CBSE Class 11 Economics Measures of Dispersion Set 02

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 MEASURES OF DISPERSION RANGE AND QUARTILE DEVIATION

Very short answer questions (1 mark each)

1. Define deviation [1 Mark]

Answer:
Deviation is the numerical difference between an individual value or observation and a fixed average, such as the mean, median, or mode.

Teacher's Note:
a) It measures how much data points spread away from a central value.
b) Always keep track of whether the deviation is positive or negative.

2. What does Range mean? [1 Mark]

Answer:
Range is the difference between the largest and smallest values in a given distribution.

Teacher's Note:
a) It is the simplest measure of dispersion.
b) Formula is Range = L - S.

3. State formula of coefficient of Range. [1 Mark]

Answer:
Coefficient of Range = \(\frac{L - S}{L + S}\) where L is the largest value and S is the smallest value.

Teacher's Note:
a) It is a relative measure of dispersion.
b) It is independent of the unit of measurement.

4. Define quartile deviation. [1 Mark]

Answer:
Quartile deviation is half of the difference between the upper quartile (Q3) and the lower quartile (Q1).

Teacher's Note:
a) Formula is \(\frac{Q_3 - Q_1}{2}\).
b) It is also known as semi-interquartile range.

5. What is inter quartile range? [1 Mark]

Answer:
Interquartile range is the difference between the upper quartile and the lower quartile, expressed as \(Q_3 - Q_1\).

Teacher's Note:
a) It covers the middle 50 percent of the data.
b) It eliminates extreme values at both tails.

6. What is the other name for semi interquartile range? [1 Mark]

Answer:
Quartile deviation.

Teacher's Note:
a) Both terms refer to half of the interquartile range.
b) Useful for skewed distributions.

7. State the formula of coefficient of quartile deviation. [1 Mark]

Answer:
Coefficient of Quartile Deviation = \(\frac{Q_3 - Q_1}{Q_3 + Q_1}\)

Teacher's Note:
a) It measures relative dispersion based on quartiles.
b) It is a pure number free from units.

Short answer questions (3/4 marks each)

RANGE

1. Calculate range and its coefficient from the following distribution.
X: 52 19 18 21 22 45 23 27 37 30 31 42 [3 Marks]

Answer:
1. Largest value (L) = 52
2. Smallest value (S) = 18
3. Range = L - S = 52 - 18 = 34
4. Coefficient of Range = \(\frac{L - S}{L + S}\) = \(\frac{52 - 18}{52 + 18}\) = \(\frac{34}{70}\) = 0.4857

Teacher's Note:
a) Identify extreme values carefully from raw data.
b) Range formula depends only on two extreme values.
c) Final coefficient is rounded to four decimal places.

2. Calculate range and its coefficient from the following distribution.
X: 6 6 7 8 9 10 11 12 13
F: 10 15 17 30 25 20 10 4 2 [3 Marks]

Answer:
1. Largest value (L) = 13 (frequencies are ignored in finding range)
2. Smallest value (S) = 6
3. Range = L - S = 13 - 6 = 7
4. Coefficient of Range = \(\frac{L - S}{L + S}\) = \(\frac{13 - 6}{13 + 6}\) = \(\frac{7}{19}\) = 0.3684

Teacher's Note:
a) Frequency values are not used when calculating range.
b) Only the extreme variable values matter.
c) Ensure correct addition in the denominator.

3. Calculate range and its coefficient from the following distribution.

[3 Marks]

Answer:
1. Lower limit of the lowest class (S) = 5
2. Upper limit of the highest class (L) = 45
3. Range = L - S = 45 - 5 = 40
4. Coefficient of Range = \(\frac{L - S}{L + S}\) = \(\frac{45 - 5}{45 + 5}\) = \(\frac{40}{50}\) = 0.8

Teacher's Note:
a) For continuous series, use extreme class boundaries.
b) Frequencies play no role in range computation.
c) Division results in a clean decimal value.

