SECTION D - QUANTITATIVE APTITUDE - CHAPTER 10

(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:8) (cid:10)(cid:11) (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:1)(cid:2)(cid:4)(cid:5)(cid:3)(cid:6) (cid:7)(cid:8)(cid:1)(cid:5)(cid:9)(cid:4)(cid:10)(cid:2)(cid:4)(cid:11)(cid:12) (cid:11)(cid:13)(cid:14)(cid:7)(cid:3)(cid:2)(cid:3) Copyright -The Institute of Chartered Accountants of India (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:1)(cid:2)(cid:4)(cid:5)(cid:3)(cid:6)(cid:7)(cid:8)(cid:9)(cid:1)(cid:5)(cid:10)(cid:4)(cid:11)(cid:2)(cid:4)(cid:12)(cid:13)(cid:7)(cid:12)(cid:14)(cid:7)(cid:8)(cid:3)(cid:2)(cid:3) LLLLLEEEEEAAAAARRRRRNNNNNIIIIINNNNNGGGGG OOOOOBBBBBJJJJJEEEEECCCCCTTTTTIIIIIVVVVVEEEEESSSSS After going through this chapter the students will be able to (cid:1) Have a broad overview of the subject of statistics and application thereof; (cid:1) Know about data collection technique including the distinction of primary and secondary data. (cid:1) Know how to present data in textual and tabular format including the technique of creating frequency distribution and working out cumulative frequency; (cid:1) Know how to present data graphically using histogram, frequency polygon and pie chart. 1111100000.....11111 IIIIINNNNNTTTTTRRRRROOOOODDDDDUUUUUCCCCCTTTTTIIIIIOOOOONNNNN OOOOOFFFFF SSSSSTTTTTAAAAATTTTTIIIIISSSSSTTTTTIIIIICCCCCSSSSS The modern development in the field of not only Management, Commerce, Economics, Social Sciences, Mathematics and so on but also in our life like public services, defence, banking, insurance sector, tourism and hospitality, police and military etc. are dependent on a particular subject known as statistics. Statistics does play a vital role in enriching a specific domain by collecting data in that field, analysing the data by applying various statistical techniques and finally making statistical inferences about the domain. In the present world, statistics has almost a universal application. Our Government applies statistics to make the economic planning in an effective and a pragmatic way. The businessman plan and expand their horizons of business on the basis of the analysis of the feedback data. The political parties try to impress the general public by presenting the statistics of their performances and accomplishments. Most of the research scholars of today also apply statistics to present their research papers in an authoritative manner. Thus the list of people using statistics goes on and on and on. Due to these factors, it is necessary to study the subject of statistics in an objective manner. HHHHHiiiiissssstttttooooorrrrryyyyy ooooofffff SSSSStttttaaaaatttttiiiiissssstttttiiiiicccccsssss Going through the history of ancient period and also that of medieval period, we do find the mention of statistics in many countries. However, there remains a question mark about the origin of the word ‘statistics’. One view is that statistics is originated from the Latin word ‘ status’. According to another school of thought, it had its origin in the Italian word ‘statista’. Some scholars believe that the German word ‘statistik’ was later changed to statistics and another suggestion is that the French word ‘statistique’ was made as statistics with the passage of time. In those days, statistics was analogous to state or, to be more precise, the data that are collected and maintained for the welfare of the people belonging to the state. We are thankful to Kautilya who had kept a record of births and deaths as well as some other precious records in his famous book ‘Arthashastra’ during Chandragupta’s reign in the fourth century B.C. During the reign of Akbar in the sixteenth century A.D. we find statistical records on agriculture in Ain-i-Akbari written by Abu Fazl. Referring to Egypt, the first census was conducted by the Pharaoh during 300 B.C. to 2000 B.C. DDDDDeeeeefffffiiiiinnnnniiiiitttttiiiiiooooonnnnn ooooofffff SSSSStttttaaaaatttttiiiiissssstttttiiiiicccccsssss We may define statistics either in a singular sense or in a plural sense Statistics, when used as a plural noun, may be defined as data qualitative as well as quantitative, that are collected, usually with a view of having statistical analysis. However, statistics, when used as a singular noun, may be defined, as the scientific method that is employed for collecting, analysing and presenting data, leading finally to drawing statistical inferences about some important characteristics it means it is ‘science of counting’ or ‘science of averages’. (cid:10)(cid:11)(cid:12)(cid:13) (cid:1)(cid:14)(cid:15)(cid:15)(cid:14)(cid:16)(cid:8) (cid:4)(cid:7)(cid:14)(cid:17)(cid:18)(cid:1)(cid:18)(cid:6)(cid:16)(cid:1)(cid:19)(cid:8) (cid:5)(cid:6)(cid:20)(cid:5) Copyright -The Institute of Chartered Accountants of India AAAAAppppppppppllllliiiiicccccaaaaatttttiiiiiooooonnnnn ooooofffff ssssstttttaaaaatttttiiiiissssstttttiiiiicccccsssss Among various applications of statistics, let us confine our discussions to the fields of Economics, Business Management and Commerce and Industry. EEEEEcccccooooonnnnnooooommmmmiiiiicccccsssss Modern developments in Economics have the root in statistics. In fact, Economics and Statistics are closely associated. Time Series Analysis , Index Numbers, Demand Analysis etc. are some overlapping areas of Economics and statistics. In this connection, we may also mention Econometrics – a branch of Economics that interact with statistics in a very positive way. Conducting socio-economic surveys and analysing the data derived from it are made with the help of different statistical methods. Regression analysis, one of the numerous applications of statistics, plays a key role in Economics for making future projection of demand of goods, sales, prices, quantities etc. which are all ingredients of Economic planning. BBBBBuuuuusssssiiiiinnnnneeeeessssssssss MMMMMaaaaannnnnaaaaagggggeeeeemmmmmeeeeennnnnttttt Gone are the days when the managers used to make decisions on the basis of hunches, intuition or trials and errors. Now a days, because of the never-ending complexity in the business and industry environment, most of the decision making processes rely upon different quantitative techniques which could be described as a combination of statistical methods and operations research techniques. So far as statistics is concerned, inferences about the universe from the knowledge of a part of it, known as sample, plays an important role in the development of certain criteria. Statistical decision theory is another component of statistics that focuses on the analysis of complicated business strategies with a list of alternatives – their merits as well as demerits. SSSSStttttaaaaatttttiiiiissssstttttiiiiicccccsssss iiiiinnnnn CCCCCooooommmmmmmmmmeeeeerrrrrccccceeeee aaaaannnnnddddd IIIIInnnnnddddduuuuussssstttttrrrrryyyyy In this age of cut-throat competition, like the modern managers, the industrialists and the businessmen are expanding their horizons of industries and businesses with the help of statistical procedures. Data on previous sales, raw materials, wages and salaries, products of identical nature of other factories etc are collected, analysed and experts are consulted in order to maximise profits. Measures of central tendency and dispersion, correlation and regression analysis, time series analysis, index numbers, sampling, statistical quality control are some of the statistical methods employed in commerce and industry. LLLLLiiiiimmmmmiiiiitttttaaaaatttttiiiiiooooonnnnnsssss ooooofffff SSSSStttttaaaaatttttiiiiissssstttttiiiiicccccsssss Before applying statistical methods, we must be aware of the following limitations: I Statistics deals with the aggregates. An individual, to a statistician has no significance except the fact that it is a part of the aggregate. II Statistics is concerned with quantitative data. However, qualitative data also can be converted to quantitative data by providing a numerical description to the corresponding qualitative data. III Future projections of sales, production, price and quantity etc. are possible under a specific set of conditions. If any of these conditions is violated, projections are likely to be inaccurate. (cid:20)(cid:5)(cid:3)(cid:5)(cid:18)(cid:20)(cid:5)(cid:18)(cid:1)(cid:20) (cid:10)(cid:11)(cid:12)(cid:21) Copyright -The Institute of Chartered Accountants of India (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:1)(cid:2)(cid:4)(cid:5)(cid:3)(cid:6)(cid:7)(cid:8)(cid:9)(cid:1)(cid:5)(cid:10)(cid:4)(cid:11)(cid:2)(cid:4)(cid:12)(cid:13)(cid:7)(cid:12)(cid:14)(cid:7)(cid:8)(cid:3)(cid:2)(cid:3) IV The theory of statistical inferences is built upon random sampling. If the rules for random sampling are not strictly adhered to, the conclusion drawn on the basis of these unrepresentative samples would be erroneous. In other words, the experts should be consulted before deciding the sampling scheme. 