A textbook of Computer Based Numerical and Statiscal Techniques part 48 pdf

11.5 TWO DIMENSIONAL DIAGRAMSIn two dimensional diagrams, the magnitude of given observations are represented by the area of the diagram.. Draw a square diagram to represent the followin

Multiple Bar Diagram

160 140 120 100 80 60 40 20 0

1 2 3 4

Sales ('000 Rs.)

Gross Profit ('000 Rs.)

Net Profit ('000 Rs.)

FIG 11.5

Example 6 Present the following data by a suitable diagram showing the sales and net profits

of private industrial companies.

Sol

60%

50%

40%

30%

20%

10%

0%

–10%

–20%

–30%

1995-1996 1996-1997 1997-1998

Series 1 Series 2

FIG 11.6

11.4 ONE DIMENSIONAL DIAGRAM

In one dimensional diagram magnitude of the observations are represented by only one of the

dimension i.e., height (length) of the bars while the widths of the bars is arbitrary and uniform.

11.5 TWO DIMENSIONAL DIAGRAMS

In two dimensional diagrams, the magnitude of given observations are represented by the area

of the diagram Thus the length as well as width of the bars will have to be considered It is also known as are diagram or surface diagram Some two dimensional diagrams are

(a)Rectangles Diagram: A rectangle is a two dimensional diagram because area of

rectangle is given by the product of its length and widths i.e., length and width of the

bars is taken into consideration

Example 7 Represent the following data on detail of cost of the two commodities by the rectangular diagram.

Sol Let us calculate the cost of material, other expenses and profit per unit

Costs and Profits per unit of Commodity A & B

180

160

140

120

100

80 60 40 20 0

Co m o

ity

Co m o

ity B

Profits

Other expenses

Value of raw material

Items

FIG 11.7

(b)Square Diagram: It is specially useful, if it is desired to compare graphically the values or quantities which differ widely from one another The method of drawing a square diagram is very simple First of all take the square root of the values of the given observations and then squares are drawn with sides proportional to these square roots, on an appropriate scale, which must be satisfied

Example 8 Draw a square diagram to represent the following data.

Sol First find out the square root of the quantities

Ratio of the sides of the square 1 25.4362 1.36

18.7083= 33.4664 1.79

18.7083=

1 Square cm = 350 kg

Square Diagram

Ratio of the sides

of the squares Square root

Yield in (kg) per

FIG 11.8

(c)Circle Diagram: Circle diagrams are alternative to square diagrams and are used for the same purpose The area of circle, which represents the given values, is given πr2,

7

π = and r is the radius of circle That is the area of circle is proportional to

the square of its radius and consequently, in the construction of the circle diagram the radius of circle is a value proportional to the square root of the given magnitude The scale to be used for constructing circle diagrams can be calculated as:

For a given magnitude ‘a’, Area = πr2 square units = a

⇒ 1 square unit = a2

r

π

Example 9 Represent the data of example 8 by a circle diagram.

Sol Above example shows as follows

Scale 1 square cm = 350 2450 111.36 kg

22

π

B C

A

FIG 11.9 (d )Pie diagram: Pie diagram are also called circular diagrams For the construction of pie diagram,

1 Each of the component values expressed by a percentage of the respective total

2 Since the angle at the center of the circle is 360º, the total magnitude of various components is taken to be equal to 360º and each component part is to be expressed proportionally in degrees

3 Since 1 per cent of the total value is equal to 360

100 = 3.6º, the percentage of the

component parts obtained in step 1 can be converted to degrees by multiplying each of them by 3.6

4 Draw a circle of appropriate radius using an appropriate scale depending on the space available

5 The degrees represented by the various component parts of given magnitude can be obtained directly without computing their percentage to the total values

Degree of any component part = component value × 360º

Total value

Example 10 Draw a pie diagram to represent the following data.

