UNIT-1 VARIABLE AND GRAPHS

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BBA1st Statistics Page 1 UNIT-1

VARIABLE AND GRAPHS

VARIABLE:- A variable is a quantity which can vary from one individual to another. variable is a symbol which can take any of a prescribed set of values ,called the domain of the variable. it is usually denoted by x, y, z. A variable is also called a variable.

Example:-barometer reading, temperature, rainfall records, wages, heights etc.

There are two types of variable

<a>continuous

<b>discrete or discontinuous variable

<a>continuous variable:-the quantity which can not assume any numerical value within a certain range are called continuous variable. Ex. Heights

<b>discrete or discontinuous variable:- the quantity which can not assume all possible values are called discrete variable . in other words a discrete variable can assume only integral values . ex. Number of student in class, no. Of employees in industry.

Constant:- a variable is called a constant if it can assume only one value.

Meaning of statistics :- the word statistics derived from the latin word “status” or the Italian word “statistica” both have a same meaning a political state. Statistic may be defined as the scientific X position of the statical method.

According to “A.L.Bowley” defined statistics may be called as the science of counting or the science of average.

Its scope is streshed overall those branches of human knowledge in which a grasp of the significance of large number is looked for.

The biologist, the customers, the administrator, businessman & the economist al make use of the statistical method & facts. it’s scope has become so wide today .few statisticiansar4e expert in all branches.

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BBA1st Statistics Page 2

Limitations:- statistical law is not a exact law, like mathematical chemical law. They are derive by taking a majority of cases are not true for every individuals.

<1> statistical method deals with the population of individuals or than with individuals.

<2>this technique apply only those data which are reducible to quantitative from consequently the characteristics which can not be majored in figures and can not be studied satisfactorily such characteristics are beauty, goodness, health, intelligence, honest etc.

<3>statistical technique is the same for the social as for the physical science while both are different in nature.

<4>only one who has in expert knowledge of statistical method can handle statistical data property.

FREQUENCY DISRIBUTION

Frequency dis:- a frequency distribution is defined when the following two information are specified.

<1>the values which the variable takes

<2>the number of items that each variable value is taken

If we are collect some values of a variable in a orbitary in order than it is difficult for one-two grasp the significance of data.

Ex. consider the marks obtained by students in a statistic paper which are arranged according to there roll number.

17,71,70,14,22,0,55,59,23,87,93,21,22,50,87,54,70,52,87,74,63,87,28,4,17,49,85,81,76,50,21,8 6,50,87,26,87,50,60,32,40,80,90,27,89,21,14,30,37,22.

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This data can be arranged in assending or desending order of magnitude. this data are arranged in the intervals0-10,10-20,20-30,30-40,40-50,50-60,60-70,70-80,80-90,90-100. This is done by the method. in this method we construct a frequency table as follows.

Marks class intervals No.of student tally marks

frequency Cumulative frequency

0-10 II 2 2

10-20 IIII 4 6

20-30 IIII IIII 10 16

30-40 IIII 4 20

40-50 II 2 22

50-60 IIII IIII 8 30

60-70 II 2 32

70-80 IIII 5 37

80-90 IIII IIII I 11 48

90-100 II 2 50

We consider every observation & draw a vertical line segment called tally marks for each entry in the suitable class.

When four variable vertical line segment are draw in a particular class than we cross it for the fifth entry &then next living a little space on the right. this type of representation of frequency is called grouped frequency distribution or simply frequency distribution.

HISTOGRAM

Histogram:-histogram is a diagram in which we draw rectangle with areas personal to the frequency bases as class interval for a given grouped frequency distribution.

To draw a histogram of given grouped frequency distribution . we first choose a suitable scale & then with this scale we marks of along the x axis all the class intervals now with the class intervals now with the class interval as bases draw rectangle with the area proportional frequency of the respective class intervals.

If the class intervals are equal the height of the rectangle are proportional to the ratio of the frequency to the width of the corresponding classes.