1111100000.....22222 CCCCCOOOOOLLLLLLLLLLEEEEECCCCCTTTTTIIIIIOOOOONNNNN OOOOOFFFFF DDDDDAAAAATTTTTAAAAA We may define ‘data’ as quantitative information about some particular characteristic(s) under consideration. Although a distinction can be made between a qualitative characteristic and a quantitative characteristic but so far as the statistical analysis of the characteristic is concerned, we need to convert qualitative information to quantitative information by providing a numeric descriptions to the given characteristic. In this connection, we may note that a quantitative characteristic is known as a variable or in other words, a variable is a measurable quantity. Again, a variable may be either discrete or continuous. When a variable assumes a finite or a countably infinite number of isolated values, it is known as a discrete variable. Examples of discrete variables may be found in the number of petals in a flower, the number of misprints a book contains, the number of road accidents in a particular locality and so on. A variable, on the other hand, is known to be continuous if it can assume any value from a given interval. Examples of continuous variables may be provided by height, weight, sale, profit and so on. Finally, a qualitative characteristic is known as an attribute. The gender of a baby, the nationality of a person, the colour of a flower etc. are examples of attributes. We can broadly classify data as (a) Primary; (b) Secondary. Collection of data plays the very important role for any statistical analysis. The data which are collected for the first time by an investigator or agency are known as primary data whereas the data are known to be secondary if the data, as being already collected, are used by a different person or agency. Thus, if Prof. Das collects the data on the height of every student in his class, then these would be primary data for him. If, however, another person, say, Professor Bhargava uses the data, as collected by Prof. Das, for finding the average height of the students belonging to that class, then the data would be secondary for Prof. Bhargava. CCCCCooooolllllllllleeeeeccccctttttiiiiiooooonnnnn ooooofffff PPPPPrrrrriiiiimmmmmaaaaarrrrryyyyy DDDDDaaaaatttttaaaaa The following methods are employed for the collection of primary data: (i) Interview method; (ii) Mailed questionnaire method; (iii) Observation method. (iv) Questionnaries filled and sent by enumerators. Interview method again could be divided into (a) Personal Interview method, (b) Indirect Interview method and (c) Telephone Interview method. In personal interview method, the investigator meets the respondents directly and collects the required information then and there from them. In case of a natural calamity like a super (cid:10)(cid:11)(cid:12)(cid:22) (cid:1)(cid:14)(cid:15)(cid:15)(cid:14)(cid:16)(cid:8) (cid:4)(cid:7)(cid:14)(cid:17)(cid:18)(cid:1)(cid:18)(cid:6)(cid:16)(cid:1)(cid:19)(cid:8) (cid:5)(cid:6)(cid:20)(cid:5) Copyright -The Institute of Chartered Accountants of India cyclone or an earthquake or an epidemic like plague, we may collect the necessary data much more quickly and accurately by applying this method. If there are some practical problems in reaching the respondents directly, as in the case of a rail accident, then we may take recourse for conducting Indirect Interview where the investigator collects the necessary information from the persons associated with the problems. Telephone interview method is a quick and rather non-expensive way to collect the primary data where the relevant information can be gathered by the researcher himself by contacting the interviewee over the phone. The first two methods, though more accurate, are inapplicable for covering a large area whereas the telephone interview, though less consistent, has a wide coverage. The amount of non-responses is maximum for this third method of data collection. Mailed questionnaire method comprises of framing a well-drafted and soundly-sequenced questionnaire covering all the important aspects of the problem under consideration and sending them to the respondents with pre-paid stamp after providing all the necessary guidelines for filling up the questionnaire. Although a wide area can be covered using the mailed questionnaire method, the amount of non-responses is likely to be maximum in this method. In observation method, data are collected, as in the case of obtaining the data on the height and weight of a group of students, by direct observation or using instrument. Although this is likely to be the best method for data collection, it is time consuming, laborious and covers only a small area. Questionnaire form of data collection is used for larger enquiries from the persons who are surveyed. Enumerators collects information directly by interviewing the persons having information : Question are explained and hence data is collected. SSSSSooooouuuuurrrrrccccceeeeesssss ooooofffff SSSSSeeeeecccccooooonnnnndddddaaaaarrrrryyyyy DDDDDaaaaatttttaaaaa There are many sources of getting secondary data. Some important sources are listed below: (a) International sources like WHO, ILO, IMF, World Bank etc. (b) Government sources like Statistical Abstract by CSO, Indian Agricultural Statistics by the Ministry of Food and Agriculture and so on. (c) Private and quasi-government sources like ISI, ICAR, NCERT etc. (d) Unpublished sources of various research institutes, researchers etc. SSSSScccccrrrrruuuuutttttiiiiinnnnnyyyyy ooooofffff DDDDDaaaaatttttaaaaa Since the statistical analyses are made only on the basis of data, it is necessary to check whether the data under consideration are accurate as well as consistence. No hard and fast rules can be recommended for the scrutiny of data. One must apply his intelligence, patience and experience while scrutinising the given information. Errors in data may creep in while writing or copying the answer on the part of the enumerator. A keen observer can easily detect that type of error. Again, there may be two or more series of figures which are in some way or other related to each other. If the data for all the series are provided, they may be checked for internal consistency. As an example, if the data for population, area and density for some places are given, then we may verify whether they are internally consistent by examining whether the relation (cid:20)(cid:5)(cid:3)(cid:5)(cid:18)(cid:20)(cid:5)(cid:18)(cid:1)(cid:20) (cid:10)(cid:11)(cid:12)(cid:23) Copyright -The Institute of Chartered Accountants of India (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:1)(cid:2)(cid:4)(cid:5)(cid:3)(cid:6)(cid:7)(cid:8)(cid:9)(cid:1)(cid:5)(cid:10)(cid:4)(cid:11)(cid:2)(cid:4)(cid:12)(cid:13)(cid:7)(cid:12)(cid:14)(cid:7)(cid:8)(cid:3)(cid:2)(cid:3) Area Density= holds. Population A good statistician can also detect whether the returns submitted by some enumerators are exactly of the same type thereby implying the lack of seriousness on the part of the enumerators. The bias of the enumerator also may be reflected by the returns submitted by him. This type of error can be rectified by asking the enumerator(s) to collect the data for the disputed cases once again. 1111100000.....33333 PPPPPRRRRREEEEESSSSSEEEEENNNNNTTTTTAAAAATTTTTIIIIIOOOOONNNNN OOOOOFFFFF DDDDDAAAAATTTTTAAAAA Once the data are collected and verified for their homogeneity and consistency, we need to present them in a neat and condensed form highlighting the essential features of the data. Any statistical analysis is dependent on a proper presentation of the data under consideration. CCCCClllllaaaaassssssssssiiiiifffffiiiiicccccaaaaatttttiiiiiooooonnnnn ooooorrrrr OOOOOrrrrrgggggaaaaannnnniiiiisssssaaaaatttttiiiiiooooonnnnn ooooofffff DDDDDaaaaatttttaaaaa It may be defined as the process of arranging data on the basis of the characteristic under consideration into a number of groups or classes according to the similarities of the observations. Following are the objectives of classification of data: (a) It puts the data in a neat, precise and condensed form so that it is easily understood and interpreted. (b) It makes comparison possible between various characteristics, if necessary, and thereby finding the association or the lack of it between them. (c) Statistical analysis is possible only for the classified data. (d) It eliminates unnecessary details and makes data more readily understandable. Data may be classified as - (i) Chronological or Temporal or Time Series Data; (ii) Geographical or Spatial Series Data; (iii) Qualitative or Ordinal Data; (iv) Quantitative or Cardinal Data. When the data are classified in respect of successive time points or intervals, they are known as time series data. The number of students appeared for CA final for the last twenty years, the production of a factory per month from 1990 to 2005 etc. are examples of time series data. Data arranged region wise are known as geographical data. If we arrange the students appeared for CA final in the year 2005 in accordance with different states, then we come across Geographical Data. Data classified in respect of an attribute are referred to as qualitative data. Data on nationality, gender, smoking habit of a group of individuals are examples of qualitative data. Lastly, when the data are classified in respect of a variable, say height, weight, profits, salaries etc., they are known as quantitative data. (cid:10)(cid:11)(cid:12)(cid:24) (cid:1)(cid:14)(cid:15)(cid:15)(cid:14)(cid:16)(cid:8) (cid:4)(cid:7)(cid:14)(cid:17)(cid:18)(cid:1)(cid:18)(cid:6)(cid:16)(cid:1)(cid:19)(cid:8) (cid:5)(cid:6)(cid:20)(cid:5) Copyright -The Institute of Chartered Accountants of India Data may be further classified as frequency data and non-frequency data. The qualitative as well as quantitative data belong to the frequency group whereas time series data and geographical data belong to the non-frequency group. MMMMMooooodddddeeeee ooooofffff PPPPPrrrrreeeeessssseeeeennnnntttttaaaaatttttiiiiiooooonnnnn ooooofffff DDDDDaaaaatttttaaaaa Next, we consider the following mode of presentation of data: (a) Textual presentation; (b) Tabular presentation or Tabulation; (c) Diagrammatic representation. (((((aaaaa))))) TTTTTeeeeexxxxxtttttuuuuuaaaaalllll ppppprrrrreeeeessssseeeeennnnntttttaaaaatttttiiiiiooooonnnnn This method comprises presenting data with the help of a paragraph or a number of paragraphs. The official report of an enquiry commission is usually made by textual presentation. Following is an example of textual presentation. ‘In 1999, out of a total of five thousand workers of Roy Enamel Factory, four thousand and two hundred were members of a Trade Union. The number of female workers was twenty per cent of the total workers out of which thirty per cent were members of the Trade Union. In 2000, the number of workers belonging to the trade union was increased by twenty per cent as compared to 1999 of which four thousand and two hundred were male. The number of workers not belonging to trade union was nine hundred and fifty of which four hundred and fifty were females.’ The merit of this mode of presentation lies in its simplicity and even a layman can present data by this method. The observations with exact magnitude can be presented with the help of textual presentation. Furthermore, this type of presentation can be taken as the first step towards the other methods of presentation. Textual presentation, however, is not preferred by a statistician simply because, it is dull, monotonous and comparison between different observations is not possible in this method. For manifold classification, this method cannot be recommended. (((((bbbbb))))) TTTTTaaaaabbbbbuuuuulllllaaaaarrrrr ppppprrrrreeeeessssseeeeennnnntttttaaaaatttttiiiiiooooonnnnn ooooorrrrr TTTTTaaaaabbbbbuuuuulllllaaaaatttttiiiiiooooonnnnn Tabulation may be defined as systematic presentation of data with the help of a statistical table having a number of rows and columns and complete with reference number, title, description of rows as well as columns and foot notes, if any. We may consider the following guidelines for tabulation : I A statistical table should be allotted a serial number along with a self-explanatory title. II The table under consideration should be divided into caption, Box-head, Stub and Body. Caption is the upper part of the table, describing the columns and sub-columns, if any. The Box-head is the entire upper part of the table which includes columns and sub-column numbers, unit(s) of measurement along with caption. Stub is the left part of the table providing the description of the rows. The body is the main part of the table that contains the numerical figures. (cid:20)(cid:5)(cid:3)(cid:5)(cid:18)(cid:20)(cid:5)(cid:18)(cid:1)(cid:20) (cid:10)(cid:11)(cid:12)(cid:25) Copyright -The Institute of Chartered Accountants of India (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:1)(cid:2)(cid:4)(cid:5)(cid:3)(cid:6)(cid:7)(cid:8)(cid:9)(cid:1)(cid:5)(cid:10)(cid:4)(cid:11)(cid:2)(cid:4)(cid:12)(cid:13)(cid:7)(cid:12)(cid:14)(cid:7)(cid:8)(cid:3)(cid:2)(cid:3) III The table should be well-balanced in length and breadth. IV The data must be arranged in a table in such a way that comparison(s) between different figures are made possible without much labour and time. Also the row totals, column totals, the units of measurement must be shown. V The data should be arranged intelligently in a well-balanced sequence and the presentation of data in the table should be appealing to the eyes as far as practicable. VI Notes describing the source of the data and bringing clarity and, if necessary, about any rows or columns known as footnotes, should be shown at the bottom part of the table. The textual presentation of data, relating to the workers of Roy Enamel Factory is shown in the following table. TTTTTaaaaabbbbbllllleeeee 1111100000.....11111 Status of the workers of Roy Enamel factory on the basis of their trade union membership for 1999 and 2000. Status Member of TU Non-member Total M F T M F T M F T Year (1) (2) (3)=(1)+ (2) (4) (5) (6)=(4)+ (5) (7) (8) (9)=(7)+ (8) 1999 3900 300 4200 300 500 800 4200 800 5000 2000 4200 840 5040 500 450 950 4700 1290 5990 SSSSSooooouuuuurrrrrccccceeeee ::::: FFFFFooooooooootttttnnnnnooooottttteeeee ::::: TU, M, F and T stand for trade union, male, female and total respectively. The tabulation method is usually preferred to textual presentation as (i) It facilitates comparison between rows and columns. (ii) Complicated data can also be represented using tabulation. (iii) It is a must for diagrammatic representation. (iv) Without tabulation, statistical analysis of data is not possible. (((((ccccc))))) DDDDDiiiiiaaaaagggggrrrrraaaaammmmmmmmmmaaaaatttttiiiiiccccc rrrrreeeeeppppprrrrreeeeessssseeeeennnnntttttaaaaatttttiiiiiooooonnnnn ooooofffff dddddaaaaatttttaaaaa Another alternative and attractive representation of statistical data is provided by charts, diagrams and pictures. Unlike the first two methods of representation of data, diagrammatic representation can be used for both the educated section and uneducated section of the society. Furthermore, any hidden trend present in the given data can be noticed only in this mode of representation. However, compared to tabulation, this is less accurate. So if there is a priority for accuracy, we have to recommend tabulation. (cid:10)(cid:11)(cid:12)(cid:26) (cid:1)(cid:14)(cid:15)(cid:15)(cid:14)(cid:16)(cid:8) (cid:4)(cid:7)(cid:14)(cid:17)(cid:18)(cid:1)(cid:18)(cid:6)(cid:16)(cid:1)(cid:19)(cid:8) (cid:5)(cid:6)(cid:20)(cid:5) Copyright -The Institute of Chartered Accountants of India We are going to consider the following types of diagrams : I Line diagram or Historiagram; II Bar diagram; III Pie chart. I LLLLLiiiiinnnnneeeee dddddiiiiiaaaaagggggrrrrraaaaammmmm ooooorrrrr HHHHHiiiiissssstttttooooorrrrriiiiiaaaaagggggrrrrraaaaammmmm When the data vary over time, we take recourse to line diagram. In a simple line diagram, we plot each pair of values of (t, y), y representing the time series at the time point t in the t t t–y plane. The plotted points are then joined successively by line segments and the resulting t chart is known as line-diagram. When the time series exhibit a wide range of fluctuations, we may think of logarithmic or ratio chart where Log y and not y is plotted against t. We use Multiple line chart for t t representing two or more related time series data expressed in the same unit and multiple – axis chart in somewhat similar situations if the variables are expressed in different units. II BBBBBaaaaarrrrr dddddiiiiiaaaaagggggrrrrraaaaammmmm There are two types of bar diagrams namely, Horizontal Bar diagram and Vertical bar diagram. While horizontal bar diagram is used for qualitative data or data varying over space, the vertical bar diagram is associated with quantitative data or time series data. Bars i.e. rectangles of equal width and usually of varying lengths are drawn either horizontally or vertically. We consider Multiple or Grouped Bar diagrams to compare related series. Component or sub-divided Bar diagrams are applied for representing data divided into a number of components. Finally, we use Divided Bar charts or Percentage Bar diagrams for comparing different components of a variable and also the relating of the components to the whole. For this situation, we may also use Pie chart or Pie diagram or circle diagram. IIIIIlllllllllluuuuussssstttttrrrrraaaaatttttiiiiiooooonnnnnsssss EEEEExxxxxaaaaammmmmpppppllllleeeee 1111100000.....11111The profits in lakhs of rupees of an industrial house for 2002, 2003, 2004, 2005, 2006, 2007 and 2008 are 5, 8, 9, 6, 12, 15 and 24 respectively. Represent these data using a suitable diagram. SSSSSooooollllluuuuutttttiiiiiooooonnnnn We can represent the profits for 7 consecutive years by drawing either a line chart or a vertical bar chart. Fig. 10.1 shows a line chart and figure 10.2 shows the corresponding vertical bar chart. (cid:20)(cid:5)(cid:3)(cid:5)(cid:18)(cid:20)(cid:5)(cid:18)(cid:1)(cid:20) (cid:10)(cid:11)(cid:12)(cid:27) Copyright -The Institute of Chartered Accountants of India (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:1)(cid:2)(cid:4)(cid:5)(cid:3)(cid:6)(cid:7)(cid:8)(cid:9)(cid:1)(cid:5)(cid:10)(cid:4)(cid:11)(cid:2)(cid:4)(cid:12)(cid:13)(cid:7)(cid:12)(cid:14)(cid:7)(cid:8)(cid:3)(cid:2)(cid:3) 25 P R 20 O 15 F I 10 T 5 S 0 TTTTTiiiiimmmmmeeeee FFFFFiiiiiggggguuuuurrrrreeeee 1111100000.....11111 Showing line chart for the Profit of an Industrial House during 1996 to 2002. P R O F I T I N L A K H R U P E E S TTTTTiiiiimmmmmeeeee FFFFFiiiiiggggguuuuurrrrreeeee 1111100000.....22222 Showing vertical bar diagram for the Profit of an Industrial house from 1996 to 2002. (cid:10)(cid:11)(cid:12)(cid:10)(cid:11) (cid:1)(cid:14)(cid:15)(cid:15)(cid:14)(cid:16)(cid:8) (cid:4)(cid:7)(cid:14)(cid:17)(cid:18)(cid:1)(cid:18)(cid:6)(cid:16)(cid:1)(cid:19)(cid:8) (cid:5)(cid:6)(cid:20)(cid:5) )seepuR hkaL ni( 2002 2003 2004 2005 2006 2007 2008 25 20 15 10 5 0 2002 2003 2004 2005 2006 2007 2008 Copyright -The Institute of Chartered Accountants of India EEEEExxxxxaaaaammmmmpppppllllleeeee 1111100000.....22222The production of wheat and rice of a region are given below : Year Production in metric tones Wheat Rice 2005 12 25 2006 15 30 2007 18 32 2008 19 36 Represent this information using a suitable diagram. SSSSSooooollllluuuuutttttiiiiiooooonnnnn We can represent this information by drawing a multiple line chart. Alternately, a multiple bar diagram may be considered. These are depicted in figure 10.3 and 10.4 respectively. 40 (Rice) S E N N 30 O T C I R T E 20 (Wheat) M N I N O I 10 T C U D O R P 0 2005 2006 2007 2008 YYYYYEEEEEAAAAARRRRR FFFFFiiiiiggggguuuuurrrrreeeee 1111100000.....33333 (cid:20)(cid:5)(cid:3)(cid:5)(cid:18)(cid:20)(cid:5)(cid:18)(cid:1)(cid:20) (cid:10)(cid:11)(cid:12)(cid:10)(cid:10) Copyright -The Institute of Chartered Accountants of India (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:1)(cid:2)(cid:4)(cid:5)(cid:3)(cid:6)(cid:7)(cid:8)(cid:9)(cid:1)(cid:5)(cid:10)(cid:4)(cid:11)(cid:2)(cid:4)(cid:12)(cid:13)(cid:7)(cid:12)(cid:14)(cid:7)(cid:8)(cid:3)(cid:2)(cid:3) Multiple line chart showing production of wheat and rice of a region during 1995–1998. (Dotted line represent production of rice and continuous line that of wheat). P 40 R O 35 D U C T 30 I O N 25 I N 20 M E T 15 R I C 10 T O 5 N N E 0 S 12909055 12909066 12909077 12909088 Rice TTTTTiiiiimmmmmeeeee 12345 12345 Wheat FFFFFiiiiiggggguuuuurrrrreeeee 1111100000.....44444 12345 12345 Multiple bar chart representing production of rice and wheat from 1995 to 1998. EEEEExxxxxaaaaammmmmpppppllllleeeee 1111100000.....33333Draw an appropriate diagram with a view to represent the following data : Source Revenue in millions of rupees Customs 80 Excise 190 Income Tax 160 Corporate Tax 75 Miscellaneous 35 (cid:10)(cid:11)(cid:12)(cid:10)(cid:13) (cid:1)(cid:14)(cid:15)(cid:15)(cid:14)(cid:16)(cid:8) (cid:4)(cid:7)(cid:14)(cid:17)(cid:18)(cid:1)(cid:18)(cid:6)(cid:16)(cid:1)(cid:19)(cid:8) (cid:5)(cid:6)(cid:20)(cid:5) Copyright -The Institute of Chartered Accountants of India SSSSSooooollllluuuuutttttiiiiiooooonnnnn Pie chart or divided bar chart would be the ideal diagram to represent this data. We consider Pie