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One of the interesting chapters of RD Sharma Class 10 is Statistics. It is the 7th chapter and comprises of six exercises. You can access the answers of the exercises here RD Sharma Solutions for Class 10. The chapter basically deals with the techniques for finding mean, median and mode of grouped data. It also includes the concept of cumulative frequency graph of a frequency distribution.

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### RD Sharma Solutions for Class 10 Maths Chapter 7 Statistics – exercise wise

### Access the RD Sharma Solutions For Class 10 Chapter 7 – Statistics

Exercise 7.1 Page No: 7.5

**1. Calculate the mean for the following distribution: **

x: |
5 |
6 |
7 |
8 |
9 |

f: |
4 |
8 |
14 |
11 |
3 |

**Solution: **

x | f | fx |

5 | 4 | 20 |

6 | 8 | 48 |

7 | 14 | 98 |

8 | 11 | 88 |

9 | 3 | 27 |

N = 40 | Î£ fx = 281 |

Mean = **Î£ **fx/ N = 281/40

âˆ´ Mean = 7.025

**2. Find the mean of the following data: **

x: |
19 |
21 |
23 |
25 |
27 |
29 |
31 |

f: |
13 |
15 |
16 |
18 |
16 |
15 |
13 |

**Solution: **

x | f | fx |

19 | 13 | 247 |

21 | 15 | 315 |

23 | 16 | 368 |

25 | 18 | 450 |

27 | 16 | 432 |

29 | 15 | 435 |

31 | 13 | 403 |

N = 106 | Î£ fx = 2620 |

Mean = **Î£ **fx/ N = 2620/106

âˆ´ Mean = 25

**3. If the mean of the following data is 20.6. Find the value of p. **

x: |
10 |
15 |
p |
25 |
35 |

f: |
3 |
10 |
25 |
7 |
5 |

**Solution: **

x | f | fx |

10 | 3 | 30 |

15 | 10 | 150 |

p | 25 | 25p |

25 | 7 | 175 |

35 | 5 | 175 |

N = 50 | Î£ fx = 530 + 25p |

We know that,

Mean = Î£ fx/ N = (2620 + 25p)/ 50

Given,

Mean = 20.6

â‡’ 20.6 = (530 + 25p)/ 50

(20.6 x 50) â€“ 530 = 25p

p = 500/ 25

âˆ´ p = 50

**4. If the mean of the following data is 15, find p. **

x: |
5 |
10 |
15 |
20 |
25 |

f: |
6 |
p |
6 |
10 |
5 |

**Solution: **

x | f | fx |

5 | 6 | 30 |

10 | p | 10p |

15 | 6 | 90 |

20 | 10 | 200 |

25 | 5 | 125 |

N = p + 27 | Î£ fx = 445 + 10p |

We know that,

Mean = Î£ fx/ N = (445 + 10p)/ (p + 27)

Given,

Mean = 15

â‡’ 15 = (445 + 10p)/ (p + 27)

15p + 405 = 445 + 10p

5p = 40

âˆ´ p = 8

**5. Find the value of p for the following distribution whose mean is 16.6**

x: |
8 |
12 |
15 |
p |
20 |
25 |
30 |

f: |
12 |
16 |
20 |
24 |
16 |
8 |
4 |

**Solution: **

x | f | fx |

8 | 12 | 96 |

12 | 16 | 192 |

15 | 20 | 300 |

P | 24 | 24p |

20 | 16 | 320 |

25 | 8 | 200 |

30 | 4 | 120 |

N = 100 | Î£ fx = 1228 + 24p |

We know that,

Mean = Î£ fx/ N = (1228 + 24p)/ 100

Given,

Mean = 16.6

â‡’ 16.6 = (1228 + 24p)/ 100

1660 = 1228 + 24p

24p = 432

âˆ´ p = 18

**6. Find the missing value of p for the following distribution whose mean is 12.58**

x: |
5 |
8 |
10 |
12 |
p |
20 |
25 |

f: |
2 |
5 |
8 |
22 |
7 |
4 |
2 |

**Solution: **

x | f | fx |

5 | 2 | 10 |

8 | 5 | 40 |

10 | 8 | 80 |

12 | 22 | 264 |

P | 7 | 7p |

20 | 4 | 80 |

25 | 2 | 50 |

N = 50 | Î£ fx = 524 + 7p |

We know that,

Mean = Î£ fx/ N = (524 + 7p)/ 50

Given,

Mean = 12.58

â‡’ 12.58 = (524 + 7p)/ 50

629 = 524 + 7p

7p = 629 â€“ 524 = 105

âˆ´ p = 15

**7. Find the missing frequency (p) for the following distribution whose mean is 7.68**

x: |
3 |
5 |
7 |
9 |
11 |
13 |

f: |
6 |
8 |
15 |
p |
8 |
4 |

**Solution: **

x | f | fx |

3 | 6 | 18 |

5 | 8 | 40 |

7 | 15 | 105 |

9 | p | 9p |

11 | 8 | 88 |

13 | 4 | 52 |

N = 41 + p | Î£ fx = 303 + 9p |

We know that,

Mean = Î£ fx/ N = (303 + 9p)/ (41 + p)

Given,

Mean = 7.68

â‡’ 7.68 = (303 + 9p)/ (41 + p)

7.68(41 + p) = 303 + 9p

7.68p + 314.88 = 303 + 9p

1.32p = 11.88

âˆ´ p = 11.88/1.32 = 9

Exercise 7.2 Page No: 7.5

**1. The number of telephone calls received at an exchange per interval for 250 successive one- minute intervals are given in the following frequency table:**

No. of calls (x): |
0 |
1 |
2 |
3 |
4 |
5 |
6 |

No. of intervals (f): |
15 |
24 |
29 |
46 |
54 |
43 |
39 |

**Compute the mean number of calls per interval.**

**Solution: **

Let the assumed mean(A) = 3

No. of calls x_{i} |
No. of intervals f_{i} |
u_{i }= x_{i }â€“ A = x_{i }â€“ 3 |
f_{i }u_{i} |

0 | 15 | -3 | -45 |

1 | 24 | -2 | -48 |

2 | 29 | -1 | -29 |

3 | 46 | 0 | 0 |

4 | 54 | 1 | 54 |

5 | 43 | 2 | 86 |

6 | 39 | 3 | 117 |

N = 250 | Î£ f_{i}x_{i }= 135 |

Mean number of calls = A + Î£ f_{i}x_{i }/ N

= 3 + 135/250

= (750 + 135)/ 250 = 885/ 250

= 3.54

**2. Five coins were simultaneously tossed 1000 times, and at each toss the number of heads was observed. The number of tosses during which 0, 1, 2, 3, 4 and 5 heads were obtained are shown in the table below. Find the mean number of heads per toss.**

No. of heads per toss (x): |
0 |
1 |
2 |
3 |
4 |
5 |

No. of tosses (f): |
38 |
144 |
342 |
287 |
164 |
25 |

Solution:

Let the assumed mean(A) = 2

No. of heads per tossÂ x_{i} |
No of intervalsÂ f_{i} |
u_{i }= x_{i }â€“ A = x_{i }â€“ 2 |
f_{i }u_{i} |

