Class | Subject | Book | Chapter | Session | Questions covered |
|---|---|---|---|---|---|
XI | Economics | Statistics for Economics (NCERT) | 5 – Measures of Central Tendency | 2026-27 | 9 of 9 |
Chapter at a glance
Measure | Formula / idea |
|---|---|
Arithmetic mean | ; assumed mean:   |
Property | Sum of deviations from the mean is zero; affected by extreme values |
Median | Middle value; grouped:  |
Quartiles | at N/4, at 3N/4 (same formula as the median)   |
Mode | Most frequent value; grouped: , ,    |
Choice | Mode for qualitative/size data, median for open-ended or skewed data, mean for most quantitative data |
Exercise Questions
Which average would be suitable in the following cases?
Answer
- Average size of readymade garments.
- Average intelligence of students in a class.
- Average production in a factory per shift.
- Average wage in an industrial concern.
- When the sum of absolute deviations from average is least.
- When quantities of the variable are in ratios.
- In case of open-ended frequency distribution.
Answer
Case | Suitable average | Reason |
|---|---|---|
(i) Size of readymade garments | Mode | The most commonly demanded size matters |
(ii) Intelligence of students | Median | Qualitative attribute; students can be ranked |
(iii) Production per shift | Arithmetic mean | Total production ÷ number of shifts |
(iv) Wage in an industrial concern | Arithmetic mean | Total wage bill ÷ number of workers |
(v) Sum of absolute deviations least | Median | Sum of absolute deviations is minimum from the median |
(vi) Quantities in ratios | Geometric mean | Suitable for ratios and rates of change |
(vii) Open-ended distribution | Median | Does not need the extreme class limits |
- Mode (ii) Median (iii) Mean (iv) Mean (v) Median (vi) Geometric mean (vii) Median
Indicate the most appropriate alternative from the multiple choices provided against each question.
Answer
- The most suitable average for qualitative measurement is (a) arithmetic mean (b) median (c) mode (d) geometric mean (e) none of the above
- Which average is affected most by the presence of extreme items? (a) median (b) mode (c) arithmetic mean (d) none of the above
- The algebraic sum of deviation of a set of n values from A.M. is (a) n (b) 0 (c) 1 (d) none of the above
Answer
- (b) median. Qualitative data can be ranked, so a positional average is used.
- (c) arithmetic mean. It uses every value, so extremes pull it up or down.
- (b) 0. .
- b (ii) c (iii) b
Comment whether the following statements are true or false.
Answer
- The sum of deviation of items from median is zero.
- An average alone is not enough to compare series.
- Arithmetic mean is a positional value.
- Upper quartile is the lowest value of top 25% of items.
- Median is unduly affected by extreme observations.
Answer
- False. The sum of deviations is zero from the arithmetic mean; from the median, the sum of absolute deviations is the least.
- True. Two series can have the same average but very different spread; measures of dispersion and skewness are also needed.
- False. The arithmetic mean is a mathematical average; median and mode are positional values.
- True. 75 % of items lie below , so it is the lowest value of the top 25 %.
- False. The median depends on position, not on the size of extreme values.
- False (ii) True (iii) False (iv) True (v) False
If the arithmetic mean of the data given below is 28, find (a) the missing frequency, and (b) the median of the series:
Answer
Profit per retail shop (₹) | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 | 50–60 |
|---|---|---|---|---|---|---|
Number of retail shops | 12 | 18 | 27 | ? | 17 | 6 |
Answer
Given: ; let the missing frequency be .
Formula:
Class | m | Frequency | fm | c.f. |
|---|---|---|---|---|
0–10 | 5 | 12 | 60 | 12 |
10–20 | 15 | 18 | 270 | 30 |
20–30 | 25 | 27 | 675 | 57 |
30–40 | 35 | f | 35f | 57 + f |
40–50 | 45 | 17 | 765 | 74 + f |
50–60 | 55 | 6 | 330 | 80 + f |
Total | 80 + f | 2100 + 35f |
(a)
(b) Now N = 100, so N/2 = 50. The c.f. just above 50 is 57, so the median class is 20–30 (L = 20, c.f. = 30, f = 27, h = 10).
- Missing frequency = 20 (b) Median = ₹27.41
The following table gives the daily income of ten workers in a factory. Find the arithmetic mean.
Answer
Workers | A | B | C | D | E | F | G | H | I | J |
|---|---|---|---|---|---|---|---|---|---|---|
Daily Income (₹) | 120 | 150 | 180 | 200 | 250 | 300 | 220 | 350 | 370 | 260 |
Answer
Formula:
Arithmetic mean = ₹240
Following information pertains to the daily income of 150 families. Calculate the arithmetic mean.