Q.1)draw a histogram for the following data. marks (class intervals)

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Sol.- Here the class interval are equal taking the scale 1c.m.=10 marks an x axis & 1c.m.=1 frequency as y axis than the histogram is.

Marks(class intervals) No. Of student

0-10 2

10-20 4

20-30 10

30-40 4

40-50 3

50-60 8

60-70 1

70-80 5

80-90 11

90-100 2

Here the class intervals are same taking the scale 1c.m.=10 marks on x axis ,

1c.m.=1 frequency on y axis

Class interval

0 2 4 6 8 10 12

0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100

series1

10 20 30 40 50 60 70 80 90 100

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Q.2)draw a histogram for following frequency distribution.

Age (in years) No.of persons

1-10 5

10-20 15

20-30 20

30-40 25

40-50 40

50-70 30

70-100 15

Sol-here the class intervals are unequal take the scale 1c.m.=110 years as x axis and 1c.m.=5 persons as y axis now we construct a first rectangle with the class 0-10 as base & the height equal to 5 person. A second rectangle with the class 10-20 as base and the height =15 person similarly construct 3^rd,4^th and 5^th rectangle4 than construct 6^th rectangle with the class 50-70 as base of the height =1/2*30=15 person proceeding in this way we construct 7 rectangle then construct 28^th rectangle width the class 70-100 as base of the height .

H 𝜶 ^{𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚}

𝒘𝒊𝒅𝒕𝒉 = ^𝟏𝟓

𝟑=5 (with the class intervals)

Age (class intervals)

0 5 10 15 20 25 30 35 40 45

0 10 20 30 40 50 60 70 80 90 100

Column1

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Frequency polygon

Freq. Polygon:-for equal class interval the frequency polygon is obtained by joining the middle point of upper sides of the rectangle of the histogram by straight lines taken in order.

Thus to draw a frequency polygon the central values of the class –intervals are plotted as x co-ordinates & the frequency of the respective classes as y-coordinate then this points are join by straight lines taken in order7 thus fdrequency polygon is obtained.

Q.3) draw a fre. Polygon for the following data?

Class interval frequency

0-10 2

10-20 4

20-30 10

30-40 4

40-50 3

50-60 8

60-70 1

70-80 5

80-90 11

90-100 2

Sol:-

Class intervals frequency Middle points

0-10 2 5

10-20 4 15

20-30 10 25

30-40 4 35

40-50 3 45

50-60 8 55

60-70 1 65

70-80 5 75

80-90 11 85

90-100 2 95

Middle point = 𝒔𝒖𝒎 𝒄𝒍𝒂𝒔𝒔 𝒊𝒏𝒕𝒆𝒓𝒗𝒂𝒍𝒔

𝟐 = ^{𝟎+𝟏𝟎}

𝟐 = ^𝟏𝟎

𝟐 = 5

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Frequency polygon

0 2 4 6 8 10 12

10 20 30 40 50 60 70 80 90 100

Column1

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FREQUENCY CURVE

Freq.curve :-it through the frequency curve is obtained by drawing a smooth free hand curve through the vertices of the frequency polygon .the frequency curve is usually drown in the case when the class interval are of small width.

Q.4) draw a fre. curve for the following data?

0-10 2

10-20 4

20-30 10

30-40 4

40-50 3

50-60 8

60-70 1

70-80 5

80-90 11

90-100 2

Sol:-

0-10 2 5

10-20 4 15

20-30 10 25

30-40 4 35

40-50 3 45

50-60 8 55

60-70 1 65

70-80 5 75

80-90 11 85

90-100 2 95

𝟐 = ^{𝟎+𝟏𝟎}

𝟐 = ^𝟏𝟎

𝟐 = 5

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Frequency curve

0 2 4 6 8 10 12

10 20 30 40 50 60 70 80 90 100

Column1

10 20 30 40 50 60 70 80 90 100

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COMULATIVE FREQUENCY DISTRIBUTION

Cumulative frequency distribution:- We frame the cumulative frequency table & then plot the point with the upper limit of the class as x coordinates & the corresponding cumulative

frequency as the y coordinate then join this points by straight lines . The figure so framed is called less than type cumulative frequency polygon. if we plot more than type of cumulative against the corresponding lower limits of each class & the points plotted are joined by straight lines . The figure to framed is called more than type cumulative freq. polygon.