0 | 38 | -2 | -76 |

1 | 144 | -1 | -144 |

2 | 342 | 0 | 0 |

3 | 287 | 1 | 287 |

4 | 164 | 2 | 328 |

5 | 25 | 3 | 75 |

N = 1000 | Î£ f_{i}x_{i }= 470 |

Mean number of heads per toss = A + Î£ f_{i}x_{i }/ N

= 2 + 470/1000

= 2 + 0.470

= 2.470

**3. The following table gives the number of branches and number of plants in the garden of a school.**

No of branches (x): |
2 |
3 |
4 |
5 |
6 |

No of plants (f): |
49 |
43 |
57 |
38 |
13 |

**Calculate the average number of branches per plant. **

**Solution: **

Let the assumed mean (A) = 4

No of branchesÂ xi | No of plantsÂ f_{i} |
u_{iÂ }= x_{iÂ }âˆ’ A = x_{iÂ }âˆ’ 4 |
f_{i }u_{i} |

2 | 49 | -2 | -98 |

3 | 43 | -1 | -43 |

4 | 57 | 0 | 0 |

5 | 38 | 1 | 38 |

6 | 13 | 2 | 26 |

N = 200 | Î£ f_{i}x_{i }= -77 |

Average number of branches per plant = A + Î£ f_{i}x_{i }/ N = 4 + (-77/200)

= 4 -77/200

= (800 -77)/200

= 3.615

**4. The following table gives the number of children of 150 families in a village**

No of children (x): |
0 |
1 |
2 |
3 |
4 |
5 |

No of families (f): |
10 |
21 |
55 |
42 |
15 |
7 |

**Find the average number of children per family.**

**Solution: **

Let the assumed mean (A) = 2

No of childrenÂ x_{i} |
No of familiesÂ f_{i} |
u_{iÂ }= x_{i}Â âˆ’ A = x_{i}Â âˆ’ 2 |
f_{i }u_{i} |

0 | 10 | -2 | -20 |

1 | 21 | -1 | -21 |

2 | 55 | 0 | 0 |

3 | 42 | 1 | 42 |

4 | 15 | 2 | 30 |

5 | 7 | 3 | 21 |

N = 150 | Î£ f_{i}x_{i }= 52 |

Average number of children for family = A + Î£ f_{i}x_{i }/ N = 2 + 52/150

= (300 +52)/150

= 352/150

= 2.35 (corrected to neat decimal)

Exercise 7.3 Page No: 7.22

**1. The following table gives the distribution of total household expenditure (in rupees) of manual workers in a city.**

Expenditure (in rupees) (x) |
Frequency (f_{i}) |
Expenditure (in rupees) (x_{i}) |
Frequency (f_{i}) |

100 â€“ 150 |
24 |
300 â€“ 350 |
30 |

150 â€“ 200 |
40 |
350 â€“ 400 |
22 |

200 â€“ 250 |
33 |
400 â€“ 450 |
16 |

250 â€“ 300 |
28 |
450 â€“ 500 |
7 |

**Find the average expenditure (in rupees) per household.**

**Solution: **

Let the assumed mean (A) = 275

Class interval | Mid value (x_{i}) |
d_{iÂ }= x_{i}Â â€“ 275 |
u_{i} = (x_{i} – 275)/50 |
Frequency f_{i} |
f_{i}u_{i} |

100 â€“ 150 | 125 | -150 | -3 | 24 | -72 |

150 â€“ 200 | 175 | -100 | -2 | 40 | -80 |

200 â€“ 250 | 225 | -50 | -1 | 33 | -33 |

250 â€“ 300 | 275 | 0 | 0 | 28 | 0 |

300 â€“ 350 | 325 | 50 | 1 | 30 | 30 |

350 â€“ 400 | 375 | 100 | 2 | 22 | 44 |

400 â€“ 450 | 425 | 150 | 3 | 16 | 48 |

450 â€“ 500 | 475 | 200 | 4 | 7 | 28 |

N = 200 | Î£ f_{i}u_{i }= -35 |

Itâ€™s seen that A = 275 and h = 50

So,

Mean = A + h x (Î£f_{i }u_{i}/N)

= 275 + 50 (-35/200)

= 275 â€“ 8.75

= 266.25

**2. A survey was conducted by a group of students as a part of their environmental awareness program, in which they collected the following data regarding the number of plants in 200 houses in a locality. Find the mean number of plants per house.**

Number of plants: |
0 – 2 |
2 – 4 |
4 – 6 |
6 – 8 |
8 – 10 |
10 – 12 |
12 – 14 |

Number of house: |
1 |
2 |
1 |
5 |
6 |
2 |
3 |

**Which method did you use for finding the mean, and why?**

**Solution: **

From the given data,

To find the class interval we know that,

Class marks (x_{i}) = (upper class limit + lower class limit)/2

Now, letâ€™s compute x_{i} and f_{i}x_{i} by the following

Number of plants | Number of house (f_{i}) |
x_{i} |
f_{i}x_{i} |

0 – 2 | 1 | 1 | 1 |

2 â€“ 4 | 2 | 3 | 6 |

4 â€“ 6 | 1 | 5 | 5 |

6 â€“ 8 | 5 | 7 | 35 |

8 â€“ 10 | 6 | 9 | 54 |

10 â€“ 12 | 2 | 11 | 22 |

12 â€“ 14 | 3 | 13 | 39 |

Total | N = 20 | Î£ f_{i}u_{i }= 162 |

Here,

Mean = Î£ f_{i}u_{i }/ N

= 162/ 20

= 8.1

Thus, the mean number of plants in a house is 8.1

We have used the direct method as the values of class mark x_{i} and f_{i} is very small.

**3. Consider the following distribution of daily wages of workers of a factory**

Daily wages (in â‚¹) |
100 – 120 |
120 – 140 |
140 – 160 |
160 – 180 |
180 – 200 |

Number of workers: |
12 |
14 |
8 |
6 |
10 |

**Find the mean daily wages of the workers of the factory by using an appropriate method.**

**Solution: **

Let the assume mean (A) = 150

Class interval | Mid value x_{i} |
d_{iÂ }= x_{i}Â – 150 |
u_{i}Â = (x_{iÂ }– 150)/20 |
Frequency f_{i} |
f_{i}u_{i} |

100 â€“ 120 | 110 | -40 | -2 | 12 | -24 |

120 – 140 | 130 | -20 | -1 | 14 | -14 |

140 – 160 | 150 | 0 | 0 | 8 | 0 |

160 – 180 | 170 | 20 | 1 | 6 | 6 |

180 – 200 | 190 | 40 | 2 | 10 | 20 |

N= 50 | Î£ f_{i}u_{i }= -12 |

Itâ€™s seen that,

A = 150 and h = 20

So,

Mean = A + h x (Î£f_{i }u_{i}/N)

= 150 + 20 x (-12/50)

= 150 â€“ 24/5

= 150 = 4.8

= 145.20

**4. Thirty women were examined in a hospital by a doctor and the number of heart beats per minute recorded and summarized as follows. Find the mean heart beats per minute for these women, choosing a suitable method. **

Number of heart beats per minute: |
65 – 68 |
68 – 71 |
71 – 74 |
74 – 77 |
77 – 80 |
80 – 83 |
83 – 86 |

Number of women: |
2 |
4 |
3 |
8 |
7 |
4 |
2 |

**Solution: **

Using the relation (x_{i}) = (upper class limit + lower class limit)/ 2

And, class size of this data = 3

Let the assumed mean (A) = 75.5

So, letâ€™s calculate d_{i}, u_{i}, f_{i}u_{i} as following:

Number of heart beats per minute | Number of women (f_{i}) |
x_{i} |
d_{i}Â = x_{i}Â â€“ 75.5 |
u_{i}Â = (x_{iÂ }– 755)/h |
f_{i}u_{i} |

65 â€“ 68 | 2 | 66.5 | -9 | -3 | -6 |

68 â€“ 71 | 4 | 69.5 | -6 | -2 | -8 |

71 â€“ 74 | 3 | 72.5 | -3 | -1 | -3 |

74 â€“ 77 | 8 | 75.5 | 0 | 0 | 0 |

77 â€“ 80 | 7 | 78.5 | 3 | 1 | 7 |

80 â€“ 83 | 4 | 81.5 | 6 | 2 | 8 |

83 â€“ 86 | 2 | 84.5 | 9 | 3 | 6 |

N = 30 | Î£ f_{i}u_{i }= 4 |

From table, itâ€™s seen that

N = 30 and h = 3

So, the mean = A + h x (Î£f_{i }u_{i}/N)

= 75.5 + 3 x (4/30

= 75.5 + 2/5

= 75.9

Therefore, the mean heart beats per minute for those women are 75.9 beats per minute.

**Find the mean of each of the following frequency distributions: (5 – 14)**

**5. **

Class interval: |
0 â€“ 6 |
6 – 12 |
12 – 18 |
18 â€“ 24 |
24 – 30 |

Frequency: |
6 |
8 |
10 |
9 |
7 |

**Solution: **

Letâ€™s consider the assumed mean (A) = 15

Class interval | Mid – value x_{i} |
d_{iÂ }= x_{iÂ }â€“ 15 |
u_{iÂ }= (x_{iÂ }â€“ 15)/6 |
f_{i} |
f_{i}u_{i} |

0 â€“ 6 | 3 | -12 | -2 | 6 | -12 |

6 – 12 | 9 | -6 | -1 | 8 | -8 |

12 – 18 | 15 | 0 | 0 | 10 | 0 |

18 â€“ 24 | 21 | 6 | 1 | 9 | 9 |

24 – 30 | 27 | 12 | 2 | 7 | 14 |

N = 40 | Î£ f_{i}u_{i }= 3 |

From the table itâ€™s seen that,

A = 15 and h = 6

Mean = A + h x (Î£f_{i }u_{i}/N)

= 15 + 6 x (3/40)

= 15 + 0.45

= 15.45

**6. **

Class interval: |
50 â€“ 70 |
70 â€“ 90 |
90 â€“ 110 |
110 â€“ 130 |
130 â€“ 150 |
150 – 170 |

Frequency: |
18 |
12 |
13 |
27 |
8 |
22 |

**Solution: **

**Letâ€™s consider the assumed mean (A) = 100**

Class interval | Mid – value x_{i} |
d_{iÂ }= x_{iÂ }â€“ 100 |
u_{iÂ }= (x_{iÂ }â€“ 100)/20 |
f_{i} |
f_{i}u_{i} |

50 â€“ 70 | 60 | -40 | -2 | 18 | -36 |

70 â€“ 90 | 80 | -20 | -1 | 12 | -12 |

90 â€“ 110 | 100 | 0 | 0 | 13 | 0 |

110 â€“ 130 | 120 | 20 | 1 | 27 | 27 |

130 â€“ 150 | 140 | 40 | 2 | 8 | 16 |

150 – 170 | 160 | 60 | 3 | 22 | 66 |

N= 100 | Î£ f_{i}u_{i }= 61 |

From the table itâ€™s seen that,

A = 100 and h = 20

Mean = A + h x (Î£f_{i }u_{i}/N)

= 100 + 20 x (61/100)

= 100 + 12.2

= 112.2

**7. **

Class interval: |
0 â€“ 8 |
8 – 16 |
16 – 24 |
24 â€“ 32 |
32 – 40 |

Frequency: |
6 |
7 |
10 |
8 |
9 |

**Solution: **

Letâ€™s consider the assumed mean (A) = 20

Class interval | Mid – value x_{i} |
d_{iÂ }= x_{iÂ }â€“ 20 |
u_{iÂ }= (x_{iÂ }â€“ 20)/8 |
f_{i} |
f_{i}u_{i} |

0 â€“ 8 | 4 | -16 | -2 | 6 | -12 |

8 â€“ 16 | 12 | -8 | -1 | 7 | -7 |

16 â€“ 24 | 20 | 0 | 0 | 10 | 0 |

24 â€“ 32 | 28 | 8 | 1 | 8 | 8 |

32 â€“ 40 | 36 | 16 | 2 | 9 | 18 |

N = 40 | Î£ f_{i}u_{i }= 7 |

From the table itâ€™s seen that,

A = 20 and h = 8

Mean = A + h x (Î£f_{i }u_{i}/N)

= 20 + 8 x (7/40)

= 20 + 1.4

= 20.4

**8. **

Class interval: |
0 â€“ 6 |
6 – 12 |
12 – 18 |
18 â€“ 24 |
24 – 30 |

Frequency: |
7 |
5 |
10 |
12 |
6 |

**Solution: **

Letâ€™s consider the assumed mean (A) = 15

Class interval | Mid – value x_{i} |
d_{iÂ }= x_{iÂ }â€“ 15 |
u_{iÂ }= (x_{iÂ }â€“ 15)/6 |
f_{i} |
f_{i}u_{i} |

0 â€“ 6 | 3 | -12 | -2 | 7 | -14 |

6 – 12 | 9 | -6 | -1 | 5 | -5 |

12 – 18 | 15 | 0 | 0 | 10 | 0 |

18 â€“ 24 | 21 | 6 | 1 | 12 | 12 |

24 – 30 | 27 | 12 | 2 | 6 | 12 |

N = 40 | Î£ f_{i}u_{i }= 5 |

From the table itâ€™s seen that,

A = 15 and h = 6

Mean = A + h x (Î£f_{i }u_{i}/N)

= 15 + 6 x (5/40)

= 15 + 0.75

= 15.75

**9. **

Class interval: |
0 – 10 |
10 – 20 |
20 – 30 |
30 â€“ 40 |
40 – 50 |

Frequency: |
9 |
12 |
15 |
10 |
14 |

**Solution: **

Letâ€™s consider the assumed mean (A) = 25

Class interval | Mid – value x_{i} |
d_{iÂ }= x_{iÂ }â€“ 25 |
u_{iÂ }= (x_{iÂ }â€“ 25)/10 |
f_{i} |
f_{i}u_{i} |

0 â€“ 10 | 5 | -20 | -2 | 9 | -18 |

10 â€“ 20 | 15 | -10 | -1 | 12 | -12 |

20 â€“ 30 | 25 | 0 | 0 | 15 | 0 |

30 â€“ 40 | 35 | 10 | 1 | 10 | 10 |

40 – 50 | 45 | 20 | 2 | 14 | 28 |

N = 60 | Î£ f_{i}u_{i }= 8 |

From the table itâ€™s seen that,

A = 25 and h = 10

Mean = A + h x (Î£f_{i }u_{i}/N)

= 25 + 10 x (8/60)