Answer
Income (₹) | More than 75 | 85 | 95 | 105 | 115 | 125 | 135 | 145 |
|---|---|---|---|---|---|---|---|---|
Number of families | 150 | 140 | 115 | 95 | 70 | 60 | 40 | 25 |
Answer
This is a “more than” cumulative distribution; convert it into class frequencies (the last class is taken as 145–155).
Income (₹) | Frequency | Mid-value m | fm |
|---|---|---|---|
75–85 | 150 − 140 = 10 | 80 | 800 |
85–95 | 140 − 115 = 25 | 90 | 2,250 |
95–105 | 115 − 95 = 20 | 100 | 2,000 |
105–115 | 95 − 70 = 25 | 110 | 2,750 |
115–125 | 70 − 60 = 10 | 120 | 1,200 |
125–135 | 60 − 40 = 20 | 130 | 2,600 |
135–145 | 40 − 25 = 15 | 140 | 2,100 |
145–155 | 25 | 150 | 3,750 |
Total | 150 | 17,450 |
Arithmetic mean ≈ ₹116.33
The size of land holdings of 380 families in a village is given below. Find the median size of land holdings.
Answer
Size of Land Holdings (acres) | Less than 100 | 100–200 | 200–300 | 300–400 | 400 and above |
|---|---|---|---|---|---|
Number of families | 40 | 89 | 148 | 64 | 39 |
Answer
Size (acres) | f | c.f. |
|---|---|---|
0–100 | 40 | 40 |
100–200 | 89 | 129 |
200–300 | 148 | 277 |
300–400 | 64 | 341 |
400 and above | 39 | 380 |
N/2 = 380/2 = 190, so the median class is 200–300 (L = 200, c.f. = 129, f = 148, h = 100).
Median size of land holding ≈ 241.22 acres
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.
Answer
Daily Income (₹) | 10–14 | 15–19 | 20–24 | 25–29 | 30–34 | 35–39 |
|---|---|---|---|---|---|---|
Number of workers | 5 | 10 | 15 | 20 | 10 | 5 |
Answer
The classes are inclusive, so convert them to continuous classes by subtracting 0.5 from lower limits and adding 0.5 to upper limits.
Income (₹) | f | 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 |
N = 65; h = 5.
(a) Highest income of the lowest 50 % = Median. N/2 = 32.5, so the median class is 24.5–29.5.
(b) Minimum income of the top 25 % = Upper quartile . 3N/4 = 48.75, so the class is 24.5–29.5.
(c) Maximum income of the lowest 25 % = Lower quartile . N/4 = 16.25, so the class is 19.5–24.5.
Note: the textbook’s answer key prints (b) ₹19.92 and (c) ₹29.19. These values are the two quartiles printed in swapped places: the top 25 % start at (₹29.19), and the lowest 25 % end at (₹19.92). The median is ₹25.125, which the key rounds to ₹25.11.
- ₹25.13 (median) (b) ₹29.19 (Q₃) (c) ₹19.92 (Q₁)
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.
Answer
Yield (kg/ha) | 50–53 | 53–56 | 56–59 | 59–62 | 62–65 | 65–68 | 68–71 | 71–74 | 74–77 |
|---|---|---|---|---|---|---|---|---|---|
Number of farms | 3 | 8 | 14 | 30 | 36 | 28 | 16 | 10 | 5 |
Answer
Yield | f | m | fm | c.f. |
|---|---|---|---|---|
50–53 | 3 | 51.5 | 154.5 | 3 |
53–56 | 8 | 54.5 | 436.0 | 11 |
56–59 | 14 | 57.5 | 805.0 | 25 |
59–62 | 30 | 60.5 | 1,815.0 | 55 |
62–65 | 36 | 63.5 | 2,286.0 | 91 |
65–68 | 28 | 66.5 | 1,862.0 | 119 |
68–71 | 16 | 69.5 | 1,112.0 | 135 |
71–74 | 10 | 72.5 | 725.0 | 145 |
74–77 | 5 | 75.5 | 377.5 | 150 |
Total | 150 | 9,573.0 |
Mean: kg/ha
Median: N/2 = 75, so the median class is 62–65 (L = 62, c.f. = 55, f = 36, h = 3).
kg/ha
Mode: the modal class is 62–65 (highest frequency 36), so , , ; , .
kg/ha
Mean = 63.82, Median = 63.67, Mode = 63.29 kg per hectare
Topic checklist
- Arithmetic mean: direct, assumed mean and step-deviation methods; weighted mean
- Median and quartiles for individual and grouped data
- Mode and its graphical location
- Relative position and choice of averages