Chapter 6 — Measures Of Dispersion

Overview

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Key Point: Range = x_max − x_min

Measures of Dispersion describe how widely values in a data set are spread around a central value (mean, median, or mode). Dispersion complements measures of central tendency: two data sets can have the same mean but different spreads. Common measures are Range, Quartile Deviation (QD) / Interquartile Range (IQR), Mean Deviation (MD), Variance, Standard Deviation (SD), and Coefficient of Variation (CV).

Why they matter: Dispersion helps compare reliability and risk — e.g., whether incomes or exam scores are tightly clustered or widely spread, whether a product’s prices are stable, or how volatile returns are.

Brief description of main measures:

• Range: Difference between maximum and minimum. Simple but sensitive to outliers.
• Quartile Deviation (QD) / IQR: QD = (Q3 − Q1)/2; IQR = Q3 − Q1. Measures spread of the middle 50% and is robust to outliers.
• Mean Deviation (MD): Average of absolute deviations from a central value (mean or median): MD = (1/n) Σ |xi − c|. Often taken about the median for robustness.
• Variance: Average of squared deviations from the mean. Population variance σ² = (1/n) Σ (xi − x̄)². For a sample, s² = (1/(n−1)) Σ (xi − x̄)².
• Standard Deviation (SD): Square root of variance. Gives dispersion in same units as the data: σ = √σ².
• Coefficient of Variation (CV): CV = (SD / mean) × 100%. Useful to compare relative variability of different-scale data sets.

Properties and useful facts:

• Range and IQR are not affected by any formula of squaring; IQR is robust to outliers.
• If every value is transformed as y = a + b x, then mean shifts and scales by a and b, variance is multiplied by b², and SD is multiplied by |b|.
• Variance has a shortcut (computational) form: σ² = (Σx_i²)/n − x̄². For grouped data use class midpoints and frequencies.

Computation for grouped data: Replace each value x_i by class midpoint m_i and use frequencies f_i. Mean: x̄ = Σ f_i m_i / Σ f_i. Variance: σ² = [Σ f_i (m_i − x̄)²] / Σ f_i (or use Σ f_i m_i² / Σ f_i − x̄²).

Interpretation tips: SD tells how far, on average, observations lie from the mean in squared-rooted units. CV helps compare dispersion across different units (e.g., incomes vs. heights). IQR and QD are preferred when data contain outliers.

📌 Examples

• Worked numerical example (ungrouped): Data = [5, 7, 8, 10, 15]. Mean = (5+7+8+10+15)/5 = 9. Range = 15 − 5 = 10. Median = 8. Q1 = median of [5,7] = 6, Q3 = median of [10,15] = 12.5, IQR = 12.5 − 6 = 6.5, QD = 6.5/2 = 3.25. Mean Deviation about mean = (|5−9|+|7−9|+|8−9|+|10−9|+|15−9|)/5 = (4+2+1+1+6)/5 = 2.8. Variance (population) = [(5−9)²+(7−9)²+(8−9)²+(10−9)²+(15−9)²]/5 = (16+4+1+1+36)/5 = 11.6. SD = √11.6 ≈ 3.405. CV = (3.405/9)×100 ≈ 37.8%.
• Real-life example (exam scores): Two classes both have mean score 70. Class A scores are mostly between 65 and 75 (low SD), Class B scores vary from 40 to 95 (high SD). Although means are equal, Class A performance is more consistent; Class B shows greater dispersion (higher risk of very low or very high scores). CV allows comparing consistency relative to the mean if class averages differ.
• Real-life example (income distribution): If one region has mean income 50,000 and SD 5,000 (CV = 10%) and another has mean 30,000 and SD 6,000 (CV = 20%), the second region has greater relative income inequality despite a lower mean.

🧮 Formulas

1. \[Range = x_max − x_min\]
2. \[IQR = Q3 − Q1\]

   \[Quartile Deviation (QD) = (Q3 − Q1)/2\]
3. \[Mean (ungrouped) x̄ = (Σ x_i)/n\]
4. \[Mean Deviation about mean MD = (1/n) Σ |x_i − x̄| (or about median M: MD = (1/n) Σ |x_i − M|)\]
5. \[Population variance σ² = (1/n) Σ (x_i − x̄)²\]
6. \[Sample variance s² = (1/(n−1)) Σ (x_i − x̄)²\]

1. What is meant by 'dispersion' and why is it studied along with measures of central tendency? / 'अपकिरण' से क्या तात्पर्य है और इसे केंद्रीय प्रवृत्ति के मापों के साथ क्यों अध्ययन किया जाता है?
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   Dispersion describes how widely the values in a data set are spread around a central value; it is studied alongside central tendency because two data sets can have the same mean but very different spreads, so dispersion reveals reliability and risk. / अपकिरण यह दर्शाता है कि आंकड़ा समूह के मान किसी केंद्रीय मान के चारों ओर कितने व्यापक रूप से फैले हैं; इसे केंद्रीय प्रवृत्ति के साथ इसलिए अध्ययन किया जाता है क्योंकि दो आंकड़ा समूहों का माध्य समान परंतु फैलाव बहुत भिन्न हो सकता है, अतः अपकिरण विश्वसनीयता और जोखिम प्रकट करता है।

