8th Class Mathematics Statistics Statistics

Statistics

Category : 8th Class

Introduction

Extraction of meaningful information by colleting, organizing, summarizing, presenting and analyzing the data is a branch of mathematics called statistics.

 

PRIMARYDATA

If the data is collected by the investigator herself/himself with the specific purpose, then such data is called the primary data.

 

SECONDARY DATA

If the data collected by someone else other than investigator are known as secondary data.

 

FREQUENCY

It is a number which tells that how many times does a particular observation appear in a given data.

 

FREQUENCY DISTRIBUTION

A tabular arrangement of data sharing their corresponding frequencies is called a frequency distribution.

 

CLASS INTERVAL

The group in which the raw data is condensed is called a Class interval. Each class is bounded by two figures.

 

GROUPED DATA

The data can be represented into classes or groups. Such a presentation is known as grouped data.

Let us observe the marks obtained by 25 students in Mathematics as follows:

56, 31, 41, 64, 53, 56, 64, 31, 88, 53, 28, 33, 70, 70, 61, 74, 74, 64, 56, 32, 53, 53, 56, 61, 53.

We observe that there are few students who get same marks, e.g., 74 marks is obtained by 2 students, 53 is obtained by 5 students etc. Let us represent them in a frequency distribution table given below:

 

Marks

Tally Marks

Frequency

28

\[|\]

1

31

\[||\]

2

32

\[|\]

1

33

\[|\]

1

41

\[|\]

1

53

\[\cancel{||||}\]

5

56

\[||||\]

4

61

\[||\]

2

64

\[|||\]

3

70

\[||\]

2

74

\[||\]

2

88

\[|\]

1

 

 

Total = 25

 

Here, we see that the lowest marks is 28 and the highest marks is 88.

We can further group them into classes as given below:

Classes

Tally Marks

Frequency

25-35  

\[\cancel{||||}\]

5

35-45   

\[|\]

1

45-55  

\[\cancel{||||}\]

5

55-65  

\[\cancel{||||}\]  \[|||\]

9

65-75   

\[||||\]

4

75-85   

Nil

0

85-95   

\[|\]

1

 

 

Total = 25

 

Class Limits: Here, marks obtained by all of the students are divided into seven classes namely, 25-35, 35-45 and so on. In class 25-35, 25 is called Lower class limit and 35 is called Upper class limit.

 

Class Size: The difference between upper and lower class limit is called Class size.

Here, class size is 35 -25 = 45 – 35 = 10.

Class Mark: \[\text{Class mark=}\frac{\text{Upper limit + lower limit}}{\text{2}}\]

 

HISTOGRAM 

Histogram is used for graphical representation of a frequency distribution.  A histogram is a graph that represents the class frequencies in a frequency distribution by vertical adjacent rectangles.

 

EXAMPLE 1:

The population of four major cities in India in a particular year is given below:

City:

Mumbai

Kolkata

Delhi

Chennai

Population (in lakhs)

120

130

150

80

Construct a bar graph to represent the above data.

Sol.

 

EXAMPLE 2:

The frequency distribution of weight (in kg) of 40 persons of a locality is given.

 

Weight (in kg)

40 - 45

45 - 50

50 - 55

55 - 60

60 - 55

Frequency

4

12

13

6

5

 Construct a bar graph to represent the above data.

Sol.

  

 

MEAN

\[\text{Mean=}\frac{\text{Sum of all obser vations}}{\text{Number of obser vations}}\]

If \[{{x}_{1}},{{x}_{2}},{{x}_{3}},....{{x}_{n}},\] are the values of n observations, then the arithmetic mean of these observations is given by

\[\frac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+....+{{x}_{n}}}{n}\]

