In order to find the mean value, we will use a direct method because the numerical value of fi and xi are small.
Find the midpoint of the given interval using the formula.
No. of plants
(Class interval)

No. of houses
Frequency (fi)

Midpoint (xi)

fixi

02

1

1

1

24

2

3

6

46

1

5

5

68

5

7

35

810

6

9

54

1012

2

11

22

1214

3

13

39


Sum fi = 20


Sum fixi = 162

The formula to find the mean is:
Mean = x̄ = ∑fi xi /∑fi
= 162/20
= 8.1
Therefore, the mean number of plants per house is 8.1
2. Consider the following distribution of daily wages of 50 workers of a factory.
Daily wages (in Rs.)

100120

120140

140160

160180

180200

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:
Find the midpoint of the given interval using the formula.
Midpoint (xi) = (upper limit + lower limit)/2
In this case, the value of midpoint (xi) is large, so let us assume the mean value, A = 150 and class interval is h = 20.
So, ui = (xi – A)/h = ui = (xi – 150)/20
Substitute and find the values as follows:
Daily wages
(Class interval)

Number of workers
frequency (fi)

Midpoint (xi)

ui = (xi – 150)/20

fiui

100120

12

110

2

24

120140

14

130

1

14

140160

8

150

0

0

160180

6

170

1

6

180200

10

190

2

20

Total

Sum fi = 50



Sum fiui = 12

So, the formula to find out the mean is:
Mean = x̄ = A + h∑fiui /∑fi =150 + (20 × 12/50) = 150 – 4.8 = 145.20
Thus, the mean daily wage of the workers = Rs. 145.20
3. The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is Rs 18. Find the missing frequency f.
Daily Pocket Allowance (in c)

1113

1315

1517

1719

1921

2123

2335

Number of children

7

6

9

13

f

5

4

Solution:
To find out the missing frequency, use the mean formula.
Here, the value of midpoint (xi) mean x̄ = 18
Class interval

Number of children (fi)

Midpoint (xi)

fixi

1113

7

12

84

1315

6

14

84

1517

9

16

144

1719

13

18 = A

234

1921

f

20

20f

2123

5

22

110

2325

4

24

96

Total

fi = 44+f


Sum fixi = 752+20f

The mean formula is
Mean = x̄ = ∑fixi /∑fi = (752+20f)/(44+f)
Now substitute the values and equate to find the missing frequency (f)
⇒ 18 = (752+20f)/(44+f)
⇒ 18(44+f) = (752+20f)
⇒ 792+18f = 752+20f
⇒ 792+18f = 752+20f
⇒ 792 – 752 = 20f – 18f
⇒ 40 = 2f
⇒ f = 20
So, the missing frequency, f = 20.
4. Thirty women were examined in a hospital by a doctor, and the number of heartbeats per minute was recorded and summarised as follows. Find the mean heartbeats per minute for these women, choosing a suitable method.
Number of heart beats per minute

6568

6871

7174

7477

7780

8083

8386

Number of women

2

4

3

8

7

4

2

Solution:
From the given data, let us assume the mean as A = 75.5
xi = (Upper limit + Lower limit)/2
Class size (h) = 3
Now, find the ui and fiui as follows:
Class Interval

Number of women (fi)

Midpoint (xi)

ui = (xi – 75.5)/h

fiui

6568

2

66.5

3

6

6871

4

69.5

2

8

7174

3

72.5

1

3

7477

8

75.5

0

0

7780

7

78.5

1

7

8083

4

81.5

3

8

8386

2

84.5

3

6


Sum fi= 30



Sum fiui = 4

Mean = x̄ = A + h∑fiui /∑fi
= 75.5 + 3×(4/30)
= 75.5 + 4/10
= 75.5 + 0.4
= 75.9
Therefore, the mean heartbeats per minute for these women is 75.9
5. In a retail market, fruit vendors were selling mangoes kept in packing boxes. These boxes contained a varying number of mangoes. The following was the distribution of mangoes according to the number of boxes.
Number of mangoes

5052

5355

5658

5961

6264

Number of boxes

15

110

135

115

25

Find the mean number of mangoes kept in a packing box. Which method of finding the mean did you choose?
Solution:
Since the given data is not continuous, we add 0.5 to the upper limit and subtract 0.45 from the lower limit as the gap between the two intervals is 1.
Here, assumed mean (A) = 57
Class size (h) = 3
Here, the step deviation method is used because the frequency values are large.
Class Interval

Number of boxes (fi)

Midpoint (xi)

di = xi – A

Fi di

49.552.5

15

51

6

90

52.555.5

110

54

3

330

55.558.5

135

57 = A

0

0

58.561.5

115

60

3

345

61.564.5

25

63

6

150


Sum fi = 400



Sum fidi = 75

The formula to find out the Mean is:
Mean = x̄ = A +h ∑fi di /∑fi
= 57 + 3(75/400)
= 57 + 0.1875
= 57.19
Therefore, the mean number of mangoes kept in a packing box is 57.19.