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

456 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Sol. 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. Year Sales Net Profits 1995–1996 14% 49% 1996–1997 10% –25% 1997–1998 13% –1% 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. STATISTICAL QUALITY CONTROL 457 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. Details Commodity A Commodity B Price per unit Rs. 4 Rs. 5 Quantity sold 40 units 30 units Value of raw material Rs. 52 Rs. 50 Other expenses of production Rs. 64 Rs. 60 Profits Rs. 44 Rs. 40 Sol. Let us calculate the cost of material, other expenses and profit per unit. Commodity A Commodity B 40 units 30 units Items Total (Rs.) Per Unit (Rs.) Total (Rs.) Per Unit (Rs.) Value of raw material 52 1.3 50 1.6 Other expenses of production 64 1.6 60 2.0 Profits 44 1.1 40 1.4 Costs and Profits per unit of Commodity A & B 180 160 140 120 100 80 60 40 20 0 Co m m o d i t y A Co m m o d i t y 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. 458 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Example 8. Draw a square diagram to represent the following data. Country A B C Yield in (kg) per hectare 350 647 1,120 Sol. First find out the square root of the quantities. Country A B C Yield in (kg) 350 647 1,120 Square root 18.7083 25.4362 33.4664 Ratio of the sides of the square 1 25.4362 1.36 18.7083 = 33.4664 1.79 18.7083 = ABC 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 πr 2 , where 22 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 = π r 2 square units = a ⇒ 1 square unit = 2 a rπ 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 STATISTICAL QUALITY CONTROL 459 B C (A) (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 360 º , 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. Items A B C D Proposed Expenditure (in million Rs.) 4,200 1,500 1,000 500 460 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Sol. Following table gives proposed expenditure in angle form Items Proposed Expenditure Angle at the centre A 4,200 ×°=° 42 360 210 72 B 1,500 ×°=° 15 360 75 72 C 1,000 ×°=° 10 360 50 72 D 500 ×°=° 10 360 25 72 Total 7,200 360 ° A B C D Pie Diagram FIG. 11.10 11.6 THREE DIMENSIONAL DIAGRAMS 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. STATISTICAL QUALITY CONTROL 461 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. 11.9 GRAPHIC REPRESENTATION OF DATA 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. Marks No. of Students Marks No. of Students 0–10 8 50–60 60 10–20 12 60–70 52 20–30 22 70–80 40 30–40 35 80–90 30 40–50 40 90–100 05 Sol. Since the class intervals are equal throughout no adjustment in frequencies are required. 462 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES 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 10–15 7 15–20 19 20–25 27 25–30 15 30–40 12 40–60 12 60–80 08 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 3 2 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. STATISTICAL QUALITY CONTROL 463 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. Marks 0–10 10–20 20–40 40–50 50–60 60–70 70–90 90–100 No. of students 4 6 14 16 14 8 16 5 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 30–40 with frequency of 7 each class. 18 16 14 12 10 8 6 4 2 0 4 6 77 14 16 888 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. 11.14 464 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES 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. 11.10 STATISTICAL QUALITY CONTROL 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 465 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 causes”. Dodge and Roming) 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 which allow for a full customer satisfaction”. A.V. Feigenbaum 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. . minute details. STATISTICAL QUALITY CONTROL 461 11.8 CARTOGRAMS Cartograms or statistical maps are used to give quantitative information on a geographical basis. Cartograms are simple and elementary. 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. 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.