= 25 + 4/3

= 79/3 = 26.333

**10. **

Class interval: |
0 – 8 |
8 – 16 |
16 â€“ 24 |
24 â€“ 32 |
32 – 40 |

Frequency: |
5 |
9 |
10 |
8 |
8 |

**Solution: **

Letâ€™s consider the assumed mean (A) = 20

Class interval | Mid – value x_{i} |
d_{iÂ }= x_{iÂ }â€“ 20 |
u_{iÂ }= (x_{iÂ }â€“ 20)/8 |
f_{i} |
f_{i}u_{i} |

0 â€“ 8 | 4 | -16 | -2 | 5 | -10 |

8 â€“ 16 | 12 | -4 | -1 | 9 | -9 |

16 â€“ 24 | 20 | 0 | 0 | 10 | 0 |

24 â€“ 32 | 28 | 4 | 1 | 8 | 8 |

32 â€“ 40 | 36 | 16 | 2 | 8 | 16 |

N = 40 | Î£ f_{i}u_{i }= 5 |

From the table itâ€™s seen that,

A = 20 and h = 8

Mean = A + h x (Î£f_{i }u_{i}/N)

= 20 + 8 x (5/40)

= 20 + 1

= 21

**11.**

Class interval: |
0 â€“ 8 |
8 – 16 |
16 – 24 |
24 â€“ 32 |
32 â€“ 40 |

Frequency: |
5 |
6 |
4 |
3 |
2 |

**Solution: **

Letâ€™s consider the assumed mean (A) = 20

Class interval | Mid – value x_{i} |
d_{iÂ }= x_{iÂ }â€“ 20 |
u_{iÂ }= (x_{iÂ }â€“ 20)/8 |
f_{i} |
f_{i}u_{i} |

0 â€“ 8 | 4 | -16 | -2 | 5 | -12 |

8 â€“ 16 | 12 | -8 | -1 | 6 | -8 |

16 â€“ 24 | 20 | 0 | 0 | 4 | 0 |

24 â€“ 32 | 28 | 8 | 1 | 3 | 9 |

32 â€“ 40 | 36 | 16 | 2 | 2 | 14 |

N = 20 | Î£ f_{i}u_{i }= -9 |

From the table itâ€™s seen that,

A = 20 and h = 8

Mean = A + h x (Î£f_{i }u_{i}/N)

= 20 + 6 x (-9/20)

= 20 â€“ 72/20

= 20 â€“ 3.6

= 16.4

**12. **

Class interval: |
10 – 30 |
30 – 50 |
50 – 70 |
70 â€“ 90 |
90 – 110 |
110 – 130 |

Frequency: |
5 |
8 |
12 |
20 |
3 |
2 |

**Solution: **

Letâ€™s consider the assumed mean (A) = 60

Class interval | Mid – value x_{i} |
d_{iÂ }= x_{iÂ }â€“60 |
u_{iÂ }= (x_{iÂ }â€“ 60)/20 |
f_{i} |
f_{i}u_{i} |

10 â€“ 30 | 20 | -40 | -2 | 5 | -10 |

30 â€“ 50 | 40 | -20 | -1 | 8 | -8 |

50 â€“ 70 | 60 | 0 | 0 | 12 | 0 |

70 â€“ 90 | 80 | 20 | 1 | 20 | 20 |

90 â€“ 110 | 100 | 40 | 2 | 3 | 6 |

110 â€“ 130 | 120 | 60 | 3 | 2 | 6 |

N = 50 | Î£ f_{i}u_{i }= 14 |

From the table itâ€™s seen that,

A = 60 and h = 20

Mean = A + h x (Î£f_{i }u_{i}/N)

= 60 + 20 x (14/50)

= 60 + 28/5

= 60 + 5.6

= 65.6

**13. **

Class interval: |
25 â€“ 35 |
35 – 45 |
45 – 55 |
55 â€“ 65 |
65 â€“ 75 |

Frequency: |
6 |
10 |
8 |
12 |
4 |

**Solution: **

Letâ€™s consider the assumed mean (A) = 50

Class interval | Mid – value x_{i} |
d_{iÂ }= x_{iÂ }â€“ 50 |
u_{iÂ }= (x_{iÂ }â€“ 50)/10 |
f_{i} |
f_{i}u_{i} |

25 – 35 | 30 | -20 | -2 | 6 | -12 |

35 – 45 | 40 | -10 | -1 | 10 | -10 |

45 – 55 | 50 | 0 | 0 | 8 | 0 |

55 – 65 | 60 | 10 | 1 | 12 | 12 |

65 – 75 | 70 | 20 | 2 | 4 | 8 |

N = 40 | Î£ f_{i}u_{i }= -2 |

From the table itâ€™s seen that,

A = 50 and h = 10

Mean = A + h x (Î£f_{i }u_{i}/N)

= 50 + 10 x (-2/40)

= 50 – 0.5

= 49.5

**14. **

Class interval: |
25 â€“ 29 |
30 â€“ 34 |
35 â€“ 39 |
40 â€“ 44 |
45 â€“ 49 |
50 â€“ 54 |
55 â€“ 59 |

Frequency: |
14 |
22 |
16 |
6 |
5 |
3 |
4 |

**Solution: **

Letâ€™s consider the assumed mean (A) = 42

Class interval | Mid – value x_{i} |
d_{iÂ }= x_{iÂ }â€“ 42 |
u_{iÂ }= (x_{iÂ }â€“ 42)/5 |
f_{i} |
f_{i}u_{i} |

25 â€“ 29 | 27 | -15 | -3 | 14 | -42 |

30 â€“ 34 | 32 | -10 | -2 | 22 | -44 |

35 â€“ 39 | 37 | -5 | -1 | 16 | -16 |

40 â€“ 44 | 42 | 0 | 0 | 6 | 0 |

45 â€“ 49 | 47 | 5 | 1 | 5 | 5 |

50 â€“ 54 | 52 | 10 | 2 | 3 | 6 |

55 â€“ 59 | 57 | 15 | 3 | 4 | 12 |

N = 70 | Î£ f_{i}u_{i }= -79 |

From the table itâ€™s seen that,

A = 42 and h = 5

Mean = A + h x (Î£f_{i }u_{i}/N)

= 42 + 5 x (-79/70)

= 42 â€“ 79/14

= 42 â€“ 5.643

= 36.357

Exercise 7.4 Page No: 7.34

**1. Following are the lives in hours of 15 pieces of the components of aircraft engine. Find the median:Â **

**715, 724, 725, 710, 729, 745, 694, 699, 696, 712, 734, 728, 716, 705, 719.**

**Solution:**

Arranging the given data in ascending order, we have

694, 696, 699, 705, 710, 712, 715, 716, 719, 721, 725, 728, 729, 734, 745

As the number of terms is an old number i.e., N = 15

We use the following procedure to find the median.

Median = (N + 1)/2 ^{th }term

= (15 + 1)/2 ^{th }term

= 8^{th}Â term

So, the 8^{th}Â term in the arranged order of the given data should be the median.

Therefore, 716 is the median of the data.

