Chapter 14

Statistics

Introduction

Class 10 Maths Statistics Notes are written in strict accordance with the NCERT Syllabus and the CBSE norms, reducing the burden on students and providing them with a straightforward method for studying or revising the subject. All statistical concepts, such as grouped and ungrouped data, measures of central tendencies such as mean and median and the relationship between these measures, as well as the important facts and formulae necessary for a proper understanding of the chapter, are covered in this section of the book.

Why do we learn statistics?

  1. Statistics aids in the understanding and correct description of natural events.

  2. Statistics aids in the effective and efficient organisation of statistical investigations in any area of study.

  3. Statistics aids in the collection of relevant quantitative data.

 

In mathematics, the mean, the median, and the mode are the three primary methods of expressing the average value of a collection of integers. When you add up all the numbers in a list, you get the arithmetic mean, which is calculated by dividing the total by the total number of items in the list. This is the most common interpretation of the term "average." The median is the value that falls in the centre of a list that is arranged from smallest to biggest. The mode value is the one that appears the most often on the list.

Mean

The arithmetic mean of a set of data is equal to the sum of all observations divided by the number of observations in the set of data. The mean is equal to the sum of all observations divided by the number of observations.

Median

In general, the median of a collection of data when it is ordered in a certain order reflects the mid-value of the data set.

Mode

A number's mode is defined as the number that appears the most often in the data collection.

As a result of the following empirical connection, the three estimates of central tendency — the mean, the median, and the mode — are connected.

The product of two times mean and one mode is three times median,

i.e., 2 Mean + Mode = 3 Median

Consider the case where it is necessary to calculate the mean, median, and mode of a continuous set of data. The values of the mean and median may be obtained using the equations shown above. The empirical formula helps us determine the value of the mode.

Refer to the Ncert Solutions Class 10 Maths Statistics below for more understanding:

1. A survey was conducted by a group of students as a part of their environment awareness program, in which they collected the following data regarding the number of plants in 20 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 Houses 1 2 1 5 6 2 3

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

Solution:

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.

Midpoint (xi) = (upper limit + lower limit)/2

 

No. of plants

(Class interval)

No. of houses

Frequency (fi)

Mid-point (xi)

fixi

0-2

1

1

1

2-4

2

3

6

4-6

1

5

5

6-8

5

7

35

8-10

6

9

54

10-12

2

11

22

12-14

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.)

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:

Find the midpoint of the given interval using the formula.

Midpoint (xi) = (upper limit + lower limit)/2

In this case, the value of mid-point (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)

Mid-point (xi)

ui = (xi – 150)/20

fiui

100-120

12

110

-2

-24

120-140

14

130

-1

-14

140-160

8

150

0

0

160-180

6

170

1

6

180-200

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)

11-13

13-15

15-17

17-19

19-21

21-23

23-35

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 mid-point (xi) mean x̄ = 18

 

Class interval

Number of children (fi)

Mid-point (xi)

        fixi   

11-13

7

12

84

13-15

6

14

84

15-17

9

16

144

17-19

13

18 = A

234

19-21

f

20

20f

21-23

5

22

110

23-25

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

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:

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)

Mid-point (xi)

ui = (xi – 75.5)/h

fiui

65-68

2

66.5

-3

-6

68-71

4

69.5

-2

-8

71-74

3

72.5

-1

-3

74-77

8

75.5

0

0

77-80

7

78.5

1

7

80-83

4

81.5

3

8

83-86

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

50-52

53-55

56-58

59-61

62-64

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)

Mid-point (xi)

di = xi – A

Fi di

49.5-52.5

15

51

-6

90

52.5-55.5

110

54

-3

-330

55.5-58.5

135

57 = A

0

0

58.5-61.5

115

60

3

345

61.5-64.5

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. 

Statistics play a critical role in observation, analysis, and mathematical prediction models. Weather forecast models are created by comparing previous weather conditions to present weather circumstances in order to anticipate future weather conditions. Statistics is the discipline of gaining knowledge from data. Statistical studies generate reliable findings when their concepts are used effectively.

Furthermore, the studies take into consideration real-world uncertainty to evaluate the likelihood of being inaccurate. Statisticians provide critical insight into whether data, analysis, and conclusions may be trusted. A statistician can steer research through a minefield of possible traps, any of which might lead to incorrect findings. It is, therefore, crucial for you to study statistics class 10 ncert solutions.

 

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