2. For the data 5, 7, 8, 10, 15, calculate the range and the interquartile range (IQR). / आंकड़ों 5, 7, 8, 10, 15 के लिए परास और अंतरचतुर्थक परास (IQR) की गणना कीजिए।
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   Range = 15 − 5 = 10; with Q1 = 6 and Q3 = 12.5, IQR = Q3 − Q1 = 12.5 − 6 = 6.5. / परास = 15 − 5 = 10; Q1 = 6 और Q3 = 12.5 के साथ, IQR = Q3 − Q1 = 12.5 − 6 = 6.5।

3. For the data 5, 7, 8, 10, 15 (mean 9), compute the population variance and standard deviation. / आंकड़ों 5, 7, 8, 10, 15 (माध्य 9) के लिए समष्टि प्रसरण और मानक विचलन की गणना कीजिए।
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   Variance σ² = [(5−9)²+(7−9)²+(8−9)²+(10−9)²+(15−9)²]/5 = (16+4+1+1+36)/5 = 11.6, so SD = √11.6 ≈ 3.41. / प्रसरण σ² = [(5−9)²+(7−9)²+(8−9)²+(10−9)²+(15−9)²]/5 = (16+4+1+1+36)/5 = 11.6, अतः मानक विचलन = √11.6 ≈ 3.41।

4. Why is the coefficient of variation (CV) preferred when comparing the variability of two data sets measured in different units or with different means? / भिन्न इकाइयों या भिन्न माध्यों वाले दो आंकड़ा समूहों की परिवर्तनशीलता की तुलना करते समय विचरण गुणांक (CV) को क्यों प्राथमिकता दी जाती है?
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   Because CV = (SD/mean) × 100 is a relative, unit-free measure, it allows fair comparison of dispersion across data sets of different scales (e.g., incomes vs heights) where the absolute standard deviation alone would be misleading. / क्योंकि CV = (मानक विचलन/माध्य) × 100 एक सापेक्ष, इकाई-रहित माप है, यह भिन्न पैमानों वाले आंकड़ा समूहों (जैसे आय बनाम ऊँचाई) में अपकिरण की निष्पक्ष तुलना की अनुमति देता है, जहाँ केवल निरपेक्ष मानक विचलन भ्रामक होता।

5. Region A has mean income 50,000 with SD 5,000 and Region B has mean income 30,000 with SD 6,000. Which region has greater relative income inequality? / क्षेत्र A की औसत आय 50,000 तथा मानक विचलन 5,000 है और क्षेत्र B की औसत आय 30,000 तथा मानक विचलन 6,000 है। किस क्षेत्र में सापेक्ष आय असमानता अधिक है?
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   CV of A = (5,000/50,000)×100 = 10% and CV of B = (6,000/30,000)×100 = 20%; therefore Region B has greater relative income inequality despite its lower mean income. / A का CV = (5,000/50,000)×100 = 10% और B का CV = (6,000/30,000)×100 = 20%; अतः कम औसत आय के बावजूद क्षेत्र B में सापेक्ष आय असमानता अधिक है।

6. Why are the range and IQR considered less informative or less robust respectively, and which is better when data contain outliers? / परास और IQR को क्रमशः कम सूचनापूर्ण या अधिक सुदृढ़ क्यों माना जाता है, और जब आंकड़ों में बाह्य मान हों तो कौन बेहतर है?
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   The range uses only the maximum and minimum and is highly sensitive to outliers, whereas the IQR measures the spread of the middle 50% and is robust to outliers, so the IQR (or quartile deviation) is the better choice when data contain extreme values. / परास केवल अधिकतम और न्यूनतम का उपयोग करता है और बाह्य मानों के प्रति अत्यधिक संवेदनशील है, जबकि IQR मध्य 50% का फैलाव मापता है और बाह्य मानों के प्रति सुदृढ़ है, अतः जब आंकड़ों में चरम मान हों तो IQR (या चतुर्थक विचलन) बेहतर विकल्प है।

7. If every observation is transformed as y = a + bx, how are the mean, variance and standard deviation affected? / यदि प्रत्येक प्रेक्षण को y = a + bx के रूप में रूपांतरित किया जाए, तो माध्य, प्रसरण और मानक विचलन किस प्रकार प्रभावित होते हैं?
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   The mean becomes a + b times the old mean, the variance is multiplied by b² (the additive constant a has no effect), and the standard deviation is multiplied by |b|. / माध्य पुराने माध्य का a + b गुना हो जाता है, प्रसरण b² से गुणित हो जाता है (योज्य अचर a का कोई प्रभाव नहीं), और मानक विचलन |b| से गुणित हो जाता है।

8. Two classes both have a mean score of 70, but Class A scores lie between 65 and 75 while Class B ranges from 40 to 95. Interpret what their dispersion tells us. / दो कक्षाओं का औसत अंक 70 है, परंतु कक्षा A के अंक 65 और 75 के बीच हैं जबकि कक्षा B के अंक 40 से 95 तक हैं। उनका अपकिरण क्या बताता है, व्याख्या कीजिए।
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   Class A has a low standard deviation, indicating consistent and reliable performance, while Class B has a high standard deviation, indicating greater variability and a higher risk of very low or very high scores—even though both share the same mean. / कक्षा A का मानक विचलन कम है, जो सुसंगत और विश्वसनीय प्रदर्शन दर्शाता है, जबकि कक्षा B का मानक विचलन अधिक है, जो अधिक परिवर्तनशीलता और बहुत कम या बहुत अधिक अंकों का अधिक जोखिम दर्शाता है—भले ही दोनों का माध्य समान हो।

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