**2. The following is the distribution of height of students of a certain class in a certain city:**

Height (in cm): |
160 – 162 |
163 – 165 |
166 – 168 |
169 – 171 |
172 – 174 |

No of students: |
15 |
118 |
142 |
127 |
18 |

**Find the median height.**

**Solution: **

Class interval (exclusive) | Class interval Â (inclusive) | Class interval frequency | Cumulative frequency |

160 â€“ 162 | 159.5 â€“ 162.5 | 15 | 15 |

163 â€“ 165 | 162.5 â€“ 165.5 | 118 | 133(F) |

166 â€“ 168 | 165.5 â€“ 168.5 | 142(f) | 275 |

169 â€“ 171 | 168.5 â€“ 171.5 | 127 | 402 |

172 â€“ 174 | 171.5 â€“ 174.5 | 18 | 420 |

N = 420 |

Here, we have N = 420,

So, N/2 = 420/ 2 = 210

The cumulative frequency just greater than N/2 is 275 then 165.5 – 168.5 is the median class such, that

L = 165.5, f = 142, F = 133 and h = (168.5 – 165.5) = 3

= 165.5 + 1.63

= 167.13

**3. Following is the distribution of I.Q of 100 students. Find the median I.Q.**

I.Q: |
55 – 64 |
65 – 74 |
75 – 84 |
85 – 94 |
95 – 104 |
105 – 114 |
115 – 124 |
125 – 134 |
135 – 144 |

No of students: |
1 |
2 |
9 |
22 |
33 |
22 |
8 |
2 |
1 |

**Solution: **

Class interval (exclusive) | Class intervalÂ (inclusive) | Class interval frequency | Cumulative frequency |

55 – 64 | 54.5 â€“ 64-5 | 1 | 1 |

65 – 74 | 64.5 â€“ 74.5 | 2 | 3 |

75 – 84 | 74.5 â€“ 84.5 | 9 | 12 |

85 – 94 | 84.5 â€“ 94.5 | 22 | 34(F) |

95 – 104 | 94.5 â€“ 104.5 | 33(f) | 67 |

105 – 114 | 104.5 â€“ 114.5 | 22 | 89 |

115 – 124 | 114.5 â€“ 124.5 | 8 | 97 |

125 – 134 | 124.5 â€“ 134.5 | 2 | 98 |

135 – 144 | 134.5 â€“ 144.5 | 1 | 100 |

N = 100 |

Here, we have N = 100,

So, N/2 = 100/ 2 = 50

The cumulative frequency just greater than N/ 2 is 67 then the median class is (94.5 – 104.5) such that L = 94.5, F = 33, h = (104.5 – 94.5) = 10

= 94.5 + 4.85

= 99.35

**4. Calculate the median from the following data:**

Rent (in Rs): |
15 – 25 |
25 – 35 |
35 – 45 |
45 – 55 |
55 – 65 |
65 – 75 |
75 – 85 |
85 – 95 |

No of houses: |
8 |
10 |
15 |
25 |
40 |
20 |
15 |
7 |

** Solution: **

Class interval | Frequency | Cumulative frequency |

15 – 25 | 8 | 8 |

25 – 35 | 10 | 18 |

35 – 45 | 15 | 33 |

45 – 55 | 25 | 58(F) |

55 – 65 | 40(f) | 98 |

65 – 75 | 20 | 118 |

75 – 85 | 15 | 133 |

85 – 95 | 7 | 140 |

N = 140 |

Here, we have N = 140,

So, N/2 = 140/ 2 = 70

The cumulative frequency just greater than N/ 2 is 98 then median class is 55 – 65 such that L = 55, f = 40, F = 58, h = 65 – 55 = 10

= 55 + 3 = 58

**5. Calculate the median from the following data:**

Marks below: |
10 â€“ 20 |
20 – 30 |
30 – 40 |
40 – 50 |
50 – 60 |
60 – 70 |
70 – 80 |
85 – 95 |

No of students: |
15 |
35 |
60 |
84 |
96 |
127 |
198 |
250 |

**Solution: **

Marks below | No. of students | Class interval | Frequency | Cumulative frequency |

10 | 15 | 0 â€“ 10 | 15 | 15 |

20 | 35 | 10 â€“ 20 | 20 | 35 |

30 | 60 | 20 â€“ 30 | 25 | 60 |

40 | 84 | 30 â€“ 40 | 24 | 84 |

50 | 96 | 40 â€“ 50 | 12 | 96(F) |

60 | 127 | 50 â€“ 60 | 31(f) | 127 |

70 | 198 | 60 â€“ 70 | 71 | 198 |

80 | 250 | 70 â€“ 80 | 52 | 250 |

N = 250 |

Here, we have N = 250,

So, N/2 = 250/ 2 = 125

The cumulative frequency just greater than N/ 2 is 127 then median class is 50 â€“ 60 such that L = 50, f = 31, F = 96, h = 60 -50 = 10

= 50 + 9.35

= 59.35

**6. Calculate the missing frequency from the following distribution, it being given that the median of the distribution is 24.**

Age in years: |
0 – 10 |
10 â€“ 20 |
20 – 30 |
30 – 40 |
40 – 50 |

No of persons: |
5 |
25 |
? |
18 |
7 |

**Solution: **

Let the unknown frequency be taken as x,

Class interval | Frequency | Cumulative frequency |

0 â€“ 10 | 5 | 5 |

10 â€“ 20 | 25 | 30(F) |

20 – 30 | x (f) | 30 + x |

30 – 40 | 18 | 48 + x |

40 – 50 | 7 | 55 + x |

N = 170 |

Itâ€™s given that

Median = 24

Then, median class = 20 – 30; L = 20,Â h = 30 -20 = 10, f = x,Â F = 30

4x = 275 + 5x – 300

4x – 5x = â€“ 25

– x = â€“ 25

x = 25

Therefore, the Missing frequency = 25

**7. The following table gives the frequency distribution of married women by age at marriage.**

Age (in years) |
Frequency |
Age (in years) |
Frequency |

15 â€“ 19 |
53 |
40 â€“ 44 |
9 |

20 â€“ 24 |
140 |
45 â€“ 49 |
5 |

25 â€“ 29 |
98 |
45 â€“ 49 |
3 |

30 â€“ 34 |
32 |
55 â€“ 59 |
3 |

35 â€“ 39 |
12 |
60 and above |
2 |

**Calculate the median and interpret the results. **

**Solution: **

Class interval (exclusive) |
Class interval (inclusive) |
Frequency |
Cumulative frequency |

15 â€“ 19 | 14.5 â€“ 19.5 | 53 | 53 (F) |

20 â€“ 24 | 19.5 â€“ 24.5 | 140 (f) | 193 |

25 â€“ 29 | 24.5 â€“ 29.5 | 98 | 291 |

30 â€“ 34 | 29.5 â€“ 34.5 | 32 | 323 |

35 â€“ 39 | 34.5 â€“ 39.5 | 12 | 335 |

40 â€“ 44 | 39.5 â€“ 44.5 | 9 | 344 |

45 â€“ 49 | 44.5 â€“ 49.5 | 5 | 349 |

50 â€“ 54 | 49.5 â€“ 54.5 | 3 | 352 |

55 – 54 | 54.5 â€“ 59.5 | 3 | 355 |

60 and above | 59.5 and above | 2 | 357 |

N =357 |

Here, we have N = 357,

So, N/2 = 357/ 2 = 178.5

The cumulative frequency just greater than N/2 is 193, so then the median class is (19.5 â€“ 24.5) such that l = 19.5, f = 140, F = 53, h = 25.5 â€“ 19.5 = 5

