Descriptive Statistics: Central Tendency & Dispersion — Previous Year Questions

📊 Topic Overview

Metric	Value
Papers Appeared	12 of 27 (44.4%)
Priority	🟠 HIGH
Section Distribution	Sec A: ~6
Most Repeated	"Describe measures of central tendency and measures of dispersion."

🎯 Must-Know Concepts

1. Measures of central tendency: mean, median, mode — definitions, computation, advantages, limitations
2. Measures of dispersion: range, variance, standard deviation — definitions and formulas
3. Descriptive vs inferential statistics distinction
4. When to use each measure

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🔥 Most Repeated Question

Common phrasings:

• "Describe the measures of central tendency and measures of dispersion."
• "Describe measures of central tendency. Elucidate advantages and disadvantages of descriptive statistics."
• "Differentiate between descriptive and inferential statistics."

Answer (Section A — ~450 words):

Descriptive statistics summarise and describe the characteristics of a dataset. They include measures of central tendency (where scores cluster) and measures of dispersion (how spread out scores are).

Measures of Central Tendency:

1. Mean (Arithmetic Mean): The sum of all scores divided by the number of scores. X̄ = ΣX/n.

   • Advantages: Uses all data; algebraically tractable; basis for further statistical analysis.
   • Limitations: Affected by extreme scores (outliers); inappropriate for skewed distributions.
   • Use: Interval and ratio data, approximately normal distributions.

2. Median: The middle score when scores are arranged in order. If n is even, median = average of two middle scores.

   • Advantages: Not affected by extreme scores; suitable for skewed distributions.
   • Limitations: Does not use all data; less suitable for further algebraic operations.
   • Use: Ordinal data, skewed distributions.

3. Mode: The most frequently occurring score. A distribution can be unimodal, bimodal, or multimodal.

   • Advantages: Unaffected by extreme scores; only measure for nominal data.
   • Limitations: May not exist; multiple modes possible; unstable in small samples.
   • Use: Nominal data, describing most typical value.

Measures of Dispersion:

Measures of dispersion indicate the spread or variability of scores around the central value.

1. Range: Highest score − Lowest score.

   • Simple but unstable; heavily influenced by extreme scores.

2. Variance (σ² or s²): The average of squared deviations from the mean. $\sigma^2 = \frac{\Sigma(X - \bar{X})^2}{n}$

   • Includes all scores; basis of many inferential statistics.
   • Expressed in squared units — harder to interpret directly.

3. Standard Deviation (SD or σ): Square root of the variance. Most widely used measure of dispersion. $SD = \sqrt{\frac{\Sigma(X-\bar{X})^2}{n}}$

   • Same units as the original data; easy to interpret.
   • Basis for z-scores, standard error, and most parametric tests.

Descriptive vs Inferential Statistics:

• Descriptive statistics summarise data from the sample itself (mean, SD, frequency tables, graphs).
• Inferential statistics use sample data to make inferences about the population (t-test, ANOVA, correlation significance).

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🧠 Memory Aid

Central tendency: Mean (equal intervals, no outliers), Median (skewed data), Mode (nominal).

Dispersion: Range (quick, rough), Variance (precise but squared units), SD (precise, same units).

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📚 Related Content

• Central Tendency, Dispersion, Skewness
• Descriptive Statistics and Data Organisation

✅ Self-Assessment

1. When is the median preferred over the mean?
2. Write the formula for standard deviation.
3. What is the relationship between variance and standard deviation?
4. List two advantages and two limitations of the mean.
5. How do descriptive statistics differ from inferential statistics?

💡 Exam Tips

• Organise the answer clearly: Central Tendency first, then Dispersion.
• For each measure: definition + formula (if applicable) + advantage + limitation + when to use.
• The descriptive vs inferential distinction is frequently added as a Section B component.
• SD formula must be written out — it earns marks.