NCERT Class 10 Chapter 14 Statistics CBSE Board Sample Problems Long Answer (For CBSE, ICSE, IAS, NET, NRA 2022)

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Question

The lengths of 50 leaves of a plant are measured correct to the nearest millimeter and the data obtained is represented in the following table

The Nearest Millimeter and the Data Obtained is Represented
Length (in mm)	109 - 117	118 - 126	127 - 135	136 - 144	145 - 153	154 - 162	163 - 171
No. of leaves	4	6	14	13	6	4	3

Find the mean length of the leaves.

Solution

The Nearest Millimeter and the Data Obtained is Represented
Class Interval
108.5 - 117.5 117.5 - 126.5 126.5 - 135.5 135.5 - 144.5 144.5 - 153.5 153.5 - 162.5 162.5 - 171.5	113 122 131 140 149 158 167	4 6 14 13 6 4 3	-3 -2 -1 0 1 2 3	-12 -12 -14 0 6 8 9
Total

Given frequency distribution is not continuous. So first we have to make it continuous.

Here assumed mean (a) = 140; class size = 9

Now,

Hence, mean length of the leaves -137.30mm.

Question

Draw a ‘less than type’ ogive for the following frequency distribution.

Draw a ‘Less Than Type’ Ogive
Class	15 - 20	20 - 25	25 - 30	30 - 35	35 - 40	40 - 45
Frequency	13	18	31	25	15	5

Solution

Draw a ‘Less Than Type’ Ogive
Class	Frequency
Less than 20 Less than 25 Less than 30 Less than 35 Less than 40 Less than 45	13

Question

In a retail market, fruit vendor were selling mangoes in packing boxes. These boxes contained varying number or mangoes. The following was the distribution.

In a Retail Market, Fruit Vendor Were Selling Mangoes in Packing Boxes
No. of mangoes	50 - 52	53 - 55	56 - 58	59 - 61	62 - 64
No. of boxes	15	110	135	115	25

Find the mean and median number of mangoes kept in a packing box.

Solution

In a Retail Market, Fruit Vendor Were Selling Mangoes in Packing Boxes
Class interval	Mid value ()	(where a = 57)
49.5 - 52.5 52.5 - 55.5 55.5 - 58.5 58.5 - 61.5 61.5 - 64.5	51 54 57 = a 60 63	-2 -1 0 1 2	15 110 135 115 25	-30 -110 0 115 50	15 125 260 375 400

Mean

For median: Median

Here

Median class is 55.5 - 58.5

Median

Question

The following table gives production yield of rice per hectare in some farms of a village:

Production Yield (In Kg/Hectare)
Production yield (in kg/hectare)	10 - 20	20 - 30	30 - 40	40 - 50	50 - 60
No. of farms	3	9	12	20	6

Draw a more than type ′ ogive. Also, find median from the curve.

Solution

Production Yield (In Kg/Hectare)
Production yield (in kg/hectare) Class interval	Frequency	Production yield more than or eual to
10 - 20 20 - 30 30 - 40 40 - 50 50 - 60	3 9 12 20 6	10 20 30 40 50	50 47 38 26 6

Question

Draw a More Than Type Ogive and from It Find Median
Life time (in hours)	More than or equal to 240	More than or equal to 280	More than or equal to 320	More than or equal to 360	More than or equal to 400	More than or equal to 440	More than or equal to 480
Number of bulbs	100	95	87	77	47	22	10

Draw a more than type ogive and from it find median. Verify it by actual calculations.

Solution

Draw a More Than Type Ogive and from It Find Median
More than or equal to	Number of bulbs
240 280 320 360 400 440 480	100 95 87 77 47 22 10

Draw a More Than Type Ogive and from It Find Median

Median from curve is 396.

Median from Curve is 396
Class interval	Frequency	Cumulative frequency
240 - 280 280 - 320 320 - 360 360 - 400 400 - 440 440 - 480 480 - 520	5 8 10 30 25 12 10	5 13 23 53 78 90 100	Median class
	100

Median

Question

Change the following distribution to a ‘more than type’ distribution. Hence draw the ‘more than type’ ogive for this distribution.

Change the Following Distribution to a ‘More Than Type’ Distribution
Class Interval	20 - 30	30 - 40	40 - 50	50 - 60	60 - 70	70 - 80	80 - 90
Frequency	10	8	12	24	6	25	15

Solution

Class Interval Cumulative Frequency
Class Interval	Cumulative Frequency
More than or equal to 20	100
More than or equal to 30	90
More than or equal to 40	82
More than or equal to 50	70
More than or equal to 60	46
More than or equal to 70	40
More than or equal to 80	15

Plot of points (20.100) , (30.90) , (40,82) , (50.70) , (60,46) , (70.40) and (80.15)

Join the points to get a curve

Question

The mean of the following distribution is 18. Find the frequency f of the class 19 — 21.

Find the Frequency F of the Class 19 — 21
Class	11 - 13	13 - 15	15 - 17	17 - 19	19 - 21	21 - 23	23 - 25
Frequency	3	6	9	13	f	5	4

Solution

Find the Frequency F of the Class 19 — 21
Class	x	f
11 - 13	12	3	36
13 - 15	14	6	84
15 - 17	16	9	144
17 - 19	18	13	234
19 - 21	20	f	20f
21 - 23	22	5	110
23 - 25	24	4	96

Mean

Question

If the median of the following frequency distribution is 32.5. Find the values of and .

If the Median of the Following Frequency Distribution is 32
Class	0 - 10	10 - 20	20 - 30	30 - 40	40 - 50	50 - 60	70 - 80	Total
Frequency		5	9	12		3	2	40

Solution

ClassFrequency Cumulative Frequency
Class	Frequency	Cumulative Frequency
0 - 10
10 - 20	5
20 - 30	9
30 - 40	12
40 - 50
50 - 60	3
60 - 70	2
	40

Median median class is 30 - 40.

Now

Also,

Question

The following distribution gives the daily income of 50 workers of a factory

Daily Income of 50 Workers of a Factory
Daily Income	400 - 420	400 - 420	400 - 420	400 - 420	400 - 420
Number of workers	12	12	12	12	12

Convert this distribution to less than a type of cumulative frequency distribution and draw its ogive.

Solution

Daily Income Number of Workers Cumulative Frequen
Daily Income	Number of workers	Cumulative Frequency
400 - 420	12	12
420 - 440	14	26
440 - 460	8	34
460 - 480	6	40
480 - 500	10	50

Correct Table

Drawing an ogive with coordinates

Question

Calculate Mean of the following distribution

Calculate Mean of the Following Distribution
C. I	10 - 19	20 - 29	30 - 39	40 - 49	50 - 59	60 - 69	70 - 79	80 - 89	90 - 99
Frequency	5	9	13	17	16	11	12	9	8

Solution

Calculate Mean of the Following Distribution
10 - 19	14.5	5	-40	-4	20
20 - 29	24.5	9	-30	-3	-27
30 - 39	34.5	13	-20	-2	-26
50 - 50	44.5	17	-10	-1	-17
40 - 49	54.5	16	0	0	0
60 - 69	64.5	11	10	1	11
70 - 79	74.5	12	20	2	24
80 - 89	84.5	9	30	3	27
90 - 99	94.5	8	40	4	32
		100

Question

If the mean of x and is M, find the mean of and

Solution

Mean of and is