chart. TTTTTaaaaabbbbbllllleeeee 1111100000.....22222 Computation for drawing Pie chart Source Revenue in Central angle (1) Million rupees (2) (2) = x360o Totalof(2) 80 Customs 80 x360o = 53o (approx) 540 190 Excise 190 x360o =127o 540 160 Income Tax 160 x360o =107o 540 75 Corporate Tax 75 x360o = 50o 540 35 Miscellaneous 35 x360o = 23o 540 Total 540 3600 EEEEExxxxxccccciiiiissssseeeee IIIIITTTTT CCCCCuuuuussssstttttooooommmmm o o o o CCCCCTTTTT ~ ~ ~ ~ MMMMMiiiiisssssccccc..... FFFFFiiiiiggggguuuuurrrrreeeee 1111100000.....55555 PPPPPiiiiieeeee ccccchhhhhaaaaarrrrrttttt ssssshhhhhooooowwwwwiiiiinnnnnggggg ttttthhhhheeeee dddddiiiiissssstttttrrrrriiiiibbbbbuuuuutttttiiiiiooooonnnnn ooooofffff RRRRReeeeevvvvveeeeennnnnuuuuueeeee (cid:20)(cid:5)(cid:3)(cid:5)(cid:18)(cid:20)(cid:5)(cid:18)(cid:1)(cid:20) (cid:10)(cid:11)(cid:12)(cid:10)(cid:21) Copyright -The Institute of Chartered Accountants of India (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:1)(cid:2)(cid:4)(cid:5)(cid:3)(cid:6)(cid:7)(cid:8)(cid:9)(cid:1)(cid:5)(cid:10)(cid:4)(cid:11)(cid:2)(cid:4)(cid:12)(cid:13)(cid:7)(cid:12)(cid:14)(cid:7)(cid:8)(cid:3)(cid:2)(cid:3) 1111100000.....44444 FFFFFRRRRREEEEEQQQQQUUUUUEEEEENNNNNCCCCCYYYYY DDDDDIIIIISSSSSTTTTTRRRRRIIIIIBBBBBUUUUUTTTTTIIIIIOOOOONNNNN As discussed in the previous section, frequency data occur when we classify statistical data in respect of either a variable or an attribute. A frequency distribution may be defined as a tabular representation of statistical data, usually in an ascending order, relating to a measurable characteristic according to individual value or a group of values of the characteristic under study. In case, the characteristic under consideration is an attribute, say nationality, then the tabulation is made by allotting numerical figures to the different classes the attribute may belong like, in this illustration, counting the number of Indian, British, French, German and so on. The qualitative characteristic is divided into a number of categories or classes which are mutually exclusive and exhaustive and the figures against all these classes are recorded. The figure corresponding to a particular class, signifying the number of times or how frequently a particular class occurs is known as the frequency of that class. Thus, the number of Indians, as found from the given data, signifies the frequency of the Indians. So frequency distribution is a statistical table that distributes the total frequency to a number of classes. When tabulation is done in respect of a discrete random variable, it is known as Discrete or Ungrouped or simple Frequency Distribution and in case the characteristic under consideration is a continuous variable, such a classification is termed as Grouped Frequency Distribution. In case of a grouped frequency distribution, tabulation is done not against a single value as in the case of an attribute or a discrete random variable but against a group of values. The distribution of the number of car accidents in Delhi during 12 months of the year 2005 is an example of a ungrouped frequency distribution and the distribution of heights of the students of St. Xavier’s College for the year 2004 is an example of a grouped frequency distribution. EEEEExxxxxaaaaammmmmpppppllllleeeee 1111100000.....44444Following are the records of babies born in a nursing home in Bangalore during a week (B denoting Boy and G for Girl) : B G G B G G B B G G G G B B B G B B G B B B G B B B G G B G Construct a frequency distribution according to gender. SSSSSooooollllluuuuutttttiiiiiooooonnnnn In order to construct a frequency distribution of babies in accordance with their gender, we count the number of male births and that of female births and present this information in the following table. (cid:10)(cid:11)(cid:12)(cid:10)(cid:22) (cid:1)(cid:14)(cid:15)(cid:15)(cid:14)(cid:16)(cid:8) (cid:4)(cid:7)(cid:14)(cid:17)(cid:18)(cid:1)(cid:18)(cid:6)(cid:16)(cid:1)(cid:19)(cid:8) (cid:5)(cid:6)(cid:20)(cid:5) Copyright -The Institute of Chartered Accountants of India TTTTTaaaaabbbbbllllleeeee 1111100000.....33333 Frequency distribution of babies according to Gender Category Number of births Boy (B) 16 Girl (G) 14 Total 30 FFFFFrrrrreeeeeqqqqquuuuueeeeennnnncccccyyyyy DDDDDiiiiissssstttttrrrrriiiiibbbbbuuuuutttttiiiiiooooonnnnn ooooofffff aaaaa VVVVVaaaaarrrrriiiiiaaaaabbbbbllllleeeee For the construction of a frequency distribution of a variable, we need to go through the following steps : I Find the largest and smallest observations and obtain the difference between them, known as Range, in case of a continuous variable. II Form a number of classes depending on the number of isolated values assumed by a discrete variable. In case of a continuous variable, find the number of class intervals using the relation, No. of class Interval X class length≅Range. III Present the class or class interval in a table known as frequency distribution table. IV Apply ‘tally mark’ i.e. a stroke against the occurrence of a particulars value in a class or class interval. V Count the tally marks and present these numbers in the next column, known as frequency column, and finally check whether the total of all these class frequencies tally with the total number of observations. EEEEExxxxxaaaaammmmmpppppllllleeeee 1111100000.....55555 A review of the first 30 pages of a statistics book reveals the following printing mistakes : 0 1 3 3 2 5 6 0 1 0 4 1 1 0 2 3 2 5 0 4 2 3 2 2 3 3 4 6 1 4 Make a frequency distribution of printing mistakes. SSSSSooooollllluuuuutttttiiiiiooooonnnnn Since x, the printing mistakes, is a discrete variable, x can assume seven values 0, 1, 2, 3, 4, 5 and 6. Thus we have 7 classes, each class comprising a single value. (cid:20)(cid:5)(cid:3)(cid:5)(cid:18)(cid:20)(cid:5)(cid:18)(cid:1)(cid:20) (cid:10)(cid:11)(cid:12)(cid:10)(cid:23) Copyright -The Institute of Chartered Accountants of India (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:1)(cid:2)(cid:4)(cid:5)(cid:3)(cid:6)(cid:7)(cid:8)(cid:9)(cid:1)(cid:5)(cid:10)(cid:4)(cid:11)(cid:2)(cid:4)(cid:12)(cid:13)(cid:7)(cid:12)(cid:14)(cid:7)(cid:8)(cid:3)(cid:2)(cid:3) TTTTTaaaaabbbbbllllleeeee 1111100000.....44444 Frequency Distribution of the number of printing mistakes of the first 30 pages of a book Printing Mistake Tally marks Frequency (No. of Pages) 0 IIII 5 1 IIII 5 2 IIII I 6 3 IIII I 6 4 IIII 4 5 II 2 6 II 2 Total – 30 EEEEExxxxxaaaaammmmmpppppllllleeeee 1111100000.....66666Following are the weights in Kgs. of 36 BBA students of St. Xavier’s College. 70 73 49 61 61 47 57 50 59 59 68 45 55 65 68 56 68 55 70 70 57 44 69 73 64 49 63 65 70 65 62 64 73 67 60 50 Construct a frequency distribution of weights, taking class length as 5. SSSSSooooollllluuuuutttttiiiiiooooonnnnn We have, Range = Maximum weight – minimum weight = 73 Kgs. – 44 Kgs. = 29 Kgs. No. of class interval × class lengths ≅ Range ⇒ No. of class interval × 5 ≅ 29 29 ⇒ No. of class interval = ≅ 6. 5 (We always take the next integer as the no. of class intervals so as to include both the minimum and maximum values). (cid:10)(cid:11)(cid:12)(cid:10)(cid:24) (cid:1)(cid:14)(cid:15)(cid:15)(cid:14)(cid:16)(cid:8) (cid:4)(cid:7)(cid:14)(cid:17)(cid:18)(cid:1)(cid:18)(cid:6)(cid:16)(cid:1)(cid:19)(cid:8) (cid:5)(cid:6)(cid:20)(cid:5) Copyright -The Institute of Chartered Accountants of India TTTTTaaaaabbbbbllllleeeee 1111100000.....55555 Frequency Distribution of weights of 36 BBA Students Weight in Kg Tally marks No. of Students (Class Interval) (Frequency) 44-48 III 3 49-53 IIII 4 54-58 IIII 5 59-63 IIII II 7 64-68 IIII IIII 9 69-73 IIII III 8 Total – 36 SSSSSooooommmmmeeeee iiiiimmmmmpppppooooorrrrrtttttaaaaannnnnttttt ttttteeeeerrrrrmmmmmsssss aaaaassssssssssoooooccccciiiiiaaaaattttteeeeeddddd wwwwwiiiiittttthhhhh aaaaa fffffrrrrreeeeeqqqqquuuuueeeeennnnncccccyyyyy dddddiiiiissssstttttrrrrriiiiibbbbbuuuuutttttiiiiiooooonnnnn CCCCClllllaaaaassssssssss LLLLLiiiiimmmmmiiiiittttt (((((CCCCCLLLLL))))) Corresponding to a class interval, the class limits may be defined as the minimum value and the maximum value the class interval may contain. The minimum value is known as the lower class limit (LCL) and the maximum value is known as the upper class limit (UCL). For the frequency distribution of weights of BBA Students, the LCL and UCL of the first class interval are 44 kgs. and 48 kgs. respectively. CCCCClllllaaaaassssssssss BBBBBooooouuuuunnnnndddddaaaaarrrrryyyyy (((((CCCCCBBBBB))))) Class boundaries may be defined as the actual class limit of a class interval. For overlapping classification or mutually exclusive classification that excludes the upper class limits like 10– 20, 20–30, 30–40, ……… etc. the class boundaries coincide with the class limits. This is usually done for a continuous variable. However, for non-overlapping or mutually inclusive classification that includes both the class limits like 0–9, 10–19, 20–29,…… which is usually applicable for a discrete variable, we have D LCB=LCL- 2 D andUCB=UCL+ 2 Where D is the difference between the LCL of the next class interval and the UCL of the given class interval. For the data presented in table 10.5, LCB of the first class interval (49-48) = 44kgs.- kgs. 2 = 43.50 kgs. (cid:20)(cid:5)(cid:3)(cid:5)(cid:18)(cid:20)(cid:5)(cid:18)(cid:1)(cid:20) (cid:10)(cid:11)(cid:12)(cid:10)(cid:25) Copyright -The Institute of Chartered Accountants of India (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:1)(cid:2)(cid:4)(cid:5)(cid:3)(cid:6)(cid:7)(cid:8)(cid:9)(cid:1)(cid:5)(cid:10)(cid:4)(cid:11)(cid:2)(cid:4)(cid:12)(cid:13)(cid:7)(cid:12)(cid:14)(cid:7)(cid:8)(cid:3)(cid:2)(cid:3) and the corresponding UCB 49-48 = 48kgs.+ kgs. 2 = 48.50 kgs. MMMMMiiiiiddddd-----pppppoooooiiiiinnnnnttttt ooooorrrrr MMMMMiiiiiddddd-----vvvvvaaaaallllluuuuueeeee ooooorrrrr ccccclllllaaaaassssssssss mmmmmaaaaarrrrrkkkkk Corresponding to a class interval, this may be defined as the total of the two class limits or class boundaries to be divided by 2. Thus, we have LCL+UCL mid-point = 2 LCB+UCB = 2 Referring to the distribution of weight of BBA students, the mid-points for the first two class intervals are 44kgs.