NCERT Solutions for Class 11th: Ch 5 Measures of Central Tendency

NCERT Solutions for Class 11th: Ch 5 Measures of Central Tendency Statistics for Economics

Page No: 71

Exercises

1. Which average would be suitable in the following cases?

(i) Average size of readymade garments.

► Mode

(ii) Average intelligence of students in a class.

► Median

(iii) Average production in a factory per shift.

► Arithmetic average

(iv) Average wages in an industrial concern.

► Arithmetic average

(v) When the sum of absolute deviations from average is least.

► Median

(vi) When quantities of the variable are in ratios.
► Arithmetic average

(vii) In case of open-ended frequency distribution.
► Median

2. Indicate the most appropriate alternative from the multiple choices provided against each question.

(i) The most suitable average for qualitative measurement is

(a) arithmetic mean
(b) median
(c) mode
(d) geometric mean
(e) none of the above

► (b) median

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(ii) Which average is affected most by the presence of extreme items?

(a) median
(b) mode
(c) arithmetic mean
(d) none of the above

► (c) arithmetic mean

(iii) 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

► (b) 0

3. Comment whether the following statements are true or false.

(i) The sum of deviation of items from median is zero.

► False

(ii) An average alone is not enough to compare series.

► True

(iii) Arithmetic mean is a positional value.

► False

(iv) Upper quartile is the lowest value of top 25% of items.

► True

(v) Median is unduly affected by extreme observations.

► 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 Rs)	0-10	10-20	20-30	30-40	40-50	50-60
Number of retail shops	12	18	27	-	17	6

Answer

(a) Let the missing frequency be x
Arithmetic mean = 28 (given)

Profit per retail shop (in Rs) Class Interval	No. of retail shops Frequency (f)	Mid Value (m)	fm
0-10	12	5	60
10-20	18	15	270
20-30	27	25	675
30-40	x	35	35x
40-50	17	45	765
50-60	6	55	330
	Σf = 80 + x		Σfx = 2100 + 35x

Mean = Σfx/Σf
⇒ 28 = 2100 + 35x/80 + x

⇒ 2240 + 28x = 2100 + 35

⇒ 2240 - 2100 = 35x - 25x

⇒ 140 = 7x

⇒  x = 140/7 = 20

Missing frequency = 20

(b)

Class Interval	Frequency (f)	Cumulative frequency (CF)
0-10	12	12
10-20	18	30
20-30	27	57
30-40	x	77
40-50	17	94
50-60	6	100
Total	Σf = 100

Median = Size of (N/2)th item
             = 100/2 = 50th item
It lies in class 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 Rs)	120	150	180	200	250	300	220	350	370	260

Answer

Workers	Daily Income (in Rs) X
A	120
B	150
C	180
D	200
E	250
F	300
G	220
H	350
I	370
J	260
Total	ΣX = 2400

N = 10
Arithmetic Mean = ΣX/N
                            = 2400/10
                            = 240
Arithmetic Mean = 240

6. Following information pertains to the daily income of 150 families. Calculate the arithmetic mean.

Income (in Rs)	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

Answer

Income	No. of families Frequency (f)	Mid Class (x)	fx
75-85	10	80	800
85-95	25	90	2250
95-105	20	100	2000
105-115	25	110	2750
115-125	10	120	1200
125-135	20	130	2600
135-145	15	140	2100
145-155	25	150	3750
	150		17450

Arithmetic Mean = Σfx/Σf
                            = 17450/150
                            = 116.33

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)	Less than 100	100-200	200-300	300-400	400 and above
Number of families	40	89	148	64	39

Answer

Size of Land Holdings        Class Interval	Number of families (f)	Cumulative frequency (CF)
0-100	40	40
100-200	89	129
200-300	148	277
300-400	64	341
400-500	39	380
Total	Σf = 380

Σf = N = 380
Median = Size of (N/2)th item
             = 380/2 = 190th item
It lies in class 200-300.

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 Rs)	10-14	15-19	20-24	25-29	30-34	35-39
Number of workers	5	10	15	20	10	5

(Hint: Compute median, lower quartile and upper quartile.)

Answer

Daily Income (in Rs)       Class Interval	No of workers (f)	Cumulative frequency (CF)
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
Total	Σf = 65

(a) Σf = N = 65
Median = Size of (N/2)th item
             = 65/2 = 32.5th item
It lies in class 24.5-29.5.

Highest income of lowest 50% workers = Rs 25.12

(b) First, we need to find Q₁
Class interval of Q₁ = (N/4)th items
                                 = (65/4)th items = 16.25th item
It lies in class 19.5-24.5.

Minimum income earned by the top 25% workers = Rs 19.92

(c) First, we need to find Q₃
Class interval of Q₃ = 3 (N/4)th items
                                 = 3 (65/4)th items = 3 × 16.25th item
                                 = 48.75th item
It lies in class 24.5-29.5.