Q.5) draw a frequency Polygon for the following data?

0-10 2

10-20 4

20-30 10

30-40 4

40-50 3

50-60 8

60-70 1

70-80 5

80-90 11

90-100 2

Sol:-

0-10 2 5

10-20 4 15

20-30 10 25

30-40 4 35

40-50 3 45

50-60 8 55

60-70 1 65

70-80 5 75

80-90 11 85

90-100 2 95

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𝟐 = ^{𝟎+𝟏𝟎}

𝟐 = ^𝟏𝟎

𝟐 = 5

Q.6)draw a histogram fre. polygon & fre. curve of the following frequency distribution class interval frequency.

0-10 2

10-20 6

20-30 10

30-40 17

40-50 30

50-60 10

60-70 14

70-80 7

80-90 4

90-100 1

0 2 4 6 8 10 12

10 20 30 40 50 60 70 80 90 100

Column1

cumulative frequency

10 20 30 40 50 60 70 80 90 100

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Sol:- histogram

0 5 10 15 20 25 30 35

10 20 30 40 50 60 70 80 90 100

frequency

10 20 30 40 50 60 70 80 90 100

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Q.7) Draw a less than type cumulative fre. polygon for the following fre. distribution?

0-10 2

10-20 4

20-30 10

30-40 4

40-50 3

50-60 8

60-70 1

70-80 5

80-90 11

90-100 2

Class interval frequency Upper limit

0-10 2 10 2

10-20 4 20 6

20-30 10 30 16

30-40 4 40 20

40-50 3 50 23

50-60 8 60 31

60-70 1 70 32

70-80 5 80 37

80-90 11 90 48

90-100 2 100 50

(addition from top)(upper limit) Less than=upper limit More than = lower limit

X- 1c.m. = 10 Y- 1c.m. = 5

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Y 50 45

40 Courve 35

30 25 20 15 10 5

0 10 20 30 40 50 60 70 80 90 100

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POPULATION

Any collection of individuals is said to be population or universe. the individuals often called the members or the units of population. They may be a physical objects or majuarment expressed numerically or otherwise .there are following 4 types of population are:-

1>Hypothetical population:-the population of concrete object is called an existence population while a hypothetical population may be defined as the collection of all possible ways in which a specified event can happen ex: the population heds & tells obtained by tossing a coin and

infinite no.of times is a hypothetical population.

2>real population:-the population of concrete individuals is called real population.ex: no.of student in a college.

3>finite population:- a population containing a finite no. of individuals is called a finite population. ex: the student in a college.

4>infinite population:- a population containing an infinite no. of individuals is called infinite population. ex: particles in a cement.

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SAMPLE

Sample:-a part of a small section selected from the population is called a sample. In the process of such selection is called sampling.

Types of sampling:-

1>random sampling:-the selection of an individual from a parent population is random when each members of population has the same chance of being choosen.

2>purposive population:-it is a random sampling in which the sample units are selected with the defined purpose in view.

3>simple sampling:-it is random sampling in which each units of the population has equal chance of being included in the sample and the probability is independent of the previous drawn.

4>satisfied sampling:-the sampling in which the aggregate of the sample unit of each of the stratum is called satisfied sample and this technique is called satisfied sampling.

Extras-

Equation:-an equation is statement of the form u=v where u is called the left hand side & v is the right hand side of the equation. Solution of equation:-the value of unknown x, which satisfied both side of the equation.

Logrithm:-if n is no. Such that 10^𝑥=n then x is called the log. Of n to the base 10 & is written as 𝑙𝑜𝑔₁₀𝑛 = 𝑥 𝑜𝑟 𝑙𝑜𝑔𝑛 = 𝑥 1^st part is decimal part 2^nd is integral part.

Graph:- a graph is pictorial representation of the relationship b/w variables. the graph called diagram or charts.