+48kgs. 49kgs.+53kgs. and 2 2 i.e. 46 kgs. and 51 kgs. respectively. WWWWWiiiiidddddttttthhhhh ooooorrrrr sssssiiiiizzzzzeeeee ooooofffff aaaaa ccccclllllaaaaassssssssss iiiiinnnnnttttteeeeerrrrrvvvvvaaaaalllll The width of a class interval may be defined as the difference between the UCB and the LCB of that class interval. For the distribution of weights of BBA students, C, the class length or width is 48.50 kgs. – 43.50 kgs. = 5 kgs. for the first class interval. For the other class intervals also, C remains same. CCCCCuuuuummmmmuuuuulllllaaaaatttttiiiiivvvvveeeee FFFFFrrrrreeeeeqqqqquuuuueeeeennnnncccccyyyyy The cumulative frequency corresponding to a value for a discrete variable and corresponding to a class boundary for a continuous variable may be defined as the number of observations less than the value or less than or equal to the class boundary. This definition refers to the less than cumulative frequency. We can define more than cumulative frequency in a similar manner. Both types of cumulative frequencies are shown in the following table. (cid:10)(cid:11)(cid:12)(cid:10)(cid:26) (cid:1)(cid:14)(cid:15)(cid:15)(cid:14)(cid:16)(cid:8) (cid:4)(cid:7)(cid:14)(cid:17)(cid:18)(cid:1)(cid:18)(cid:6)(cid:16)(cid:1)(cid:19)(cid:8) (cid:5)(cid:6)(cid:20)(cid:5) Copyright -The Institute of Chartered Accountants of India TTTTTaaaaabbbbbllllleeeee 1111100000.....66666 Cumulative Frequency Distribution of weights of 36 BBA students Weight in kg Cumulative Frequency (CB) Less than More than 43.50 0 33 + 3 or 36 48.50 0 + 3 or 3 29 + 4 or 33 53.50 3 + 4 or 7 24 + 5 or 29 58.50 7 + 5 or 12 17 + 7 or 24 63.50 12 + 7 or 19 8 + 9 or 17 68.50 19 + 9 or 28 0 + 8 or 8 73.50 28 + 8 or 36 0 FFFFFrrrrreeeeeqqqqquuuuueeeeennnnncccccyyyyy dddddeeeeennnnnsssssiiiiitttttyyyyy ooooofffff aaaaa ccccclllllaaaaassssssssss iiiiinnnnnttttteeeeerrrrrvvvvvaaaaalllll It may be defined as the ratio of the frequency of that class interval to the corresponding class length. The frequency densities for the first two class intervals of the frequency distribution of weights of BBA students are 3/5 and 4/5 i.e. 0.60 and 0.80 respectively. RRRRReeeeelllllaaaaatttttiiiiivvvvveeeee fffffrrrrreeeeeqqqqquuuuueeeeennnnncccccyyyyy aaaaannnnnddddd pppppeeeeerrrrrccccceeeeennnnntttttaaaaagggggeeeee fffffrrrrreeeeeqqqqquuuuueeeeennnnncccccyyyyy ooooofffff aaaaa ccccclllllaaaaassssssssss iiiiinnnnnttttteeeeerrrrrvvvvvaaaaalllll Relative frequency of a class interval may be defined as the ratio of the class frequency to the total frequency. Percentage frequency of a class interval may be defined as the ratio of class frequency to the total frequency, expressed as a percentage. For the last example, the relative frequencies for the first two class intervals are 3/36 and 4/36 respectively and the percentage frequencies are 300/36 and 400/36 respectively. It is quite obvious that whereas the relative frequencies add up to unity, the percentage frequencies add up to one hundred. 1111100000.....55555 GGGGGRRRRRAAAAAPPPPPHHHHHIIIIICCCCCAAAAALLLLL RRRRREEEEEPPPPPRRRRREEEEESSSSSEEEEENNNNNTTTTTAAAAATTTTTIIIIIOOOOONNNNN OOOOOFFFFF AAAAA FFFFFRRRRREEEEEQQQQQUUUUUEEEEENNNNNCCCCCYYYYY DDDDDIIIIISSSSSTTTTTRRRRRIIIIIBBBBBUUUUUTTTTTIIIIIOOOOONNNNN We consider the following types of graphical representation of frequency distribution : (i) Histogram or Area diagram; (ii) Frequency Polygon; (iii) Ogives or cumulative Frequency graphs. (i) HHHHHiiiiissssstttttooooogggggrrrrraaaaammmmm ooooorrrrr AAAAArrrrreeeeeaaaaa dddddiiiiiaaaaagggggrrrrraaaaammmmm This is a very convenient way to represent a frequency distribution. Histogram helps us to get an idea of the frequency curve of the variable under study. Some statistical measure can be obtained using a histogram. A comparison among the frequencies for different class intervals is possible in this mode of diagrammatic representation. In order to draw a histogram, the class limits are first converted to the corresponding class boundaries and a series of adjacent rectangles, one against each class interval, with the (cid:20)(cid:5)(cid:3)(cid:5)(cid:18)(cid:20)(cid:5)(cid:18)(cid:1)(cid:20) (cid:10)(cid:11)(cid:12)(cid:10)(cid:27) Copyright -The Institute of Chartered Accountants of India (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:1)(cid:2)(cid:4)(cid:5)(cid:3)(cid:6)(cid:7)(cid:8)(cid:9)(cid:1)(cid:5)(cid:10)(cid:4)(cid:11)(cid:2)(cid:4)(cid:12)(cid:13)(cid:7)(cid:12)(cid:14)(cid:7)(cid:8)(cid:3)(cid:2)(cid:3) class interval as base or breadth and the frequency or frequency density usually when the class intervals are not uniform as length or altitude, is erected. The histogram for the distribution of weight of 36 BBA students is shown below. The mode of the weights has also been determined using the histogram. i.e. Mode = 66.50 kgs. 10 N O. O F 8 S T U D E N 6 T S O R 4 F R E Q 2 U E N C Y 0 4433..550 5533..550 6633..55 0 ((MMoOd =e =6 66.65.)5 0 )7 3 7.35.50 WWWWWeeeeeiiiiiggggghhhhhttttt iiiiinnnnn kkkkkgggggsssss..... (((((ccccclllllaaaaassssssssss bbbbbooooouuuuunnnnndddddaaaaarrrrryyyyy))))) FFFFFiiiiiggggguuuuurrrrreeeee 1111100000.....66666 Showing histogram for the distribution of weight of 36 BBA students (cid:10)(cid:11)(cid:12)(cid:13)(cid:11) (cid:1)(cid:14)(cid:15)(cid:15)(cid:14)(cid:16)(cid:8) (cid:4)(cid:7)(cid:14)(cid:17)(cid:18)(cid:1)(cid:18)(cid:6)(cid:16)(cid:1)(cid:19)(cid:8) (cid:5)(cid:6)(cid:20)(cid:5) Copyright -The Institute of Chartered Accountants of India (ii) FFFFFrrrrreeeeeqqqqquuuuueeeeennnnncccccyyyyy PPPPPooooolllllyyyyygggggooooonnnnn Usually frequency polygon is meant for single frequency distribution. However, we also apply it for grouped frequency distribution provided the width of the class intervals remains the same. A frequency curve can be regarded as a limiting form of frequency polygon. In order to draw a frequency polygon, we plot (x, f) for i = 1, 2, 3, ……….. n with x denoting i i i the mid-point of the its class interval and f, the corresponding frequency, n being the i number of class intervals. The plotted points are joined successively by line segments and the figure, so drawn, is given the shape of a polygon, a closed figure, by joining the two extreme ends of the drawn figure to two additional points (x ,0) and (x ,0). 0 n+1 The frequency polygon for the distribution of weights of BBA students is shown in Figure 10.7. We can also obtain a frequency polygon starting with a histogram by adding the mid-points of the upper sides of the rectangles successively and then completing the figure by joining the two ends as before. Mid-points No. of Students (Frequency) 46 3 51 4 56 5 61 7 66 9 71 8 10 F 8 R E Q 6 U E N 4 C Y 2 0 46 56 66 76 WWWWWeeeeeiiiiiggggghhhhhttttt (((((MMMMMiiiiiddddd-----vvvvvaaaaallllluuuuueeeee))))) FFFFFiiiiiggggguuuuurrrrreeeee 1111100000.....77777 Showing frequency polygon for the distribution of height of 36 BBA students (cid:20)(cid:5)(cid:3)(cid:5)(cid:18)(cid:20)(cid:5)(cid:18)(cid:1)(cid:20) (cid:10)(cid:11)(cid:12)(cid:13)(cid:10) Copyright -The Institute of Chartered Accountants of India (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:1)(cid:2)(cid:4)(cid:5)(cid:3)(cid:6)(cid:7)(cid:8)(cid:9)(cid:1)(cid:5)(cid:10)(cid:4)(cid:11)(cid:2)(cid:4)(cid:12)(cid:13)(cid:7)(cid:12)(cid:14)(cid:7)(cid:8)(cid:3)(cid:2)(cid:3) (((((iiiiiiiiiiiiiii))))) OOOOOgggggiiiiivvvvveeeeesssss ooooorrrrr CCCCCuuuuummmmmuuuuulllllaaaaatttttiiiiivvvvveeeee FFFFFrrrrreeeeeqqqqquuuuueeeeennnnncccccyyyyy GGGGGrrrrraaaaappppphhhhh By plotting cumulative frequency against the respective class boundary, we get ogives. As such there are two ogives – less than type ogives, obtained by taking less than cumulative frequency on the vertical axis and more than type ogives by plotting more than type cumulative frequency on the vertical axis and thereafter joining the plotted points successively by line segments. Ogives may be considered for obtaining quartiles graphically. If a perpendicular is drawn from the point of intersection of the two ogives on the horizontal axis, then the x-value of this point gives us the value of median, the second or middle quartile. Ogives further can be put into use for making short term projections. Figure 10.8 depicts the ogives and the determination of the quartiles. This figure give us the following information. 1st quartile or lower quartile (Q ) = 55 kgs. 1 2nd quartile or median (Q or Me) = 62.50 kgs. 2 3rd quartile or upper quartile (Q ) = 68 kgs. 3 FFFFFiiiiiggggguuuuurrrrreeeee 1111100000.....88888 Showing the ogives for the distribution of weights of 36 BBA students (cid:10)(cid:11)(cid:12)(cid:13)(cid:13) (cid:1)(cid:14)(cid:15)(cid:15)(cid:14)(cid:16)(cid:8) (cid:4)(cid:7)(cid:14)(cid:17)(cid:18)(cid:1)(cid:18)(cid:6)(cid:16)(cid:1)(cid:19)(cid:8) (cid:5)(cid:6)(cid:20)(cid:5) C U M U L A T I V E F R E Q U E N C Y 5.37 5.86 )Q( 5.36 )Q( 5.85 )Q( 5.35 5.84 5.34 3 2 1 ) B C ( g k n i t h g ei W 05.37 05.86 Q 05.36 Q 05.85 Q 05.35 05.84 05.34 3 2 1 )BC( .sgk ni thgieW Copyright -The Institute of Chartered Accountants of India We find Q = 55 kgs. 1 Q = Me = 62.50 kgs. 2 Q = 68 kgs. 3 FFFFFrrrrreeeeeqqqqquuuuueeeeennnnncccccyyyyy CCCCCuuuuurrrrrvvvvveeeee A frequency curve is a smooth curve for which the total area is taken to be unity. It is a limiting form of a histogram or frequency polygon. The frequency curve for a distribution can be obtained by drawing a smooth and free hand curve through the mid-points of the upper sides of the rectangles forming the histogram. There exist four types of frequency curves namely (a) Bell-shaped curve; (b) U-shaped curve; (c) J-shaped curve; (d) Mixed curve. Most of the commonly used distributions provide bell-shaped curve, which, as suggested by the name, looks almost like a bell. The distribution of height, weight, mark, profit etc. usually belong to this category. On a bell-shaped curve, the frequency, starting from a rather low value, gradually reaches the maximum value, somewhere near the central part and then gradually decreases to