Maximum income earned by lowest 25% workers = Rs 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)	50-53	53-56	56-59	59-62	62-65	65-68	68-71	71-74	74-77
Number of workers	3	8	14	30	36	28	16	10	5

Answer

Production Yield  (kg. per hectare)	No. of farms Frequency (f)	Mid Class (x)	fx	Cumulative frequency (CF)
50-53	3	51.5	154.5	3
53-56	8	54.5	436	11
56-59	14	57.5	805	25
59-62	30	60.5	1815	55
62-65	36	63.5	2286	91
65-68	28	66.5	1862	119
68-71	16	69.5	1112	135
71-74	10	72.5	725	145
74-77	5	75.5	377.5	150
	Σf = 150		Σfx = 9573

Mean = Σfx/Σf  = 9573/150 = 63.82 hectare

Modal Class = 62-65


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NCERT Solutions of Class 11th: Ch 4 Presentation of Data

NCERT Solutions for Class 11th: Ch 3 Presentation of Data Statistics of Economics

Exercises

Page No: 56

Answer the following questions, 1 to 10, choosing the correct answer

1. Bar diagram is a
(i) one-dimensional diagram
(ii) two-dimensional diagram
(iii) diagram with no dimension
(iv) none of the above

► (i) one-dimensional diagram

2. Data represented through a histogram can help in finding graphically the
(i) mean
(ii) mode
(iii) median
(iv) all the above

► (ii) mode

3. Ogives can be helpful in locating graphically the
(i) mode
(ii) mean
(iii) median
(iv) none of the above

► (iii) median

4. Data represented through arithmetic line graph help in understanding
(i) long-term trend
(ii) cyclicity in data
(iii) seasonality in data
(iv) all the above

► (i) long-term trend

5. Width of bars in a bar diagram need not be equal (True/False).

► False. Width of bars in a bar diagram need to be equal.

6. Width of rectangles in a histogram should essentially be equal (True/False).

► False. Width of rectangles in a histogram may or may not be equal.

7. Histogram can only be formed with continuous classification of data (True/False).

► True

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8. Histogram and column diagram are the same method of presentation of data (True/False).

► False. Histogram and column diagram are the different method of presentation of data.

9. Mode of a frequency distribution can be known graphically with the help of histogram (True/False).

► True

10. Median of a frequency distribution cannot be known from the ogives (True/False).

► False. Median of a frequency distribution can be known from the ogives.

11. What kinds of diagrams are more effective in representing the following?
(i) Monthly rainfall in a year
(ii) Composition of the population of Delhi by religion
(iii) Components of cost in a factory

Answer

(i) Monthly rainfall in a year - simple bar diagram as only one variable i.e. monthly rainfall is to be present and compared visually.

(ii) Composition of the population of Delhi by religion - Simple bar diagram Plotting different religion on the x-axis and the number of people on the y-axis, one can easily compare the number of the population religion-wise.

(iii) Components of cost in a factory - Pie chart as entire circle represents the total cost and various components of costsare shown by different portions of the circle.

12. Suppose you want to emphasise the increase in the share of urban non-workers and lower level of urbanisation in India as shown in Example 4.2. How would you do it in the tabular form?

Answer

Urban non workers	Rural Non Workers	Total Non workers
19 crores	42 crores	62 crores

The number of rural non-workers is greater than urban non-workers. The higher number of rural non-workers indicates lower level of urbanisation in India according to 2001 Census data.

13. How does the procedure of drawing a histogram differ when class intervals are unequal in comparison to equal class intervals in a frequency table?

Answer

When the class intervals are equal then by normal method we can make histogram which has equal width of rectangle. When the class intervals are unequal, heights of rectangles are to be adjusted to yield comparable measurements by using frequency density (class frequency divided by width of the class interval) instead of absolute frequency.

14. The Indian Sugar Mills Association reported that, 'Sugar production during the first fortnight of December 2001 was about 3,87,000 tonnes, as against 3,78,000 tonnes during the same fortnight last year (2000). The off-take of sugar from factories during the first fortnight of December 2001 was 2,83,000 tonnes for internal consumption and 41,000 tonnes for exports as against 1,54,000 tonnes for internal consumption and nil for exports during the same fortnight last season.'
(i) Present the data in tabular form.
(ii) Suppose you were to present these data in diagrammatic form which of the diagrams would you use and why?
(iii) Present these data diagrammatically.

Answer

(i)

Sugar Production in India
Date	Total Production (tonnes)	Internal Consumption (tonnes)	Export of sugar (tonnes)
December, 2000	3,78,000	1,54,000	-
December, 2000	3,27,000	2,83,000	41,000

(ii) To present these data in diagrammatic form we can use multiple bar diagram because these are effective in comparing two or more sets of data.

(iii)

15. The following table shows the estimated sectoral real growth rates (percentage change over the previous year) in GDP at factor cost.

Represent the data as multiple time series graphs.

Answer

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