STATISTICS FOR AGRICULTURAL SCIENCES THIRD EDITION

PART I

STATISTICAL METHODS

CHAPTER 1

INTRODUCTION

In recent days we hear talking about ‘Statistics’ from a common person to highly qualified person. It only shows how ‘Statistics’ has been intimately connected with wide range of activities in daily life.

Statistics can be used either as plural or singular. When it is used as plural, it is a systematic presentation of facts and figures. It is in this context that majority of people use the word ‘Statistics’. They only meant mere facts and figures. These figures may be with regard to production of foodgrains in different years, area under cereal crops in different years, per capita income in a particular state at different times, etc., and these are generally published in Trade Journals, Economics and Statistics Bulletins. Newspapers, etc. When statistics is used as singular, it is a science which deals with collection, classification, tabulation, analysis and interpretation of data.

Statistics as a science is of recent origin. The word ‘Statistics’ has been derived from a Latin word which means ‘State’ which in turn means ‘politically organised people’ i.e., government. Since governments used to collect the relevant data on births and deaths, defence personnel, financial status of the peoples, import and export, etc. Statistics was identified with Government. Recently, it pervades all branches of sciences, social sciences and even in Humanities like English literature. For example, in English literature the style of a particular poet or an author can be assessed with the help of statistical tools.

In the opinion of Fisher ‘Statistics’ has got three important functions to play (i) Study of statistical populations (ii) study of the variation within the statistical populations (iii) study of the methods of reduction of data.

P.C. Mahalanobis compares ‘Statistician’ with a ‘Doctor’ where Doctor prescribes medicine according to the disease of the patient whereas statistician suggests statistical technique according to the data in hand for proper analysis and interpretation.

Bowley defined statistics as ‘the science of measurements of the social organism regarded as a whole in all its manifestations.’ Another definition says that it is ‘quantitative data affected to a marked extent by a multiplicity of causes.’ Y.t another definition says that it is a ‘Science of counting’ or ‘Science of averages’ and so on. But all these definitions are incomplete and are complementary to each other.

There are some of the limitations of ‘Statistics’ also when the data are not properly handled. People start disbelieving in statistics when the (1) data are not reliable (2) computing spurious relationships between variables (3) generalizing from a small sample to a population without taking care of error involved.

If one is ensured that data are reliable and is properly handled by a ‘skilled statistician’, the mistrust of statistics will disappear and in place of it precise and exact revelation of data will come up for reasonable conclusions.

CHAPTER 2

COLLECTION, CLASSIFICATION AND TABULATION OF DATA

2.1 COLLECTION OF DATA

The data are of two kinds: (i) Primary data (ii) Secondary data.

Primary data are based on primary source of information and the secondary data are based on secondary source of information.

2.1.1 Primary Data

Primary data are collected by the following methods: (i) By the investigator himself, (ii) By conducting a large scale survey with the help of field investigators. (iii) By sending questionnaires by post.

(i) The first method is limited in scope since the investigator himself cannot afford to bear the expenses of a large scale survey and also the time involved therein. Therefore, this method is of much use only in small pilot surveys like case studies. This method is being adopted by individual Investigators who submit dissertation for Master’s and Doctoral degrees in rural sociology, Ag. Extension, Ag. Economics, Home Management, etc.

(ii) In this method the schedules which elicit comprehensive information will be framed by the Chief Investigator with the help of other experts based on objectives of the survey. The field investigators would be trained with the methodology and survey, mode of filling the schedules and the skill of conducting interviews with the respondents, etc. The field investigators will furnish the schedules by personal interview method and submit the schedules to the Chief Investigator for further statistical analysis. This method of collecting data requires more money and time since wide range of information covering large area is to be collected. But the findings based on the large scale survey will be more comprehensive and helpful for policy making decisions. The Decennial Census in India, National sample survey rounds conducted by Govt. of India, Cost of Cultivation schemes, PL 480 schemes, etc., are some of the examples of this method.

(iii) In the third method, the questionnaire containing different types of questions on a particular topic or topics systematically arranged in order which elicit answers of the type yes or no or multiple choice will be sent by post and will be obtained by post. This method is easy for collecting the data with minimum expenditure but the respondents must be educated enough so as to fill the questionnaires properly and send them back realizing the importance of a survey. The Council of Scientific and Industrial Research (CSIR) conducted a survey recently by adopting this method for knowing the status of scientific personnel in India.

2.1.2

The secondary data can be collected from secondary source of information like newspapers, journals and from third person where first hand knowledge is not available. Journals like Trade Statistics, Statistical Abstracts published by State Bureau of Economics and Statistics, Agricultural situation in India, import and export statistics and Daily Economic times are some of the main sources of information providing secondary data.

