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NCERT Class 11 Economics Chapter 5 Measures of Central Tendency
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Measures of Central Tendency
Chapter: 5
PART – (A) STATISTICS FOR ECONOMICS
TEXTUAL QUESTION ANSWERS
1. Which average would be suitable in the following cases?
(a) Average size of readymade garments.
Ans: Mode.
(b) Average intelligence of students in a class.
Ans: Median.
(c) Average production in a factory per shift.
Ans: Arithmetic Mean.
(d) Average wage in an industrial concern.
Ans: (d) Arithmetic Mean.
(e) When the sum of absolute deviations from average is least.
Ans: Arithmetic Mean.
(f) When quantities of the variable are in ratios.
Ans: Geometric mean.
(g) In case of open-ended frequency distribution.
Ans: Median or Mode.
2. Indicate the most appropriate alternative from the multiple choices provided against each question.
(a) The most suitable average for qualitative measurement is:
(i) Arithmetic means.
(ii) Median.
(iii) Mode.
(iv) Geometric mean.
(v) None of the above.
Ans: (ii) Median.
(b) Which average is affected most by the presence of extreme items?
(i) Median.
(ii) Mode.
(iii) Arithmetic mean.
(iv) None of the above.
Ans: (iii) Arithmetic mean.
(c) The algebraic sum of deviation of a set of n values from A.M. is:
(i) n
(ii) 0
(iii) 1
(iv) None of the above.
Ans: (ii) 0
3. Comment whether the following statements are true or false.
(a) The sum of deviation of items from median is zero.
Ans: False.
(b) An average alone is not enough to compare series.
Ans: True.
(c) Arithmetic mean is a positional value.
Ans: False.
(d) Upper quartile is the lowest value of the top 25% of items.
Ans: True.
(e) Median is unduly affected by extreme observations.
Ans: False.
4. If the arithmetic mean of the data given below is 28, find (a) the missing frequency, and (b) the median of the series:
| Profit per retail shop (in ₹) | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 |
| Number of retail shops | 12 | 18 | 27 | – | 17 | 6 |
Ans: (a) Let the missing frequency be f₁.
Arithmetic Mean = 28
| Profit Per Retail Shop (in₹) Class Interval | Number of Retail Shops (f) | Mid Value (m) | fm |
| 0-10 | 12 | 5 | 60 |
| 10-20 | 18 | 15 | 270 |
| 20-30 | 27 | 25 | 675 |
| 30-40 | f₁ | 35 | 35f₁ |
| 40-50 | 17 | 45 | 765 |
| 50-60 | 6 | 55 | 330 |
| ∑f = 80 + f₁ | ∑fm = 2100 + 35 f₁ |
2240 + 28f₁ = 2100 + 35f₁
2240 – 2100 = 35f₁ = 28f₁
140 = 7f₁
f₁ = 20
Hence, the missing frequency is 20.
(b)
| Class Interval | Frequency | Cumulative Frequency |
| 0-10 | 12 | 12 |
| 10-20 | 18 | 30 |
| 20-30 | 27 | 57 |
| 30-40 | 20 | 77 |
| 40-50 | 17 | 94 |
| 50-60 | 6 | 1000 |
| ∑f = 100 |
50th item lies in the 57th cumulative frequency and the corresponding class interval is 20-30.
5. The following table gives the daily income of ten workers in a factory. Find the arithmetic mean.
| Workers | A | B | C | D | E | F | G | H | I | J |
| Daily Income (in ₹) | 120 | 150 | 180 | 200 | 250 | 300 | 220 | 350 | 370 | 260 |
Ans:
| Workers | Daily Income (in ₹) (x) |
| A | 120 |
| B | 150 |
| C | 180 |
| D | 200 |
| E | 250 |
| F | 300 |
| G | 220 |
| H | 350 |
| I | 370 |
| J | 260 |
| ∑X = 2,400 |
N = 10
∴ Arithmetic Mean = ₹ 240
6. Following information pertains to the daily income of 150 families. Calculate the arithmetic mean.
| Income (in₹) | Number of families |
| More than 75 | 150 |
| More than 85 | 140 |
| More than 95 | 115 |
| More than 105 | 95 |
| More than 115 | 70 |
| More than 125 | 60 |
| More than 135 | 40 |
| More than 145 | 25 |
Ans:
| Income Class Interval | Number of Families (c.f.) | Frequency (f) | Mid Value (m) | fm |
| 75 – 85 | 150 | 150 – 140 = 10 | 80 | 800 |
| 85 – 95 | 140 | 140 – 115 = 25 | 90 | 2,250 |
| 95 – 105 | 115 | 115 – 95 = 20 | 100 | 2,000 |
| 105 – 115 | 95 | 95 – 70 = 25 | 110 | 2,750 |
| 115 – 125 | 70 | 70 – 60 = 10 | 120 | 1,200 |
| 125 – 135 | 60 | 60 – 40 = 20 | 130 | 2,600 |
| 135 – 145 | 40 | 40 – 25 = 15 | 140 | 2,100 |
| 145 – 155 | 25 | 25 | 150 | 3,750 |
| Σf = 150 | Σfm = 17,450 |
7. The size of land holdings of 380 families in a village is given below. Find the median size of land holdings.
| Size of Land Holdings (in acres) | Number of families |
| Less than 100 | 40 |
| 100-200 | 89 |
| 200-300 | 148 |
| 300-400 | 64 |
| 400 and above | 39 |
Ans:
| Size of Land Holdings Class Interval | Number of Families (f) | Cumulative Frequency (c.f.) |
| 0 – 100 | 40 | 40 |
| 100 – 200 | 89 | 129 |
| 200 – 300 | 148 | 277 |
| 300 – 400 | 64 | 341 |
| 400 – 500 | 39 | 380 |
| Σf = 380 |
190ᵗʰ item lies in The 129ᵗʰcumulative frequency and the corresponding class interval is 200-300.
= 200 + 41.22 = 241.22
Median size of land holdings = 241.22 acres.
8. The following series relates to the daily income of workers employed in a firm. Compute (a) highest income of lowest 50% workers (b) minimum income earned by the top 25% workers and (c) maximum income earned by lowest 25% workers.
| Daily Income (in₹) | 10-14 | 15-19 | 20-24 | 25-29 | 30-34 | 35-39 |
| Number of workers | 5 | 10 | 15 | 20 | 10 | 5 |
Ans:
| Daily Income (in) Class Interval | Number of Workers(f) | Cumulative Frequency (c.f.) |
| 9.5 – 14.5 | 5 | 5 |
| 14.5 – 19.5 | 10 | 15 |
| 19.5 – 24.5 | 15 | 30 |
| 24.5 – 29.5 | 20 | 50 |
| 29.5 – 34.5 | 10 | 60 |
| 34.5 – 39.5 | 5 | 65 |
| Σf = 65 |
(a) Highest income of lowest 50% workers will be given by the Median. Σf = N = 65
32.5ᵗʰ item he’s in the 50% cumulative frequency and the corresponding class interval is 24.5-29.5.
= ₹ 25.13
(b) Minimum income earned by top 25% workers will be given by the lower quartile Q₁.
= 16.25ᵗʰ item
16.25ᵗʰ item lies in the 30ᵗʰ cumulative frequency and the corresponding class interval is 19.5-24.5
= ₹19.92
(c) Maximum income earned by lowest 25% workers will be given by the upper quartile Q₃.
= 3 × 16.25ᵗʰ item = 48.75ᵗʰ item
48.75ᵗʰ item lines in 50ᵗʰ item and the corresponding class interval is 24.5-29.5
= ₹ 29.19
9. The following table gives production yield in kg per hectare of wheat of 150 farms in a village. Calculate the mean, median and mode values.
| Production yield (kg per hectare) | Number of farms |
| 50-53 | 3 |
| 53-56 | 8 |
| 56-59 | 14 |
| 59-62 | 30 |
| 62-65 | 36 |
| 65-68 | 28 |
| 68-71 | 16 |
| 71-74 | 10 |
| 74-77 | 5 |
Ans:
| Production yield (kg per hectare) | f | c.f. | M.V.(x) | dx | dx’ | fdx’ |
| 50-53 | 3 | 3 | 51.5 | -12 | -4 | -12 |
| 53-56 | 8 | 11 | 54.5 | -9 | -3 | -24 |
| 56-59 | 14 | 25 | 57.5 | -6 | -2 | -28 |
| 59-62 | 30 | 55 | 60.5 | -3 | -1 | -30 |
| 62-65 | 36 | 91 | 63.5 | 0 | 0 | 0 |
| 65-68 | 28 | 119 | 66.5 | 3 | 1 | 28 |
| 68-71 | 16 | 135 | 69.5 | 6 | 2 | 32 |
| 71-74 | 10 | 145 | 72.5 | 9 | 3 | 30 |
| 74-77 | 5 | 150 | 75.5 | 12 | 4 | 20 |
| Σf = 150 | Σfdx’ = 16 |
By interpretation:
By inspection we find that mode lies in the class 62-65, because this class has maximum frequency, i.e., 36.