Median = 23.98

Which means nearly half the women were married between the ages of 15 and 25

**8. The following table gives the distribution of the life time of 400 neon lamps:**

Life time: (in hours) |
Number of lamps |

1500 – 2000 |
14 |

2000 – 2500 |
56 |

2500 â€“ 3000 |
60 |

3000 â€“ 3500 |
86 |

3500 â€“ 4000 |
74 |

4000 â€“ 4500 |
62 |

4500 â€“ 5000 |
48 |

**Find the median life. **

**Solution: **

Life time | Number of lamps fi | Cumulative frequency (cf) |

1500 â€“ 2000 | 14 | 14 |

2000 â€“ 2500 | 56 | 70 |

2500 â€“ 3000 | 60 | 130(F) |

3000 â€“ 3500 | 86(f) | 216 |

3500 â€“ 4000 | 74 | 290 |

4000 â€“ 4500 | 62 | 352 |

4500 â€“ 5000 | 48 | 400 |

N = 400 |

Itâ€™s seen that, the cumulative frequency just greater than n/2 (400/2 = 200) is 216 and it belongs to the class interval 3000 â€“ 3500 which becomes the Median class = 3000 – 3500

Lower limits (l) of median class = 3000 and,

Frequency (f) of median class = 86

Cumulative frequency (cf) of class preceding median class = 130

And, the Class size (h) = 500

Thus, calculating the median by the formula, we get

= 3000 + (35000/86)

= 3406.98

Thus, the median life time of lamps is 3406.98 hours

**9. The distribution below gives the weight of 30 students in a class. Find the median weight of students:**

Weight (in kg): |
40 – 45 |
45 – 50 |
50 – 55 |
55 – 60 |
60 – 65 |
65 – 70 |
70 – 75 |

No of students: |
2 |
3 |
8 |
6 |
6 |
3 |
2 |

**Solution: **

Weight (in kg) | Number of students fi | Cumulative frequency (cf) |

40 â€“ 45 | 2 | 2 |

45 â€“ 50 | 3 | 5 |

50 â€“ 55 | 8 | 13 |

55 â€“ 60 | 6 | 19 |

60 â€“ 65 | 6 | 25 |

65 â€“ 70 | 3 | 28 |

70 â€“ 75 | 2 | 30 |

Itâ€™s seen that, the cumulative frequency just greater than n/ 2 (i.e. 30/ 2 = 15) is 19, belongs to class interval 55Â â€“ 60.

So, itâ€™s chosen that

Median class = 55 â€“ 60

Lower limit (l) of median class = 55

Frequency (f) of median class = 6

Cumulative frequency (cf) = 13

And, Class size (h) = 5

Thus, calculating the median by the formula, we get

= 55 + 10/6 = 56.666

So, the median weight is 56.67 kg.

**10. Find the missing frequencies and the median for the following distribution if the mean is 1.46**

No. of accidents: |
0 |
1 |
2 |
3 |
4 |
5 |
Total |

Frequencies (no. of days): |
46 |
? |
? |
25 |
10 |
5 |
200 |

**Solution: **

No. of accidents (x) | No. of days (f) | fx |

0 | 46 | 0 |

1 | x | x |

2 | y | 2y |

3 | 25 | 75 |

4 | 10 | 40 |

5 | 5 | 25 |

N = 200 | Sum = x + 2y + 140 |

Itâ€™s given that, N = 200

â‡’ 46 + x + y + 25 + 10 + 5 = 200

â‡’ x + y = 200 â€“ 46 â€“ 25 â€“ 10 â€“ 5

â‡’ x + y = 114 —- (i)

And also given, Mean = 1.46

â‡’ Sum/ N = 1.46

â‡’ (x + 2y + 140)/ 200 = 1.46

â‡’ x + 2y = 292 â€“ 140

â‡’ x + 2y = 152 —- (ii)

Subtract equation (i) from equation (ii), we get

x + 2y â€“ x â€“ y = 152 â€“ 114

â‡’ y = 38

Now, on putting the value of y in equation (i), we find x = 114 â€“ 38 = 76

Thus, the table become:

No. of accidents (x) | No. of days (f) | Cumulative frequency |

0 | 46 | 46 |

1 | 76 | 122 |

2 | 38 | 160 |

3 | 25 | 185 |

4 | 10 | 195 |

5 | 5 | 200 |

N = 200 |

Itâ€™s seen that,

N = 200 N/2 = 200/2 = 100

So, the cumulative frequency just more than N/2 is 122

Therefore, the median is 1.

Exercise 7.5 Page No: 7.34

**1. Find the mode of the following data:Â **

**(i) 3, 5, 7, 4, 5, 3, 5, 6, 8, 9, 5, 3, 5, 3, 6, 9, 7, 4Â **

**(ii) 3, 3, 7, 4, 5, 3, 5, 6, 8, 9, 5, 3, 5, 3, 6, 9, 7, 4Â **

**(iii) 15, 8, 26, 25, 24, 15, 18, 20, 24, 15, 19, 15Â **

**Solution:**

(i)

Value (x) | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

Frequency (f) | 4 | 2 | 5 | 2 | 2 | 1 | 2 |

Thus, the mode = 5 since it occurs the maximum number of times.

(ii)

Value (x) | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

Frequency (f) | 5 | 2 | 4 | 2 | 2 | 1 | 2 |

Thus, the mode = 3 since it occurs the maximum number of times.

(iii)

Value (x) | 8 | 15 | 18 | 19 | 20 | 24 | 25 |

Frequency (f) | 1 | 4 | 1 | 1 | 1 | 2 | 1 |

Thus, the mode = 15 since it occurs the maximum number of times.

**2. The shirt size worn by a group of 200 persons, who bought the shirt from a store, are as follows:**

Shirt size: |
37 |
38 |
39 |
40 |
41 |
42 |
43 |
44 |

Number of persons: |
15 |
25 |
39 |
41 |
36 |
17 |
15 |
12 |

**Find the model shirt size worn by the group.**

**Solution: **

Shirt size: | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 |

Number of persons: | 15 | 25 | 39 | 41 | 36 | 17 | 15 | 12 |

From the data its observed that,

Model shirt size = 40 since it was the size which occurred for the maximum number of times.

**3. Find the mode of the following distribution. **

**(i)**

Class interval: |
0 â€“ 10 |
10 â€“ 20 |
20 â€“ 30 |
30 â€“ 40 |
40 â€“ 50 |
50 â€“ 60 |
60 â€“ 70 |
70 â€“ 80 |

Frequency: |
5 |
8 |
7 |
12 |
28 |
20 |
10 |
10 |

**Solution: **

Class interval: | 0 â€“ 10 | 10 â€“ 20 | 20 â€“ 30 | 30 â€“ 40 | 40 â€“ 50 | 50 â€“ 60 | 60 â€“ 70 | 70 â€“ 80 |

Frequency: | 5 | 8 | 7 | 12 | 28 | 20 | 10 | 10 |

Itâ€™s seen that the maximum frequency is 28.

So, the corresponding class i.e., 40 â€“ 50 is the modal class.

And,

l = 40, h = 50 40 = 10, f = 28, f_{1}Â = 12, f_{2Â }= 20

Using the formula for finding mode, we get

= 40 + 160/ 24

= 40 + 6.67

= 46.67

**(ii)**

Class interval |
10 â€“ 15 |
15 â€“ 20 |
20 â€“ 25 |
25 â€“ 30 |
30 â€“ 35 |
35 â€“ 40 |

Frequency |
30 |
45 |
75 |
35 |
25 |
15 |

**Solution: **

Class interval | 10 â€“ 15 | 15 â€“ 20 | 20 â€“ 25 | 25 â€“ 30 | 30 â€“ 35 | 35 â€“ 40 |

Frequency | 30 | 45 | 75 | 35 | 25 | 15 |

Itâ€™s seen that the maximum frequency is 75.