reach its lowest value at the other extremity. For a U-shaped curve, the frequency is minimum near the central part and the frequency slowly but steadily reaches its maximum at the two extremities. The distribution of Kolkata bound commuters belongs to this type of curve as there are maximum number of commuters during the peak hours in the morning and in the evening. The J-shaped curve starts with a minimum frequency and then gradually reaches its maximum frequency at the other extremity. The distribution of commuters coming to Kolkata from the early morning hour to peak morning hour follows such a distribution. Sometimes, we may also come across an inverted J-shaped frequency curve. Lastly, we may have a combination of these frequency curves, known as mixed curve. These are exhibited in the following figures. (cid:20)(cid:5)(cid:3)(cid:5)(cid:18)(cid:20)(cid:5)(cid:18)(cid:1)(cid:20) (cid:10)(cid:11)(cid:12)(cid:13)(cid:21) Copyright -The Institute of Chartered Accountants of India (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:1)(cid:2)(cid:4)(cid:5)(cid:3)(cid:6)(cid:7)(cid:8)(cid:9)(cid:1)(cid:5)(cid:10)(cid:4)(cid:11)(cid:2)(cid:4)(cid:12)(cid:13)(cid:7)(cid:12)(cid:14)(cid:7)(cid:8)(cid:3)(cid:2)(cid:3) CCCCCLLLLLAAAAASSSSSSSSSS BBBBBOOOOOUUUUUNNNNNDDDDDAAAAARRRRRYYYYY FFFFFiiiiiggggguuuuurrrrreeeee 1111100000.....99999 BBBBBeeeeellllllllll-----ssssshhhhhaaaaapppppeeeeeddddd cccccuuuuurrrrrvvvvveeeee CCCCCLLLLLAAAAASSSSSSSSSS BBBBBOOOOOUUUUUNNNNNDDDDDAAAAARRRRRYYYYY FFFFFiiiiiggggguuuuurrrrreeeee 1111100000.....1111100000 UUUUU-----ssssshhhhhaaaaapppppeeeeeddddd cccccuuuuurrrrrvvvvveeeee (cid:10)(cid:11)(cid:12)(cid:13)(cid:22) (cid:1)(cid:14)(cid:15)(cid:15)(cid:14)(cid:16)(cid:8) (cid:4)(cid:7)(cid:14)(cid:17)(cid:18)(cid:1)(cid:18)(cid:6)(cid:16)(cid:1)(cid:19)(cid:8) (cid:5)(cid:6)(cid:20)(cid:5) Copyright -The Institute of Chartered Accountants of India CCCCCLLLLLAAAAASSSSSSSSSS BBBBBOOOOOUUUUUNNNNNDDDDDAAAAARRRRRYYYYY FFFFFiiiiiggggguuuuurrrrreeeee 1111100000.....1111111111 JJJJJ-----ssssshhhhhaaaaapppppeeeeeddddd cccccuuuuurrrrrvvvvveeeee CCCCCLLLLLAAAAASSSSSSSSSS BBBBBOOOOOUUUUUNNNNNDDDDDAAAAARRRRRYYYYY FFFFFiiiiiggggguuuuurrrrreeeee 1111100000.....1111122222 MMMMMiiiiixxxxxeeeeeddddd CCCCCuuuuurrrrrvvvvveeeee (cid:20)(cid:5)(cid:3)(cid:5)(cid:18)(cid:20)(cid:5)(cid:18)(cid:1)(cid:20) (cid:10)(cid:11)(cid:12)(cid:13)(cid:23) Copyright -The Institute of Chartered Accountants of India (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:1)(cid:2)(cid:4)(cid:5)(cid:3)(cid:6)(cid:7)(cid:8)(cid:9)(cid:1)(cid:5)(cid:10)(cid:4)(cid:11)(cid:2)(cid:4)(cid:12)(cid:13)(cid:7)(cid:12)(cid:14)(cid:7)(cid:8)(cid:3)(cid:2)(cid:3) EEEEEXXXXXEEEEERRRRRCCCCCIIIIISSSSSEEEEE SSSSSeeeeettttt AAAAA Answer the following questions. Each question carries 1 mark. 1. Which of the following statements is false? (a) Statistics is derived from the Latin word ‘Status’ (b) Statistics is derived from the Italian word ‘Statista’ (c) Statistics is derived from the French word ‘Statistik’ (d) None of these. 2. Statistics is defined in terms of numerical data in the (a) Singular sense (b) Plural sense (c) Either (a) or (b) (d) Both (a) and (b). 3. Statistics is applied in (a) Economics (b) Business management (c) Commerce and industry (d) All these. 4. Statistics is concerned with (a) Qualitative information (b) Quantitative information (c) (a) or (b) (d) Both (a) and (b). 5. An attribute is (a) A qualitative characteristic (b) A quantitative characteristic (c) A measurable characteristic (d) All these. 6. Annual income of a person is (a) An attribute (b) A discrete variable (c) A continuous variable (d) (b) or (c). 7. Marks of a student is an example of (a) An attribute (b) A discrete variable (c) A continuous variable (d) None of these. 8. Nationality of a student is (a) An attribute (b) A continuous variable (c) A discrete variable (d) (a) or (c). 9. Drinking habit of a person is (a) An attribute (b) A variable (c) A discrete variable (d) A continuous variable. (cid:10)(cid:11)(cid:12)(cid:13)(cid:24) (cid:1)(cid:14)(cid:15)(cid:15)(cid:14)(cid:16)(cid:8) (cid:4)(cid:7)(cid:14)(cid:17)(cid:18)(cid:1)(cid:18)(cid:6)(cid:16)(cid:1)(cid:19)(cid:8) (cid:5)(cid:6)(cid:20)(cid:5) Copyright -The Institute of Chartered Accountants of India 10. Age of a person is (a) An attribute (b) A discrete variable (c) A continuous variable (d) A variable. 11. Data collected on religion from the census reports are (a) Primary data (b) Secondary data (c) Sample data (d) (a) or (b). 12. The data collected on the height of a group of students after recording their heights with a measuring tape are (a) Primary data (b) Secondary data (c) Discrete data (d) Continuous data. 13. The primary data are collected by (a) Interview method (b) Observation method (c) Questionnaire method (d) All these. 14. The quickest method to collect primary data is (a) Personal interview (b) Indirect interview (c) Telephone interview (d) By observation. 15. The best method to collect data, in case of a natural calamity, is (a) Personal interview (b) Indirect interview (c) Questionnaire method (d) Direct observation method. 16. In case of a rail accident, the appropriate method of data collection is by (a) Personal interview (b) Direct interview (c) Indirect interview (d) All these. 17. Which method of data collection covers the widest area? (a) Telephone interview method (b) Mailed questionnaire method (c) Direct interview method (d) All these. 18. The amount of non-responses is maximum in (a) Mailed questionnaire method (b) Interview method (c) Observation method (d) All these. 19. Some important sources of secondary data are (a) International and Government sources (b) International and primary sources (c) Private and primary sources (d) Government sources. (cid:20)(cid:5)(cid:3)(cid:5)(cid:18)(cid:20)(cid:5)(cid:18)(cid:1)(cid:20) (cid:10)(cid:11)(cid:12)(cid:13)(cid:25) Copyright -The Institute of Chartered Accountants of India (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:1)(cid:2)(cid:4)(cid:5)(cid:3)(cid:6)(cid:7)(cid:8)(cid:9)(cid:1)(cid:5)(cid:10)(cid:4)(cid:11)(cid:2)(cid:4)(cid:12)(cid:13)(cid:7)(cid:12)(cid:14)(cid:7)(cid:8)(cid:3)(cid:2)(cid:3) 20. Internal consistency of the collected data can be checked when (a) Internal data are given (b) External data are given (c) Two or more series are given (d) A number of related series are given. 21. The accuracy and consistency of data can be verified by (a) Internal checking (b) External checking (c) Scrutiny (d) Both (a) and (b). 22. The mode of presentation of data are (a) Textual, tabulation and diagrammatic (b) Tabular, internal and external (c) Textual, tabular and internal (d) Tabular, textual and external. 23. The best method of presentation of data is (a) Textual (b) Tabular (c) Diagrammatic (d) (b) and (c). 24. The most attractive method of data presentation is (a) Tabular (b) Textual (c) Diagrammatic (d) (a) or (b). 25. For tabulation, ‘caption’ is (a) The upper part of the table (b) The lower part of the table (c) The main part of the table (d) The upper part of a table that describes the column and sub-column. 26. ‘Stub’ of a table is the (a) Left part of the table describing the columns (b) Right part of the table describing the columns (c) Right part of the table describing the rows (d) Left part of the table describing the rows. 27. The entire upper part of a table is known as (a) Caption (b) Stub (c) Box head (d) Body. 28. The unit of measurement in tabulation is shown in (a) Box head (b) Body (c) Caption (d) Stub. (cid:10)(cid:11)(cid:12)(cid:13)(cid:26) (cid:1)(cid:14)(cid:15)(cid:15)(cid:14)(cid:16)(cid:8) (cid:4)(cid:7)(cid:14)(cid:17)(cid:18)(cid:1)(cid:18)(cid:6)(cid:16)(cid:1)(cid:19)(cid:8) (cid:5)(cid:6)(cid:20)(cid:5) Copyright -The Institute of Chartered Accountants of India 29. In tabulation source of the data, if any, is shown in the (a) Footnote (b) Body (c) Stub (d) Caption. 30. Which of the following statements is untrue for tabulation? (a) Statistical analysis of data requires tabulation (b) It facilitates comparison between rows and not columns (c) Complicated data can be presented (d) Diagrammatic representation of data requires tabulation. 31. Hidden trend, if any, in the data can be noticed in (a) Textual presentation (b) Tabulation (c) Diagrammatic representation (d) All these. 32. Diagrammatic representation of data is done by (a) Diagrams (b) Charts (c) Pictures (d) All these. 33. The most accurate mode of data presentation is (a) Diagrammatic method (b) Tabulation (c) Textual presentation (d) None of these. 34. The chart that uses logarithm of the variable is known as (a) Line chart (b) Ratio chart (c) Multiple line chart (d) Component line chart. 35. Multiple line chart is applied for (a) Showing multiple charts (b) Two or more related time series when the variables are expressed in the same unit (c) Two or more related time series when the variables are expressed in different unit (d) Multiple variations in the time series. 36. Multiple axis line chart is considered when (a) There is more than one time series (b) The units of the variables are different (c) (a) or (b) (d) (a) and (b). 37. Horizontal bar diagram is used for (a) Qualitative data (b) Data varying over time (c) Data varying over space (d) (a) or (c). (cid:20)(cid:5)(cid:3)(cid:5)(cid:18)(cid:20)(cid:5)(cid:18)(cid:1)(cid:20) (cid:10)(cid:11)(cid:12)(cid:13)(cid:27) Copyright -The Institute of Chartered Accountants of India (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:1)(cid:2)(cid:4)(cid:5)(cid:3)(cid:6)(cid:7)(cid:8)(cid:9)(cid:1)(cid:5)(cid:10)(cid:4)(cid:11)(cid:2)(cid:4)(cid:12)(cid:13)(cid:7)(cid:12)(cid:14)(cid:7)(cid:8)(cid:3)(cid:2)(cid:3) 38. Vertical bar diagram is applicable when (a) The data are qualitative (b) The data are quantitative (c) When the data vary over time (d) (a) or (c). 39. Divided bar chart is considered for (a) Comparing different components of a variable (b) The relation of different components to the table (c) (a) or (b) (d) (a) and (b). 