Mean of frequency distribution

If \[{{x}_{1}},{{x}_{2}},{{x}_{3}},....{{x}_{n}},\] are n values of a variable with corresponding frequencies \[{{f}_{1}},{{f}_{2}},{{f}_{3}},....{{f}_{n}}\]respectively, then the arithmetic mean of these values is defined as

\[Mean=\frac{{{f}_{1}}{{x}_{1}}+{{f}_{2}}{{x}_{2}}+....+{{f}_{n}}{{x}_{n}}}{{{f}_{1}}+{{f}_{2}}+{{f}_{3}}+....{{f}_{n}}}\]

\[Mean=\frac{\sum{{{f}_{i}}{{x}_{i}}}}{\sum{{{f}_{i}}}}\]

or \[=\frac{\sum{{{f}_{i}}{{x}_{i}}}}{\sum{{{f}_{i}}}}\]  or  \[=\frac{\sum{fx}}{\sum{f}}\]

 

EXAMPLES 3:

Find the mean of the following distribution:

x:

4

6

9

10

15

f:

5

10

10

7

8

Sol.        

               

\[{{x}_{i}}\]

\[{{f}_{i}}\]

\[{{f}_{i}}{{x}_{i}}\]

4

5

20

6

10

60

9

10

90

10

7

70

15

8

120

Total

40

360

\[Mean=\frac{\sum{{{f}_{i}}{{x}_{i}}}}{\sum{{{f}_{i}}}}=\frac{360}{40}=9\]

 

RANGE

The difference between the highest and lowest observations of a group of observations is defined as its range.

 

MEDIAN

When the data is arranged in asending or desending order, the middle observation (or variable) is called Median.

Process to find the median:

(i)   Arrange the observations in ascending or descending order of magnitude.

(ii)  Determine the total number of observations (say n).

(iii) If n is odd, then Median \[=\left( \frac{n+1}{2} \right)\]th observation.

(iv) If n is even, then \[\text{Mesian=}\frac{\frac{\text{n}}{\text{2}}\text{th obser vation+}\left( \frac{\text{n}}{\text{2}}\text{+1} \right)\text{th obser vation}}{\text{2}}\]

 

EXAMPLE 4:

Find the median of the following data

24,36,46,17,18,25,35.

Sol.

Arranging the data in ascending order of magnitude, we have

17,18,24,25,35,36,46

n=7(odd)

Median \[=\frac{n+1}{2}th\] observation \[=\frac{8}{2}=4th\] observation = 25.

 

EXAMPLE 5:

Find the median of the following observation

11, 12, 14, 18, x + 2, x + 4, 30,32,35,41,

If median is 24 find the value ofx.

Sol.         Here n == 10 (even)

                \[\text{Median=}\frac{\frac{\text{n}}{\text{2}}\text{th observation+}\left( \frac{\text{n}}{\text{2}}\text{+1} \right)\text{th observation}}{\text{2}}\]

\[\text{24=}\frac{\frac{\text{10}}{\text{2}}\text{th observation+}\left( \frac{\text{10}}{\text{2}}\text{+1} \right)\text{th observation}}{\text{2}}\]

\[\text{24=}\frac{\text{5th observation+6th observation}}{\text{2}}\]

\[24=\frac{(x+2)+(x+4)}{2}\]

\[\Rightarrow \]  \[24=\frac{(x+2)+(x+4)}{2}\]

\[\Rightarrow \]\[x=21\]

 

SIMPLE PIE CHART

This is also a graphical mode of representing data, where frequency is related with the central angle of sector of the circle.

\[\text{Central angle=}\frac{\text{Frequency}}{\text{Total frequency}}\text{ }\!\!\times\!\!\text{ 36}{{\text{0}}^{\text{o}}}\]

EXAMPLE 6:

The numbers of cars produced by a car factory in four years are as follows:

Years:

2002

2003

2004

2005

No. of cars:

2000

2400

900

1900

 

Draw a pie chart to represent it

Sol.         To draw the pie-chart, we prepare the following table:

 

Hence, the required pie chart is drawn alongside.

 

Mode: The mode of a set of observations, is the value which occurs most frequently.

For examples: Find the mode of the given data.

3,4,5,3,3,2,5,3,1,2

Solution: Since, 3 has occured maximum 4 times or 3 has maximum frequency. So mode is 3s.

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