So, the corresponding class i.e., 20 – 25 is the modal class.

And,

l = 20, h = 25 – 20 = 5, f = 75, f_{1}Â = 45, f_{2Â }= 35

Using the formula for finding mode, we get

= 20 + 150/70

= 20 + 2.14

= 22.14

**(iii)**

Class interval |
25 â€“ 30 |
30 â€“ 35 |
35 â€“ 40 |
40 â€“ 45 |
45 â€“ 50 |
50 â€“ 55 |

Frequency |
25 |
34 |
50 |
42 |
38 |
14 |

**Solution: **

Class interval | 25 â€“ 30 | 30 â€“ 35 | 35 â€“ 40 | 40 â€“ 45 | 45 â€“ 50 | 50 â€“ 55 |

Frequency | 25 | 34 | 50 | 42 | 38 | 14 |

Itâ€™s seen that the maximum frequency is 50.

So, the corresponding class i.e., 35 – 40 is the modal class.

And,

l = 35, h = 40 – 35 = 5, f = 50, f_{1}Â = 34, f_{2Â }= 42

Using the formula for finding mode, we get

= 35 + 80/24

= 35 + 3.33

= 38.33

**4. Compare the modal ages of two groups of students appearing for an entrance test:**

Age in years |
16 â€“ 18 |
18 â€“ 20 |
20 â€“ 22 |
22 â€“ 24 |
24 â€“ 26 |

Group A |
50 |
78 |
46 |
28 |
23 |

Group B |
54 |
89 |
40 |
25 |
17 |

**Solution: **

Age in years | 16 â€“ 18 | 18 â€“ 20 | 20 â€“ 22 | 22 â€“ 24 | 24 â€“ 26 |

Group A | 50 | 78 | 46 | 28 | 23 |

Group B | 54 | 89 | 40 | 25 | 17 |

For Group A:

Itâ€™s seen that the maximum frequency is 78.

So, the corresponding class 18 â€“ 20 is the model class.

And,

l = 18, h = 20 – 18 = 2, f = 78, f_{1}Â = 50, f_{2Â }= 46

Using the formula for finding mode, we get

= 18 + 56/60

= 18 + 0.93

= 18.93 years

For group B:

Itâ€™s seen that the maximum frequency is 89

So, the corresponding class 18 – 20 is the modal class.

And,

l = 18, h = 20 – 18 = 2, f = 89, f_{1}Â = 54, f_{2Â }= 40

Using the formula for finding mode, we get

Mode

= 18 + 70/84

= 18 + 0.83

= 18.83 years

Therefore, the modal age of the Group A is higher than that of Group B.

**5. The marks in science of 80 students of class X are given below. Find the mode of the marks obtained by the students in science.
**

Marks |
0 â€“ 10 |
10 â€“ 20 |
20 â€“ 30 |
30 â€“ 40 |
40 â€“ 50 |
50 â€“ 60 |
60 â€“ 70 |
70 â€“ 80 |
80 â€“ 90 |
90 â€“ 100 |

Frequency |
3 |
5 |
16 |
12 |
13 |
20 |
5 |
4 |
1 |
1 |

**Solution: **

Marks | 0 â€“ 10 | 10 â€“ 20 | 20 â€“ 30 | 30 â€“ 40 | 40 â€“ 50 | 50 â€“ 60 | 60 â€“ 70 | 70 â€“ 80 | 80 â€“ 90 | 90 â€“ 100 |

Frequency | 3 | 5 | 16 | 12 | 13 | 20 | 5 | 4 | 1 | 1 |

Itâ€™s seen that the maximum frequency is 20.

So, the corresponding class 50 – 60 is the modal class.

And,

l = 50, h = 60 â€“ 50 = 10, f = 20, f_{1}Â = 13, f_{2Â }= 5

Using the formula for finding mode, we get

= 50 + 70/22

= 50 + 3.18

= 53.18

**6. The following is the distribution of height of students of a certain class in a city:**

Height (in cm): |
160 â€“ 162 |
163 â€“ 165 |
166 â€“ 168 |
169 â€“ 171 |
172 â€“ 174 |

No of students: |
15 |
118 |
142 |
127 |
18 |

**Find the average height of maximum number of students.Â **

**Solution: **

Heights(exclusive) | 160 â€“ 162 | 163 â€“ 165 | 166 â€“ 168 | 169 â€“ 171 | 172 â€“ 174 |

Heights (inclusive) | 159.5 â€“ 162.5 | 162.5 â€“ 165.5 | 165.5 â€“ 168.5 | 168.5 â€“ 171.5 | 171.5 â€“ 174.5 |

No of students | 15 | 118 | 142 | 127 | 18 |

Itâ€™s seen that the maximum frequency is 142.

So, the corresponding class 165.5 â€“ 168.5 is the modal class.

And,

l = 165.5, h = 168.5 – 165.5 = 3, f = 142, f_{1}Â = 118, f_{2}Â = 127

Using the formula for finding mode, we get

= 165.5 + 72/39

= 165.5 + 1.85

= 167.35 cm

**7. The following table shows the ages of the patients admitted in a hospital during a year:**

Ages (in years): |
5 â€“ 15 |
15 â€“ 25 |
25 â€“ 35 |
35 â€“ 45 |
45 â€“ 55 |
55 â€“ 65 |

No of students: |
6 |
11 |
21 |
23 |
14 |
5 |

**Find the mode and the mean of the data given above. Compare and interpret the two measures of central tendency.Â **

**Solution: **

To find the mean:

For the given data let the assumed mean (A) = 30

Age (in years) | Number of patients f_{i} |
Class marks x_{i} |
d_{i} = x_{i} â€“ 275 |
fidi |

5 â€“ 15 | 6 | 10 | – 20 | -120 |

15 â€“ 25 | 11 | 20 | – 10 | -110 |

25 â€“ 35 | 21 | 30 | 0 | 0 |

35 â€“ 45 | 23 | 40 | 10 | 230 |

45 â€“ 55 | 14 | 50 | 20 | 280 |

55 â€“ 65 | 5 | 60 | 30 | 150 |

N = 80 | Î£f_{iÂ }d_{i}Â = 430 |

Itâ€™s observed from the table thatÂ Î£f_{i} = N = 80 andÂ Î£f_{iÂ }d_{i}Â = 430.

Using the formula for mean,

= 30 + 430/80

= 30 + 5.375

= 35.375

= 35.38

Thus, the mean of this data is 35.38. It can also be interpreted as that on an average the age of a patients admitted to hospital was 35.38 years.

It is also observed that maximum class frequency is 23 and it belongs to class interval 35 â€“ 45

So, modal class is 35 â€“ 45 with the Lower limit (l) of modal class = 35

And, Frequency (f) of modal class = 23

Class size (h) = 10

Frequency (f_{1}) of class preceding the modal class = 21

Frequency (f_{2}) of class succeeding the modal class = 14

Mode

Therefore, the mode is 36.8. This represents that maximum number of patients admitted in hospital were of 36.8 years.

Hence, itâ€™s seen that mode is greater than the mean.