40. In order to compare two or more related series, we consider (a) Multiple bar chart (b) Grouped bar chart (c) (a) or (b) (d) (a) and (b). 41. Pie-diagram is used for (a) Comparing different components and their relation to the total (b) Representing qualitative data in a circle (c) Representing quantitative data in circle (d) (b) or (c). 42. A frequency distribution (a) Arranges observations in an increasing order (b) Arranges observation in terms of a number of groups (c) Relaters to a measurable characteristic (d) all these. 43. The frequency distribution of a continuous variable is known as (a) Grouped frequency distribution (b) Simple frequency distribution (c) (a) or (b) (d) (a) and (b). (cid:10)(cid:11)(cid:12)(cid:21)(cid:11) (cid:1)(cid:14)(cid:15)(cid:15)(cid:14)(cid:16)(cid:8) (cid:4)(cid:7)(cid:14)(cid:17)(cid:18)(cid:1)(cid:18)(cid:6)(cid:16)(cid:1)(cid:19)(cid:8) (cid:5)(cid:6)(cid:20)(cid:5) Copyright -The Institute of Chartered Accountants of India 44. The distribution of shares is an example of the frequency distribution of (a) A discrete variable (b) A continuous variable (c) An attribute (d) (a) or (c). 45. The distribution of profits of a blue-chip company relates to (a) Discrete variable (b) Continuous variable (c) Attributes (d) (a) or (b). 46. Mutually exclusive classification (a) Excludes both the class limits (b) Excludes the upper class limit but includes the lower class limit (c) Includes the upper class limit but excludes the upper class limit (d) Either (b) or (c). 47. Mutually inclusive classification is usually meant for (a) A discrete variable (b) A continuous variable (c) An attribute (d) All these. 48. Mutually exclusive classification is usually meant for (a) A discrete variable (b) A continuous variable (c) An attribute (d) Any of these. 49. The LCB is (a) An upper limit to LCL (b) A lower limit to LCL (c) (a) and (b) (d) (a) or (b). (cid:20)(cid:5)(cid:3)(cid:5)(cid:18)(cid:20)(cid:5)(cid:18)(cid:1)(cid:20) (cid:10)(cid:11)(cid:12)(cid:21)(cid:10) Copyright -The Institute of Chartered Accountants of India (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:1)(cid:2)(cid:4)(cid:5)(cid:3)(cid:6)(cid:7)(cid:8)(cid:9)(cid:1)(cid:5)(cid:10)(cid:4)(cid:11)(cid:2)(cid:4)(cid:12)(cid:13)(cid:7)(cid:12)(cid:14)(cid:7)(cid:8)(cid:3)(cid:2)(cid:3) 50. The UCL is (a) An upper limit to UCL (b) A lower limit to LCL (c) Both (a) and (b) (d) (a) or (b). 51. length of a class is (a) The difference between the UCB and LCB of that class (b) The difference between the UCL and LCL of that class (c) (a) or (b) (d) Both (a) and (b). 52. For a particular class boundary, the less than cumulative frequency and more than cumulative frequency add up to (a) Total frequency (b) Fifty per cent of the total frequency (c) (a) or (b) (d) None of these. 53. Frequency density corresponding to a class interval is the ratio of (a) Class frequency to the total frequency (b) Class frequency to the class length (c) Class length to the class frequency (d) Class frequency to the cumulative frequency. 54. Relative frequency for a particular class (a) Lies between 0 and 1 (b) Lies between 0 and 1, both inclusive (c) Lies between –1 and 0 (d) Lies between –1 to 1. 55. Mode of a distribution can be obtained from (a) Histogram (b) Less than type ogives (c) More than type ogives (d) Frequency polygon. 56. Median of a distribution can be obtained from (a) Frequency polygon (b) Histogram (c) Less than type ogives (d) None of these. 57. A comparison among the class frequencies is possible only in (a) Frequency polygon (b) Histogram (c) Ogives (d) (a) or (b). 58. Frequency curve is a limiting form of (a) Frequency polygon (b) Histogram (c) (a) or (b) (d) (a) and (b). (cid:10)(cid:11)(cid:12)(cid:21)(cid:13) (cid:1)(cid:14)(cid:15)(cid:15)(cid:14)(cid:16)(cid:8) (cid:4)(cid:7)(cid:14)(cid:17)(cid:18)(cid:1)(cid:18)(cid:6)(cid:16)(cid:1)(cid:19)(cid:8) (cid:5)(cid:6)(cid:20)(cid:5) Copyright -The Institute of Chartered Accountants of India 59. Most of the commonly used frequency curves are (a) Mixed (b) Inverted J-shaped (c) U-shaped (d) Bell-shaped. 60. The distribution of profits of a company follows (a) J-shaped frequency curve (b) U-shaped frequency curve (c) Bell-shaped frequency curve (d) Any of these. SSSSSeeeeettttt BBBBB Answer the following questions. Each question carries 2 marks. 1. Out of 1000 persons, 25 per cent were industrial workers and the rest were agricultural workers. 300 persons enjoyed world cup matches on TV. 30 per cent of the people who had not watched world cup matches were industrial workers. What is the number of agricultural workers who had enjoyed world cup matches on TV? (a) 260 (b) 240 (c) 230 (d) 250 2. A sample study of the people of an area revealed that total number of women were 40% and the percentage of coffee drinkers were 45 as a whole and the percentage of male coffee drinkers was 20. What was the percentage of female non-coffee drinkers? (a) 10 (b) 15 (c) 18 (d) 20 3. Cost of sugar in a month under the heads Raw Materials, labour, direct production and others were 12, 20, 35 and 23 units respectively. What is the difference between the central angles for the largest and smallest components of the cost of sugar? (a) 72o (b) 48o (c) 56o (d) 92o 4. The number of accidents for seven days in a locality are given below : No. of accidents: 0 1 2 3 4 5 6 Frequency : 15 19 22 31 9 3 2 What is the number of cases when 3 or less accidents occurred? (a) 56 (b) 6 (c) 68 (d) 87 5. The following data relate to the incomes of 86 persons : Income in Rs. : 500–999 1000–1499 1500–1999 2000–2499 No. of persons : 15 28 36 7 What is the percentage of persons earning more than Rs. 1500? (a) 50 (b) 45 (c) 40 (d) 60 (cid:20)(cid:5)(cid:3)(cid:5)(cid:18)(cid:20)(cid:5)(cid:18)(cid:1)(cid:20) (cid:10)(cid:11)(cid:12)(cid:21)(cid:21) Copyright -The Institute of Chartered Accountants of India (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:1)(cid:2)(cid:4)(cid:5)(cid:3)(cid:6)(cid:7)(cid:8)(cid:9)(cid:1)(cid:5)(cid:10)(cid:4)(cid:11)(cid:2)(cid:4)(cid:12)(cid:13)(cid:7)(cid:12)(cid:14)(cid:7)(cid:8)(cid:3)(cid:2)(cid:3) 6. The following data relate to the marks of a group of students: Marks : Below 10 Below 20 Below 30 Below 40 Below 50 No. of students : 15 38 65 84 100 How many students got marks more than 30? (a) 65 (b) 50 (c) 35 (d) 43 7. Find the number of observations between 250 and 300 from the following data : Value : More than 200 More than 250 More than 300 More than 350 No. of observations: 56 38 15 0 (a) 56 (b) 23 (c) 15 (d) 8 SSSSSeeeeettttt CCCCC Answer the following questions. Each question carries 5 marks. 1. In a study about the male and female students of commerce and science departments of a college in 5 years, the following datas were obtained : 1995 2000 70% male students 75% male students 65% read Commerce 40% read Science 20% of female students read Science 50% of male students read Commerce 3000 total No. of students 3600 total No. of students. After combining 1995 and 2000 if x denotes the ratio of female commerce student to female Science student and y denotes the ratio of male commerce student to male Science student, then (a) x = y (b) x > y (c) x < y (d) x ≥ y 2. In a study relating to the labourers of a jute mill in West Bengal, the following information was collected. ‘Twenty per cent of the total employees were females and forty per cent of them were married. Thirty female workers were not members of Trade Union. Compared to this, out of 600 male workers 500 were members of Trade Union and fifty per cent of the male workers were married. The unmarried non-member male employees were 60 which formed ten per cent of the total male employees. The unmarried non-members of the employees were 80’. On the basis of this information, the ratio of married male non-members to the married female non-members is (a) 1 : 3 (b) 3 : 1 (c) 4 : 1 (d) 5 : 1 (cid:10)(cid:11)(cid:12)(cid:21)(cid:22) (cid:1)(cid:14)(cid:15)(cid:15)(cid:14)(cid:16)(cid:8) (cid:4)(cid:7)(cid:14)(cid:17)(cid:18)(cid:1)(cid:18)(cid:6)(cid:16)(cid:1)(cid:19)(cid:8) (cid:5)(cid:6)(cid:20)(cid:5) Copyright -The Institute of Chartered Accountants of India 3. The weight of 50 students in pounds are given below : 82, 95, 120, 174, 179, 176, 159, 91, 85, 175 88, 160, 97, 133, 159, 176, 151, 115, 105, 172, 170, 128, 112, 101, 123, 117, 93, 117, 99, 90, 113, 119, 129, 134, 178, 105, 147, 107, 155, 157, 98, 117, 95, 135, 175, 97, 160, 168, 144, 175 If the data are arranged in the form of a frequency distribution with class intervals as 81-100, 101-120, 121-140, 141-160 and 161-180, then the frequencies for these 5 class intervals are (a) 6, 9, 10, 11, 14 (b) 12, 8, 7, 11, 12 (c) 10, 12, 8, 11, 9 (d) 12, 11, 6, 9, 12 4. The following data relate to the marks of 48 students in statistics : 56, 10, 54, 38, 21, 43, 12, 22, 48, 51, 39, 26, 12, 17, 36, 19, 48, 36, 15, 33, 30, 62, 57, 17, 5, 17, 45, 46, 43, 55, 57, 38, 43, 28, 32, 35, 54, 27, 17, 16, 11, 43, 45, 2, 16, 46, 28, 45, What are the frequency densities for the class intervals 30-39, 40-49 and 50-59 (a) 0.20, 0.50, 0.90 (b) 0.70, 0.90, 1.10 (c) 0.1875, 0.1667, 0.2083 (d) 0.90, 1.00, 0.80 5. The following information relates to the age of death of 50 persons in an area : 36, 48, 50, 45, 49, 31, 50, 48, 42, 57 43, 40, 32, 41, 39, 39, 43, 47, 45, 52 47, 48, 53, 37, 48, 50, 41, 49, 50, 53 38, 41, 49, 45, 36, 39, 31, 48, 59, 48 37, 49, 53, 51, 54, 59, 48, 38, 39, 45 If the class intervals are 31-33, 34-36, 37-39, …. Then the percentage frequencies for the last five class intervals are (a) 18, 18, 10, 2 and 4. (b) 10, 15, 18, 4 and 2. (c) 14, 18, 20, 10 and 2. (d) 10, 12, 16, 4 and 6. (cid:20)(cid:5)(cid:3)(cid:5)(cid:18)(cid:20)(cid:5)(cid:18)(cid:1)(cid:20) (cid:10)(cid:11)(cid:12)(cid:21)(cid:23) Copyright -The Institute of Chartered Accountants of India (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:1)(cid:2)(cid:4)(cid:5)(cid:3)(cid:6)(cid:7)(cid:8)(cid:9)(cid:1)(cid:5)(cid:10)(cid:4)(cid:11)(cid:2)(cid:4)(cid:12)(cid:13)(cid:7)(cid:12)(cid:14)(cid:7)(cid:8)(cid:3)(cid:2)(cid:3) AAAAANNNNNSSSSSWWWWWEEEEERRRRRSSSSS SSSSSeeeeettttt AAAAA 1. (c) 2. (b) 3. (d) 4. (d) 5. (a) 6. (b) 7. (b) 8. (a) 9. (a) 10. (c) 11. (b) 12. (a) 13. (d) 14. (c) 15. (a) 16. (c) 17. (b) 18. (a) 19. (a) 20. (d) 21. (c) 22. (a) 23. (b) 24. (c) 25. (d) 26. (d) 27. (c) 28. (a) 29. (a) 30. (b) 31. (c) 32. (d) 33. (b) 34. (b) 35. (b) 36. (d) 37. (d) 38. (b) 39. (d) 40. (c) 41. (a) 42. (d) 43. (a) 44. (a) 45. (b) 46. (b) 47. (a) 48. (b) 49. (b) 50. (a) 51. (a) 52. (a) 53. (b) 54. (a) 55. (a) 56. (c) 57. (b) 58. (d) 59. (d) 60. (c) SSSSSeeeeettttt BBBBB 1. (a) 2. (b) 3. (d) 4. (d) 5. (a) 6. (c) 7. (b) SSSSSeeeeettttt CCCCC 1. (b) 2. (c) 3. (d) 4. (d) 5. (a) (cid:10)(cid:11)(cid:12)(cid:21)(cid:24) (cid:1)(cid:14)(cid:15)(cid:15)(cid:14)(cid:16)(cid:8) (cid:4)(cid:7)(cid:14)(cid:17)(cid:18)(cid:1)(cid:18)(cid:6)(cid:16)(cid:1)(cid:19)(cid:8) (cid:5)(cid:6)(cid:20)(cid:5) Copyright -The Institute of Chartered Accountants of India AAAAADDDDDDDDDDIIIIITTTTTIIIIIOOOOONNNNNAAAAALLLLL QQQQQUUUUUEEEEESSSSSTTTTTIIIIIOOOOONNNNN BBBBBAAAAANNNNNKKKKK 1. Graph is a (a) Line diagram (b) Bar diagram (c) Pie diagram (d) Pictogram 2. Details are shown by (a) Charts (b) Tabular presentation (c) both (d) none 3. The relationship between two variables are shown in (a) Pictogram (b) Histogram (c) Bar diagram (d) Line diagram 4. In general the number of types of tabulation are (a) two (b) three (c) one (d) four 5. A table has (a) four (b) two (c) five (d) none parts. 