Chapter 7: Measures of Dispersion – Questions & Answers

📌 Part A – 1 Mark Questions (Very Short Answer)

1. What is dispersion?
   Dispersion measures the degree of variation or spread in a data set.
2. Define Range.
   Range is the difference between the highest and lowest value in a series.
3. Write the formula for Range.
   Range = L – S, where L = Largest value, S = Smallest value.
4. Write the formula for Coefficient of Range.
   Coefficient of Range = (L – S) / (L + S).
5. What is Quartile Deviation?
   Quartile Deviation is half the difference between the third quartile (Q₃) and the first quartile (Q₁).
6. Write the formula for Quartile Deviation (QD).
   QD = (Q₃ – Q₁) / 2.
7. Write the formula for Coefficient of Quartile Deviation.
   Coefficient of QD = (Q₃ – Q₁) / (Q₃ + Q₁).
8. What does a low dispersion indicate?
   Low dispersion means data points are close to the average, so the average is more reliable.
9. What does a high dispersion indicate?
   High dispersion means data points are spread far from the average, so the average is less reliable.
10. Which measure of dispersion uses only two values from the data?
    Range.
11. Which measure of dispersion is based on the middle 50% of the data?
    Quartile Deviation.
12. Why is Range considered a simple measure?
    Because it is easy to calculate and understand.
13. Why is Quartile Deviation considered better than Range?
    Because it is not affected by extreme values (outliers).
14. If Q₃ = 40 and Q₁ = 20, find Quartile Deviation.
    QD = (40 – 20) / 2 = 20 / 2 = 10.
15. If L = 100 and S = 20, find the Coefficient of Range.
    Coefficient = (100 – 20) / (100 + 20) = 80 / 120 = 0.667.
16. What is the main limitation of Range?
    It ignores all intermediate values and is affected by outliers.
17. Name the two measures of dispersion covered in this chapter.
    Range and Quartile Deviation.
18. What does 'L' stand for in the Range formula?
    L stands for the Largest value in the series.
19. What does 'S' stand for in the Range formula?
    S stands for the Smallest value in the series.
20. Quartile Deviation is also known as ________.
    Semi-interquartile range.

📌 Part B – 2/4 Marks Questions (Short Answer)

1. Define Range and write its formula. Calculate Range for the data: 25, 32, 85, 42, 10, 20, 18, 28.

   Definition: Range is the simplest measure of dispersion. It is the difference between the largest and the smallest value in a data set.
   Formula: Range = L – S

   Calculation:
   Largest (L) = 85, Smallest (S) = 10
   Range = 85 – 10 = 75

2. Calculate the Coefficient of Range for the data: 25, 32, 85, 42, 10, 20, 18, 28.

   Formula: Coefficient of Range = (L – S) / (L + S)
   L = 85, S = 10
   Coefficient = (85 – 10) / (85 + 10) = 75 / 95 = 0.79

3. What is Quartile Deviation? Explain its importance.

   Quartile Deviation (QD) is a measure of dispersion based on quartiles. It is calculated as half the difference between the third quartile (Q₃) and the first quartile (Q₁). It measures the spread of the middle 50% of the data.

   Importance: Unlike Range, QD is not affected by extreme values (outliers) because it ignores the top 25% and bottom 25% of the data. It gives a better idea of the spread of the central data.

4. Find Quartile Deviation and its coefficient for the marks: 20, 28, 40, 12, 30, 15, 50.

   Step 1: Arrange in ascending order.
   12, 15, 20, 28, 30, 40, 50

   Step 2: Find Q₁ and Q₃.
   N = 7
   Q₁ = Size of (N+1)/4 th item = (7+1)/4 = 8/4 = 2nd item = 15.
   Q₃ = Size of 3(N+1)/4 th item = 3×2 = 6th item = 40.

   Step 3: Calculate QD.
   QD = (Q₃ – Q₁) / 2 = (40 – 15) / 2 = 25 / 2 = 12.5

   Step 4: Calculate Coefficient of QD.
   Coefficient = (Q₃ – Q₁) / (Q₃ + Q₁) = (40 – 15) / (40 + 15) = 25 / 55 = 0.455

5. Distinguish between Range and Quartile Deviation.

   • Range: Based on only two values (largest and smallest). Affected by outliers. Simple to calculate. Measures the total spread of data.
   • Quartile Deviation: Based on quartiles (Q₁ and Q₃). Not affected by outliers. Uses middle 50% of data. More reliable for skewed data.

6. Calculate Range for the following continuous series:

   Salary (₹)	No. of Employees
   5000 - 10000	180
   10000 - 15000	220
   15000 - 20000	300
   20000 - 25000	160
   25000 - 30000	140

   Largest value (L) = Upper limit of last class = 30000
   Smallest value (S) = Lower limit of first class = 5000
   Range = 30000 – 5000 = ₹25,000
   Coefficient of Range = (30000 – 5000) / (30000 + 5000) = 25000 / 35000 = 0.83

📌 Part C – 6/8 Marks Questions (Long Answer)

1. Explain the concept of dispersion. Calculate Range and Quartile Deviation from the following individual data: 100, 120, 80, 150, 200, 90, 130.