2.2 CLASSIFICATION OF DATA

The data can be classified into two ways: (i) classification according to attributes (Descriptive classification) (ii) classification according to measurements (Numerical classification).

2.2.1 Descriptive Classification

The classification of individuals (or subjects) according to qualitative characteristic (or characteristics) is known as descriptive classification.

(a) Classification by Dichotomy: The classification of individuals (or objects) according to one attribute is known as simple classification. Classification of fields according to irrigated and unirrigated, population into employed and unemployed, students as hostellers and not-hostellers, etc., are some of the examples of simple classification.

(b) Manifold Classification: Classification of individuals (or objects) according to more than one attribute is known as manifold classification. For example, flowers can be classified according to colour and shape; students can be classified according to class, residence and sex, etc.

2.2.2 Numerical Classification

Classification of individuals (or objects) according to quantitative characteristics such as height, weight, income, yield, age, etc., is called as numerical classification.

EXAMPLE: 227 students are classified according to weight as follows:

TABLE 2.1

2.3 TABULATION OF DATA

Tabulation facilitates the presentation of large information into concise way under different titles and sub-titles so that the data in the table can further be subjected to statistical analysis. The following are the different types of tabulation:

2.3.1 Simple Tabulation

Tabulation of data according to one characteristic (or variable) is called as simple tabulation.

Tabulation of different high yielding varieties of wheat in a particular state, area under different types of soils are some of the examples.

2.3.2 Double Tabulation

Tabulation of data according to two attributes (or variables) is called double tabulation. For example, tabulation can be done according to crops under irrigated and unirrigated conditions.

2.3.3 Triple Tabulation

Tabulation of data according to three characteristics (or variables) is called triple tabulation.

For example, population tabulated according to sex, literacy and employment.

2.3.4 Manifold Tabulation

Tabulation of data according to more than three characteristics is called manifold tabulation.

EXAMPLE: Tabulated data of students in a college according to native place, class, residence and sex is given in Table 2.2.

TALBLE 2.2 STUDENTS

2.3.5

The following are some of the precautions to be taken in tabulation of data.

(a) The title of the table should be short and precise as far as possible and should convey the general contents of the table.

(b) The sub-titles also should be given so that whenever a part of information is required it can be readily obtained from the marginal totals of the table.

(c) The various items in a table should follow in a logical sequence. For example, the names of the states can be put in an alphabetical order, the crops can be written according to importance on the basis of consumption pattern, the age of students in an ascending order, etc.

(d) Footnotes should be given at the end of a table whenever a word or figure has to be explained more elaborately.

(e) Space should be left after every five items in each column of the table. This will not only help in understanding of the items for comparison but also contributes for the neatness of the table.

EXERCISES

1. Draw up two independent blank tables, giving rows, columns and totals in each case, summarising the details about the members of a number of families, distinguishing males from females, earners from dependants and adults from children.

2. At an examination of 600 candidates, boys outnumber girls by 16 per cent. Also those passing the examination exceed the number of those failing by 310. The number of successful boys choosing science subjects was 300 while among the girls offering arts subjects there were 25 failures. Altogether only 135 offered arts and 33 among them failed. Boys failing the examination numbered 18. Obtain all the class frequencies.

3. In an Agricultural University 1200 teachers are to be classified into 600 Agricultural, 340 Veterinary, 200 Home Science and 60 Agricultural Engineering Faculties. In each Faculty there are three cadres such as Professors, Associate Professors and Assistant Professors and in each Cadre there are three types of activity as teaching, Research and Extension. Draw the appropriate table by filling up the data.

4. Classify the population into Male and Female; Rural and Urban, Employed and unemployed, Private and Government and draw the table by filling up with hypothetical or original data.

CHAPTER 3

FREQUENCY DISTRIBUTION

3.1 FREQUENCY DISTRIBUTION

Frequency may be defined as the number of individuals (or objects) having the same measurement or enumeration count or lies in the same measurement group. Frequency distribution is the distribution of frequencies over different measurements (or measurement groups). The forming of frequency distribution is illustrated here.

EXAMPLE: Below are the heights (in inches) of 75 plants in a field of a paddy crop.

The difference between highest and lowest heights is 40 - 1 = 39.

Supposing that 10 groups are to be formed, the class interval for each class would be 39/10 = 3.9. The groups (or classes) will be formed with a class interval of 4 starting from 1continuing upto 40. The number of plants will be accounted in each class with the help of vertical line called ‘tally mark’ After every fourth tally mark the fifth mark is indicated by crossing the earlier four marks. This procedure is shown in the following Table 3.1.

TABLE 3.1

3.1.1 Inclusive Method of Grouping

The different groups formed in Table 3.1 belong to inclusive method of grouping since both upper and lower limits are included in each class. For example, in the first group, plants having heights 1 and 4 are included in that group itself. The width of each class is called class interval. The midvalue of the class interval is called class mark.