6. The number of errors in Statistics are (a) one (b) two (c) three (d) four 7. The number of “Frequency distribution“ is (a) two (b) one (c) five (d) four 8. (Class frequency)/(Width of the class ) is defined as (a) Frequency density (b) Frequency distribution (c) both (d) none 9. Tally marks determines (a) class width (b) class boundary (c) class limit (d) class frequency 10. Cumulative Frequency Distribution is a (a) graph (b) frequency (c) Statistical Table (d) distribution 11. To find the number of observations less than any given value (a) Single frequency distribution (b) Grouped frequency distribution (c) Cumulative frequency distribution (d) None is used. 12. An area diagram is (a) Histogram (b) Frequency Polygon (c) Ogive (d) none (cid:20)(cid:5)(cid:3)(cid:5)(cid:18)(cid:20)(cid:5)(cid:18)(cid:1)(cid:20) (cid:10)(cid:11)(cid:12)(cid:21)(cid:25) Copyright -The Institute of Chartered Accountants of India (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:1)(cid:2)(cid:4)(cid:5)(cid:3)(cid:6)(cid:7)(cid:8)(cid:9)(cid:1)(cid:5)(cid:10)(cid:4)(cid:11)(cid:2)(cid:4)(cid:12)(cid:13)(cid:7)(cid:12)(cid:14)(cid:7)(cid:8)(cid:3)(cid:2)(cid:3) 13. When all classes have a common width (a) Pie Chart (b) Frequency Polygon (c) both (d) none is used. 14. An approximate idea of the shape of frequency curve is given by (a) Ogive (b) Frequency Polygon (c) both (d) none 15. Ogive is a (a) line diagram (b) Bar diagram (c) both (d) none 16. Unequal widths of classes in the frequency distribution do not cause any difficulty in the construction of (a) Ogive (b) Frequency Polygon (c) Histogram (d) none 17. The graphical representation of a cumulative frequency distribution is called (a) Histogram (b) Ogive (c) both (d) none. 18. The most common form of diagrammatic representation of a grouped frequency distribution is (a) Ogive (b) Histogram (c) Frequency Polygon (d) none 19. Vertical bar chart may appear somewhat alike (a) Histogram (b) Frequency Polygon (c) both (d) none 20. The number of types of cumulative frequency is (a) one (b) two (c) three (d) four 21. A representative value of the class interval for the calculation of mean, standard deviation, mean deviation etc. is (a) class interval (b) class limit (c) class mark (d) none 22. The no. of observations falling within a class is called (a) density (b) frequency (c) both (d) none 23. Classes with zero frequencies are called (a) nil class (b) empty class (c) class (d) none 24. For determining the class frequencies it is necessary that these classes are (a) mutually exclusive (b) not mutually exclusive (c) independent (d) none (cid:10)(cid:11)(cid:12)(cid:21)(cid:26) (cid:1)(cid:14)(cid:15)(cid:15)(cid:14)(cid:16)(cid:8) (cid:4)(cid:7)(cid:14)(cid:17)(cid:18)(cid:1)(cid:18)(cid:6)(cid:16)(cid:1)(cid:19)(cid:8) (cid:5)(cid:6)(cid:20)(cid:5) Copyright -The Institute of Chartered Accountants of India 25. Most extreme values which would ever be included in a class interval are called (a) class limits (b) class interval (c) class boundaries (d) none 26. The value exactly at the middle of a class interval is called (a) class mark (b) mid value (c) both (d ) none 27. Difference between the lower and the upper class boundaries is (a) width (b) size (c) both (d) none 28. In the construction of a frequency distribution , it is generally preferable to have classes of (a) equal width (b) unequal width (c) maximum (d) none 29. Frequency density is used in the construction of (a) Histogram (b) Ogive (c) Frequency Polygon (d) none when the classes are of unequal width. 30. “Cumulative Frequency“ only refers to the (a) less-than type (b) more-than type (c) both (d) none 31. For the construction of a grouped frequency distribution (a) class boundaries (b) class limits (c) both (d) none are used. 32. In all Statistical calculations and diagrams involving end points of classes (a) class boundaries (b) class value (c) both (d) none are used. 33. Upper limit of any class is (a) same (b) different (c) both (d) none from the lower limit of the next class. 34. Upper boundary of any class coincides with the Lower boundary of the next class. (a) true (b) false (c) both (d) none. 35. Excepting the first and the last, all other class boundaries lie midway between the upper limit of a class and the lower limit of the next higher class. (a) true (b) false (c) both (d) none 36. The lower extreme point of a class is called (a) lower class limit (b) lower class boundary (c) both (d) none 37. For the construction of grouped frequency distribution from ungrouped data (a) class limits (b) class boundaries (c) class width (d) none are used. (cid:20)(cid:5)(cid:3)(cid:5)(cid:18)(cid:20)(cid:5)(cid:18)(cid:1)(cid:20) (cid:10)(cid:11)(cid:12)(cid:21)(cid:27) Copyright -The Institute of Chartered Accountants of India (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:1)(cid:2)(cid:4)(cid:5)(cid:3)(cid:6)(cid:7)(cid:8)(cid:9)(cid:1)(cid:5)(cid:10)(cid:4)(cid:11)(cid:2)(cid:4)(cid:12)(cid:13)(cid:7)(cid:12)(cid:14)(cid:7)(cid:8)(cid:3)(cid:2)(cid:3) 38. When one end of a class is not specified, the class is called (a) closed- end class (b) open- end class (c) both (d) none 39. Class boundaries should be considered to be the real limits for the class interval. (a) true (b) false (c) both (d) none 40. Difference between the maximum & minimum value of a given data is called (a) width (b) size (c) range (d) none 41. In Histogram if the classes are of unequal width then the heights of the rectangles must be proportional to the frequency densities. (a) true (b) false (c) both (d) none 42. When all classes have equal width, the heights of the rectangles in Histogram will be numerically equal to the (a) class frequencies (b) class boundaries (c) both (d) none 43. Consecutive rectangles in a Histogram have no space in between (a) true (b) false (c) both (d) none 44. Histogram emphasizes the widths of rectangles between the class boundaries . (a) false (b) true (c) both (d) none 45. To find the mode graphically (a) Ogive (b) Frequency Polygon (c) Histogram (d) none may be used. 46. When the width of all classes is same, frequency polygon has not the same area as the Histogram. (a) True (b) false (c) both (d) none 47. For obtaining frequency polygon we join the successive points whose abscissa represent the corresponding class frequency_____ (a) true (b) false (c) both (d) none 48. In representing simple frequency distributions of a discrete variable (a) Ogive (b) Histogram (c) Frequency Polygon (d) both is useful. 49. Diagrammatic representation of the cumulative frequency distribution is (a) Frequency Polygon (b) Ogive (c) Histogram (d) none 50. For the overlapping classes 0—10 , 10—20 , 20—30 etc.the class mark of the class 0—10 is (a) 5 (b) 0 (c) 10 (d) none 51. For the non-overlapping classes 0—19 , 20—39 , 40—59 the class mark of the class 0—19 is (a) 0 (b) 19 (c) 9.5 (d) none (cid:10)(cid:11)(cid:12)(cid:22)(cid:11) (cid:1)(cid:14)(cid:15)(cid:15)(cid:14)(cid:16)(cid:8) (cid:4)(cid:7)(cid:14)(cid:17)(cid:18)(cid:1)(cid:18)(cid:6)(cid:16)(cid:1)(cid:19)(cid:8) (cid:5)(cid:6)(cid:20)(cid:5) Copyright -The Institute of Chartered Accountants of India 52. Class : 0—10 10—20 20—30 30—40 40—50 Frequency : 5 8 15 6 4 For the class 20—30 , cumulative frequency is (a) 20 (b) 13 (c) 15 (d) 28 53. An Ogive can be prepared in _____________ different ways. (a) 2 (b) 3 (c) 4 (d) none 54. The curve obtained by joining the points, whose x- coordinates are the upper limits of the class-intervals and y coordinates are corresponding cumulative frequencies is called (a) Ogive (b) Histogram (c) Frequency Polygon (d) Frequency Curve 55. The breadth of the rectangle is equal to the length of the class-interval in (a) Ogive (b) Histogram (c) both (d) none 56. In Histogram, the classes are taken (a) overlapping (b) non-overlapping (c) both (d) none 57. For overlapping class-intervals the class limit & class boundary are (a) same (b) not same (c) zero (d) none 58. Classification is of (a) four (b) Three (c) two (d) five kinds. AAAAANNNNNSSSSSWWWWWEEEEERRRRRSSSSS 1 (a) 2 (b) 3 (d) 4 (a) 5 (c) 6 (b) 7 (a) 8 (a) 9 (d) 10 (c) 11 (c) 12 (a) 13 (b) 14 (b) 15 (a) 16 (a) 17 (b) 18 (b) 19 (a) 20 (b) 21 (c) 22 (b) 23 (b) 24 (a) 25 (c) 26 (c) 27 (c) 28 (a) 29 (a) 30 (a) 31 (b) 32 (a) 33 (b) 34 (a) 35 (a) 36 (b) 37 (a) 38 (b) 39 (a) 40 (c) 41 (a) 42 (a) 43 (a) 44 (b) 45 (c) 46 (b) 47 (b) 48 (c) 49 (b) 49 (b) 51 (c) 52 (d) 53 (a) 54 (a) 55 (b) 56 (a) 57 (a) 58 (a) (cid:20)(cid:5)(cid:3)(cid:5)(cid:18)(cid:20)(cid:5)(cid:18)(cid:1)(cid:20) (cid:10)(cid:11)(cid:12)(cid:22)(cid:10) Copyright -The Institute of Chartered Accountants of India