   Meaning of Dispersion: Dispersion measures the spread or variability in a data set. It tells us how much the data points differ from the central value (like mean or median). Low dispersion means data is clustered around the center; high dispersion means data is scattered.

   Step 1: Arrange data in ascending order.
   80, 90, 100, 120, 130, 150, 200. N = 7.

   Step 2: Calculate Range.
   L = 200, S = 80
   Range = 200 – 80 = 120
   Coefficient of Range = (200 – 80) / (200 + 80) = 120 / 280 = 0.429

   Step 3: Calculate Quartile Deviation.
   Q₁ = Size of (N+1)/4 th item = (7+1)/4 = 8/4 = 2nd item = 90.
   Q₃ = Size of 3(N+1)/4 th item = 3×2 = 6th item = 150.
   QD = (Q₃ – Q₁) / 2 = (150 – 90) / 2 = 60 / 2 = 30.
   Coefficient of QD = (150 – 90) / (150 + 90) = 60 / 240 = 0.25

   Conclusion: For this data, Range = 120, QD = 30. QD is much smaller because it ignores the extreme value (200) and focuses on the middle spread.

2. Calculate Quartile Deviation and its coefficient for the following continuous series:

   Wages (₹)	No. of Workers
   20-25	2
   25-30	10
   30-35	25
   35-40	16
   40-45	7

   Step 1: Calculate cumulative frequency (cf).

   Class	f	cf
   20-25	2	2
   25-30	10	12
   30-35	25	37
   35-40	16	53
   40-45	7	60

   Total N = 60.

   Step 2: Find Q₁.
   N/4 = 60/4 = 15. The class where cf just exceeds 15 is 30-35 (cf = 37).
   Q₁ Class: 30-35. L₁ = 30, cf₁ = 12, f₁ = 25, i = 5.
   Q₁ = L₁ + [(N/4 – cf₁) / f₁] × i
   Q₁ = 30 + [(15 – 12) / 25] × 5
   Q₁ = 30 + (3/25) × 5 = 30 + 0.6 = 30.6

   Step 3: Find Q₃.
   3N/4 = 180/4 = 45. The class where cf just exceeds 45 is 35-40 (cf = 53).
   Q₃ Class: 35-40. L₃ = 35, cf₃ = 37, f₃ = 16, i = 5.
   Q₃ = L₃ + [(3N/4 – cf₃) / f₃] × i
   Q₃ = 35 + [(45 – 37) / 16] × 5
   Q₃ = 35 + (8/16) × 5 = 35 + 2.5 = 37.5

   Step 4: Calculate QD and Coefficient.
   QD = (Q₃ – Q₁) / 2 = (37.5 – 30.6) / 2 = 6.9 / 2 = 3.45
   Coefficient of QD = (37.5 – 30.6) / (37.5 + 30.6) = 6.9 / 68.1 = 0.101

3. Explain the advantages and disadvantages of Range and Quartile Deviation.

   Range:

   • Advantages: Simplest to understand and calculate. Easy to interpret. Useful in quality control where extreme values matter.
   • Disadvantages: Only uses two values, ignoring all others. Highly affected by outliers. Not suitable for open-ended distributions.

   Quartile Deviation:

   • Advantages: Not affected by extreme values. Good for skewed data. Based on middle 50% of data, so more reliable than Range.
   • Disadvantages: Ignores the first and last 25% of data. Does not consider all observations. Less sensitive to changes in data.

4. Calculate Range and Quartile Deviation for the discrete series:

   Marks	No. of Students
   10	4
   20	7
   30	15
   40	8
   50	7
   60	2

   Step 1: Calculate Range.
   Largest mark = 60, Smallest mark = 10
   Range = 60 – 10 = 50

   Step 2: Calculate cumulative frequency.

   Marks	f	cf
   10	4	4
   20	7	11
   30	15	26
   40	8	34
   50	7	41
   60	2	43

   N = 43.

   Step 3: Find Q₁.
   Q₁ item = (N+1)/4 = 44/4 = 11th item. The 11th item falls in the cf of 11, which corresponds to marks = 20.
   So, Q₁ = 20.

   Step 4: Find Q₃.
   Q₃ item = 3(N+1)/4 = 3×44/4 = 132/4 = 33rd item. The 33rd item falls in the cf of 34, which corresponds to marks = 40.
   So, Q₃ = 40.

   Step 5: Calculate QD.
   QD = (Q₃ – Q₁) / 2 = (40 – 20) / 2 = 20 / 2 = 10
   Coefficient of QD = (40 – 20) / (40 + 20) = 20 / 60 = 0.33