**8. The following data gives the information on the observed lifetimes (in hours) of 225 electrical components:**

Lifetimes (in hours): |
0 â€“ 20 |
20 â€“ 40 |
40 â€“ 60 |
60 â€“ 80 |
80 â€“ 100 |
100 â€“ 120 |

No. of components: |
10 |
35 |
52 |
61 |
38 |
29 |

**Determine the modal lifetimes of the components.**

**Solution: **

From the data given as above its observed that maximum class frequency is 61 which belongs to class interval 60 â€“ 80.

So, modal class limit (l) of modal class = 60

Frequency (f) of modal class = 61

Frequency (f_{1}) of class preceding the modal class = 52

Frequency (f_{2}) of class succeeding the modal class = 38

Class sizeÂ (h) = 20

Using the formula for find mode, we have

Mode

Thus, the modal lifetime of electrical components is 65.625 hours

**9. The following table gives the daily income of 50 workers of a factory:**

Daily income |
100 â€“ 120 |
120 â€“ 140 |
140 â€“ 160 |
160 â€“ 180 |
180 â€“ 200 |

Number of workers |
12 |
14 |
8 |
6 |
10 |

**Find the mean, mode and median of the above data.**

**Solution: **

Class interval | Mid value (x) | Frequency (f) | fx | Cumulative frequency |

100 â€“ 120 | 110 | 12 | 1320 | 12 |

120 â€“ 140 | 130 | 14 | 1820 | 26 |

140 â€“ 160 | 150 | 8 | 1200 | 34 |

160 â€“ 180 | 170 | 6 | 1000 | 40 |

180 â€“ 200 | 190 | 10 | 1900 | 50 |

N = 50 | Î£fx = 7260 |

We know that,

Mean = Î£fx / N

= 7260/ 50

= 145.2

Then,

We have, N = 50

â‡’ N/2 = 50/2 = 25

So, the cumulative frequency just greater than N/2 is 26, then the median class is 120 – 140

Such that l = 120, h = 140 – 120 = 20, f = 14, F = 12

= 120 + 260/14

= 120 + 18.57

= 138.57

From the data, its observed that maximum frequency is 14, so the corresponding class 120 – 140 is the modal class

And,

l = 120, h = 140 – 120 = 20, f = 14, f_{1Â }= 12, f_{2Â }= 8

= 120 + 5

= 125

Therefore, mean = 145.2, median = 138.57 and mode = 125

Exercise 7.6 Page No: 7.62

**1. Draw an ogive by less than the method for the following data:**

No. of rooms |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |

No. of houses |
4 |
9 |
22 |
28 |
24 |
12 |
8 |
6 |
5 |
2 |

**Solution: **

No. of rooms | No. of houses | Cumulative Frequency |

Less than or equal to 1 | 4 | 4 |

Less than or equal to 2 | 9 | 13 |

Less than or equal to 3 | 22 | 35 |

Less than or equal to 4 | 28 | 63 |

Less than or equal to 5 | 24 | 87 |

Less than or equal to 6 | 12 | 99 |

Less than or equal to 7 | 8 | 107 |

Less than or equal to 8 | 6 | 113 |

Less than or equal to 9 | 5 | 118 |

Less than or equal to 10 | 2 | 120 |

Itâ€™s required to plot the points (1, 4), (2, 3), (3, 35), (4, 63), (5, 87), (6, 99), (7, 107), (8, 113), (9, 118), (10, 120), by taking upper class limit over the x-axis and cumulative frequency over the y-axis.

**2. The marks scored by 750 students in an examination are given in the form of a frequency distribution table:**

Marks |
No. of Students |

600 â€“ 640 |
16 |

640 â€“ 680 |
45 |

680 â€“ 720 |
156 |

720 â€“ 760 |
284 |

760 â€“ 800 |
172 |

800 â€“ 840 |
59 |

840 â€“ 880 |
18 |

**Prepare a cumulative frequency distribution table by less than method and draw an ogive. **

**Solution: **

Marks | No. of Students | Marks Less than | Cumulative Frequency |

600 â€“ 640 | 16 | 640 | 16 |

640 â€“ 680 | 45 | 680 | 61 |

680 â€“ 720 | 156 | 720 | 217 |

720 â€“ 760 | 284 | 760 | 501 |

760 â€“ 800 | 172 | 800 | 673 |

800 â€“ 840 | 59 | 840 | 732 |

840 â€“ 880 | 18 | 880 | 750 |

Plot the points (640, 16), (680, 61), (720, 217), (760, 501), (800, 673), (840, 732), (880, 750) by taking upper class limit over the x-axis and cumulative frequency over the y-axis.

**3. Draw an Ogive to represent the following frequency distribution:**

Class-interval |
0 â€“ 4 |
5 â€“ 9 |
10 â€“ 14 |
15 â€“ 19 |
20 â€“ 24 |

No. of students |
2 |
6 |
10 |
5 |
3 |

**Solution: **

Since the given frequency distribution is not continuous we will have to first make it continuous and then prepare the cumulative frequency:

Class-interval | No. of Students | Less than | Cumulative frequency |

0.5 â€“ 4.5 | 2 | 4.5 | 2 |

4.5 â€“ 9.5 | 6 | 9.5 | 8 |

9.5 â€“ 14.5 | 10 | 14.5 | 18 |

14.5 â€“ 19.5 | 5 | 19.5 | 23 |

19.5 â€“ 24.5 | 3 | 24.5 | 26 |

Plot the points (4.5, 2), (9.5, 8), (14.5, 18), (19.5, 23), (24.5, 26) by taking the upper class limit over the x-axis and cumulative frequency over the y-axis.

**4. The monthly profits (in Rs) of 100 shops are distributed as follows:**

Profit per shop |
No of shops: |

0 â€“ 50 |
12 |

50 â€“ 100 |
18 |

100 â€“ 150 |
27 |

150 â€“ 200 |
20 |

200 â€“ 250 |
17 |

250 â€“ 300 |
6 |

**Draw the frequency polygon for it. **

**Solution: **

Doing for the less than method, we have

Profit per shop | Mid-value | No of shops: |

Less than 0 | 0 | 0 |

Less than 0 â€“ 50 | 25 | 12 |

Less than 50 â€“ 100 | 75 | 18 |

Less than 100 â€“ 150 | 125 | 27 |

Less than 150 â€“ 200 | 175 | 20 |

Less than 200 â€“ 250 | 225 | 17 |

Less than 250 â€“ 300 | 275 | 6 |

Above 300 | 300 | 0 |

By plotting the respectively coordinates we can get the frequency polygon.

**5. The following distribution gives the daily income of 50 workers of a factory:**

Daily income (in Rs): |
No of workers: |

100 â€“ 120 |
12 |

120 â€“ 140 |
14 |

140 â€“ 160 |
8 |

160 â€“ 180 |
6 |

180 â€“ 200 |
10 |

**Convert the above distribution to a ‘less than’ type cumulative frequency distribution and draw its ogive.**

**Solution: **

Firstly, we prepare the cumulative frequency table by less than method as given below:

Daily income | Cumulative frequency |

Less than 120 | 12 |

Less than 140 | 26 |

Less than 160 | 34 |

Less than 180 | 40 |

Less than 200 | 50 |

Now we mark on x-axis upper class limit, y-axis cumulative frequencies. Thus we plot the point (120, 12), (140, 26), (160, 34), (180, 40), (200, 50).