QUARTILE DEVIATION

1. Calculate the upper and lower quartiles, quartile deviation and its coefficient from the following distribution.
139 140 145 144 140 141 144 143 141 143 142 145 [4 Marks]

Answer:
1. Arrange data in ascending order: 139, 140, 140, 141, 141, 142, 143, 143, 144, 144, 145, 145 (Total N = 12)
2. Lower quartile Q1 = size of \(\frac{N + 1}{4}\) th item = size of \(\frac{13}{4}\) th = 3.25th item = 140 + 0.25(140 - 140) = 140 (or standard convention: position 3 = 140)
3. Upper quartile Q3 = size of \(\frac{3(N + 1)}{4}\) th item = size of 9.75th item = 144
4. Quartile Deviation = \(\frac{Q_3 - Q_1}{2}\) = \(\frac{144 - 140}{2}\) = 2
5. Coefficient of Quartile Deviation = \(\frac{Q_3 - Q_1}{Q_3 + Q_1}\) = \(\frac{144 - 140}{144 + 140}\) = \(\frac{4}{284}\) = 0.0141

Teacher's Note:
a) Sorting raw data accurately is the primary step.
b) Use standard positional formulas for individual series.
c) Check integer positions if fractional interpolation is skipped.
d) Final coefficient represents relative spread around median.

2. Calculate quartile deviation and its coefficient from the following distribution.

[4 Marks]

Answer:
1. Cumulative frequencies (cf): 2, 5, 11, 26, 36, 41, 45, 48, 49. Total N = 49.
2. Q1 = size of \(\frac{N + 1}{4}\) th item = \(\frac{50}{4}\) th = 12.5th item, which lies in cf 26 corresponding to value 61.
3. Q3 = size of \(\frac{3(N + 1)}{4}\) th item = 37.5th item, which lies in cf 41 corresponding to value 63.
4. Quartile Deviation = \(\frac{63 - 61}{2}\) = \(\frac{2}{2}\) = 1
5. Coefficient of Quartile Deviation = \(\frac{63 - 61}{63 + 61}\) = \(\frac{2}{124}\) = 0.0161

Teacher's Note:
a) Construct cumulative frequency table correctly.
b) Match position values carefully with cumulative totals.
c) Watch out for discrete series indexing rules.
d) Final calculation yields a compact deviation measure.

3. Calculate quartile deviation and its coefficient from the following distribution.

[4 Marks]

Answer:
1. Cumulative frequencies (cf): 11, 28, 60, 88, 100. Total N = 100.
2. Q1 position = \(\frac{N}{4}\) = \(\frac{100}{4}\) = 25th item, which lies in class 60-70.
3. Q1 = \(L + \frac{\frac{N}{4} - cf}{f} \times h\) = \(60 + \frac{25 - 11}{17} \times 10\) = \(60 + \frac{140}{17}\) = 68.24
4. Q3 position = \(\frac{3N}{4}\) = 75th item, which lies in class 80-90.
5. Q3 = \(L + \frac{\frac{3N}{4} - cf}{f} \times h\) = \(80 + \frac{75 - 60}{28} \times 10\) = \(80 + \frac{150}{28}\) = 85.36
6. Quartile Deviation = \(\frac{85.36 - 68.24}{2}\) = \(\frac{17.12}{2}\) = 8.56
7. Coefficient of Quartile Deviation = \(\frac{85.36 - 68.24}{85.36 + 68.24}\) = \(\frac{17.12}{153.60}\) = 0.1115

Teacher's Note:
a) Use continuous series interpolation formula for Q1 and Q3.
b) cf is taken from the preceding class interval.
c) Check interval width h carefully.
d) Verify both quartile computations before finding half-difference.

4. Calculate quartile deviation and its coefficient from the following distribution.

[4 Marks]

Answer:
1. Convert 'less than' cumulative frequencies into normal class intervals and frequencies:
0-10: 5
10-20: 8 - 5 = 3
20-30: 15 - 8 = 7
30-40: 20 - 15 = 5
40-50: 30 - 20 = 10
50-60: 33 - 30 = 3
60-70: 35 - 33 = 2
2. Total N = 35. Q1 position = \(\frac{35}{4}\) = 8.75th item, lying in 20-30 class.
3. Q1 = \(20 + \frac{8.75 - 5}{7} \times 10\) = \(20 + \frac{3.75 \times 10}{7}\) = 25.36
4. Q3 position = \(\frac{3 \times 35}{4}\) = 26.25th item, lying in 40-50 class.
5. Q3 = \(40 + \frac{26.25 - 20}{10} \times 10\) = \(40 + 6.25\) = 46.25
6. Quartile Deviation = \(\frac{46.25 - 25.36}{2}\) = \(\frac{20.89}{2}\) = 10.45
7. Coefficient of Quartile Deviation = \(\frac{46.25 - 25.36}{46.25 + 25.36}\) = \(\frac{20.89}{71.61}\) = 0.2917

Teacher's Note:
a) Convert cumulative frequency series into simple frequency distribution first.
b) Pay close attention to subtraction of cumulative values.
c) Apply class interpolation formulas precisely.
d) Double-check all intermediate decimal calculations.