Proposed Expenditure (in million Rs.) 4,200 1,500 1,000 500

Sol Following table gives proposed expenditure in angle form

72

72

72

72

A B

C

D

Pie Diagram

FIG 11.10

Three dimensional diagrams are also known as volume diagrams, consists of cubes, cylinders spheres etc length, width and height have to be taken into account Such diagrams are used where the range of difference between the smallest and the largest value is very large Of the various three dimensional diagrams, ‘cubes’ are the smallest and most commonly used devices

of diagrammatic presentation of the data

11.7 PICTOGRAMS

Pictograms is the technique of presenting statistical data through appropriate pictures and is one of very important key particularly when the statistical facts are to be presented to a layman without any mathematical background Pictograms have some limitations also They are difficult

to construct and time consuming Besides, it is necessary to one symbol to represent a fixed number of units, which may create difficulties It gives only an overall picture, not give minute details

11.8 CARTOGRAMS

Cartograms or statistical maps are used to give quantitative information on a geographical basis Cartograms are simple and elementary forms of visual presentation and are easy to understand Normally it is used when the regional or geographical comparisons are to be required to highlight

Graphs is used to study the relationship between the variables Graphs are more obvious, precise and accurate than diagrams and can be effectively used for further statistical analysis, viz., to study slopes, forecasting whenever possible Graphs are drawn on a special type of paper known as graph paper Graph paper has a finite network of horizontal and vertical lines; the thick lines for each division of a centimeter or an inch measure and thin lines for small parts of the same Graphs are classified in two parts

1 Graphs of frequency distribution

2 Graphs of time series

11.9.1 Graphs of Frequency Distribution

The so-called frequency graphs are designed to reveal clearly the characteristic features of a frequency data The most commonly graph for charting a frequency distribution of the data are: (a)Histogram: A frequency density diagram is a histogram According to Opermann, “A histogram is a bar chart or graph showing the frequency of occurrence of each value

of the variable being analyzed” In another way we say that, a histogram is a set of vertical bars whose areas are proportional to the frequencies represented While constructing histogram the variable is always taken on the x-axis and the frequencies

depending on it on the y-axis It applies in general or when class intervals are equal.

In each case the height of the rectangle will be proportional to the frequencies When class intervals are unequal, a correction for unequal class intervals is required For making the correction we take that class which has lowest class interval and adjust the frequencies of other classes If one class interval is twice as wide as the one having lowest class interval we divide the height of its rectangle by two, if it is three times more we divide the height of its rectangle by three and so on

Example 11 Represent the following data by a histogram.

Sol Since the class intervals are equal throughout no adjustment in frequencies are required

70 60 50 40 30 20 10 0

8 12

22

35 40

60 52

40 30

5

Marks

FIG 11.11

Example 12 Represent the following data by a histogram.

Weekly Wages in Rs No of Workers

Sol Since class intervals are unequal, frequencies are required to adjust The adjustment

is done as follows The lowest class interval is 5 therefore the frequencies of class 30–40 shall

be divided by two since the class interval is double, that of 40–60 by 4 etc

30

25

20

15

10

5

0 7

19

27

15

6

FIG 11.12

(b)Frequency Polygon: ‘Polygon’ literally means ‘many-angled’ diagram A frequency polygon is a graph of frequency distribution It is particularly effective in comparing two or more frequency distribution There are two ways for constructing frequency polygon

1 Draw a histogram for a given data and then join by straight lines the midpoints of the upper horizontal sides of each rectangle with the adjacent once The figure so formed is called frequency polygon To close the polygon at both ends of the distribution, extending them to the base line

2 Take midpoints of the various class-intervals and then plot the frequency corresponding

to each point and to join all these points by a straight lines The figure obtained would exactly be the same as obtained by method no 1 The only difference is that here we have not to construct a histogram

Example 13 Draw a frequency polygon from the following data.

Sol Since class intervals are unequal, so we have to adjust the frequencies The class

20-40 would be divided into two parts 20–30 and 30–20-40 with frequency of 7 each class

18

16

14

12

10

8

6

4

2

0

4 6

7 7

14 16

8 8 8

5

FIG 11.13

(c)Frequency Curve: A frequency curve is a smooth free hand curve drawn through the vertices of a frequency polygon The area enclosed by the frequency curve is same as that of the histogram or frequency polygon but its shape is smooth one and not with sharp edges Smoothing should be done very carefully so that the curve looks as regular as possible and sudden and sharp turns should be avoided Though different types of data may give rise to a variety of frequency curves

