What is the difference between sample and population standard deviation?
Population standard deviation divides the sum of squared deviations by N (when measuring the entire population), while sample standard deviation divides by n - 1 (Bessel's correction) to prevent underestimation bias in sample subsets.