Symmetrical Curve Asymmetrical Curve

FIG 11.14

1 Symmetrical Curve: In this type of curve, the class frequencies first rise steadily, reach

a maximum and then fall in the same identical manner

2 Asymmetrical (skewed) frequency curves: A frequency curve is said to be skewed if

it is not symmetrical

3 U-Curve: The frequency distributions in which the maximum frequency occurs at the extremes (i.e., both ends) of the range and frequency keeps on falling symmetrically (about the middle), the minimum frequency being attained at the centre, give rise to

a U-shaped curve

U-Shapped Curve J-Shaped Curve Inverted J- Shaped

FIG 11.15

4 J-Shaped Curve: In a J-shaped curve the distribution starts with low frequencies in the lower classes and then frequencies increase steadily as the variable value increases and finally the maximum frequency is attained in the last class Such curves are not regular but become unavoidable in certain situations

(d )Cumulative frequency curve or Ogive: Ogive, pronounced Ojive, is a graphic presentation of the cumulative frequency distribution There are two types of cumulative frequency distributions One is ‘less than’ ogive and second is ‘more than’ ogive The curve obtained by plotting cumulative frequencies (less than or more than) is called a cumulative frequency curve of an ogive

1 ‘Less than’ method: In this method we start with the upper limits of the classes and

go on adding the frequencies When these frequencies are plotted we get a rising curve

2 ‘More than’ method: In this method we start with the lower limits of the classes and from the frequencies we subtract the frequency of each class When these frequencies are plotted we get a declining curve

11.9.2 Graphs of Time-Series

A time series is an arrangement of statistical data in a chronological order i.e., with respect to occurrence of time The time series data are represented geometrically by means of time series graph, which is also known as Historigram The various types of time series graphs are

1 Horizontal line graph or historigrams

2 Net balance graphs

3 Range or variation graphs

4 Components or band graphs

Statistical quality control abbreviated as SQC involves the statistical analysis of the inspection data, which is based on sampling and the principles involved in normal curve The origin of

Statistical Quality Control is only recent Walter A Shewhart and Harold F Dodge of the Bell Laboratories (U.S.A) introduced it after the First World War They used probability theory to developed methods for predicting the quality of the products by conducting tests of the quality

on samples of products turned out from the factory During the Second World War these methods were used for testing war equipment Today the methods of SQC are used widely in production, storage, aircraft, automobile, textile, plastic, petroleum, electrical equipment, telephones, transportation, chemical, medicine and so on In fact, it is impossible to think of any industrial field where statistical techniques are not used Also it has become an integral and permanent part of management controls

The makers of the product normally set the quality standards The quality consciousness amongst producer is always more than there is competition from rival producers Also when consumers are quality conscious The need for quality control arises because of the fact that even after the quality standards have been specified some variation in quality is unavoidable Further, the SQC is only diagnostic It can only indicate whether the standard is being

maintained The re-medical action rests with the technician It is therefore remarked, “Quality control is achieved most efficiently, of course, not by the inspection operation itself, but by getting at

Statistician’s role is there because the analysis is probabilistic There is use of sampling and rules of statistical inference Also SQC refers to the statistical techniques employed for the maintenance of uniform quality in a continous flow of manufactured products

“SQC is an effective system for co-ordinating the quality maintenance and quality improvement efforts of the various graphs in an organization so as to enable production at the most economical levels

Advantages and Uses of SQC: SQC is a very important technique, which is used to assess the causes of variation in the quality of the manufactured product It enables us to determine whether the quality standards are being met without inspecting every unit produced

in the process It primarily aims at the isolation of the chance and assignable causes of variation and consequently helps in the detection, identification and elimination of the assignable causes

of erratic fluctuations whenever they are present

“A production process is said to be in a state of statistical control if it is operating

in the presence of chance causes only and is free from assignable causes of variation” There are some advantages, when a manufacturing process is operating in a state of statistical control

1 The important use and advantage of SQC is the control, maintenance and improvement

in the quality standards

2 Since only a fraction of output is inspected, costs of inspection are greatly reduced

3 SQC have greater efficiency because much of the boredom is avoided, the work of inspection being considerable reduced

4 An excellent feature of quality control is that it is easy to apply One the system is established person who have not had extensive specialized training can operate it

5 It ensures an early detection of faults and hence a minimum waste of rejects production

6 From SQC charts one can easily detach whether or not a change in the production process results in a significant change in quality

7 The diagnosis of the assignable causes of variation gives us an early and timely warningabout the occurrence of defects These are help in reduction in, waste and scrap, cost per unit etc