3.1.2 Exclusive Method of Grouping

In this method the upper limit of each group is excluded in that group and included in the next higher group. The inclusive method of grouping in Table 3.1 can be converted to exclusive method of grouping by modifying the classes 1-4, 5-8, 9-12, 13-16…, to 0.5-4.5, 4.5-8.5, 8.5-12.5, 12.5-16.5. However, the class interval in each group in exclusive method increased is 4. Here the upper limit, 4.5 is excluded in the first group and included in the next higher group 4.5-8.5. In other words, plants having. heights between 0.5 to 4.4 are included in the group 0.5-4.5and having heights from 4.5 to 8.4 are included in the next group 4.5 to 8.5 and so on.

3.1.3 Discrete Variable

A variable which can take only fixed number of values is known as discrete variable. In other words, there will be a definite gap between any two values. The number of children per family, the number of petals per flower, the number of tillers per plant, etc., are discrete variables. This variable is also called as ‘discontinuous variable’.

3.1.4 Discrete Distribution

The distribution of frequencies of discrete variable is called discrete distribution. The frequency distribution of plants according to number of tillers is given in Table 3.2.

TABLE 3.2

3.1.5 Continuous Variable

A variable which can assume any value between two fixed limits is known as continuous variable. The height of plant, the weight of an animal, the income of an individual, the yield per hectare of paddy, crop etc., are continuous variables.

3.1.6 Continuous Distribution

The distribution of’ frequencies according to continuous variable is called continuous distribution. For example, the distribution of students according to weights, is given in Table 3.3.

TABLE 3.3

3.2 DIAGRAMMATIC REPRESENTATION

The representation of data with the help of a diagram is called diagrammatic representation.

(i) Bar Diagram: In this diagram, the height of each bar is directly proportional to the magnitude of the variable. The width of each bar and the space between bars should be same.

EXAMPLE: The yearwise data on area under irrigation in a particular state is represented by bar diagram in Fig. 3.1.

TABLE 3.4

Fig. 3.1 Bar diagram.

(ii) Component Bar Diagram: In this case, the heights of the component parts of the bar are directly proportional to the magnitude of the constituent parts of the variable. Here also the width of the bars and the space between the bars should be same. This diagram would not be much of advantage if the component parts are more than three.

EXAMPLE: The area under irrigation in Table 3.4 is further subdivided according to source of irrigation and is presented in Table 3.5.

TABLE 3.5

The component bar diagram representing the data in Table 3.5 are given in Fig. 3.2.

Fig. 3.2 Component bar diagram.

(iii) Multiple Bar Diagram: In this diagram, the height of each bar in a group of bars is directly proportional to the magnitude of individual item in a group of items. For example, yearwise cereal production, sex-wise literacy in different years, election yearwise, number of seats secured by different political parties in a Parliament (or State assembly) can be represented by Multiple bar diagram

EXAMPLE: The following is the data on wages for different categories of agriculture labour in different years.

TABLE 3.6 LABOUR WAGES

The multiple bar diagram representing the data in Table 3.7 is given in Fig. 3.3.

(iv) Pie Diagram: This is also known as Pie-chart. It is useful when the number of component parts of the variable is more than three. Here the areas of different sectors of a circle is directly proportional to the magnitudes of the different component parts of the variable.

Fig. 3.3 Multiple bar diagram.

Let m1 be the magnitude of the first component out of m, the total magnitude of the variable.

Then,

Where θ1 is the angle of a first sector. Similarly θ2, θ3 can be obtained by multiplying 2π with , etc.

After obtaining θ1, θ2, the different sectors can be drawn on a circle each representing the individual component. The radius of the circle is proportional to the total magnitude of the variable.

EXAMPLE: Represent the expenditure of a salaried employee on different items by Pie-diagram. The details are given in Table 3.7.

The Pie-diagram representing the data in Table 3.7 is given in Fig. 3.4.

TABLE 3.7

Fig. 3.4 Pie diagram.

It may be noted that the expenditure on food and house rent is accounted for a major share of the employee’s salary. Also the expenditure targets on different items in five year plans can be purposefully represented by pie diagram. If more than one employee is involved in the above example, as many circles may be drawn representing as many employees with the radius of each circle is proportional to the square root of the total salary.of the corresponding employee.

(v) Pictograms: These are also called as pictorial charts. In this each variable is represented by the corresponding picture and the volume of a picture is directly proportional to the magnitude of the variable. For example, wheat production can be represented by the size of the wheat bag (or wheat ear or the number of wheat bags of the same size) according to particular scale, the size of the army by the size of the soldier (or soldiers of same size), the strength of navy by the size of battleship (or the battle ships of same size), number of tractors by the size of tractor (or the tractors of same size) according to particular scale, etc.

Advantages: A diagram is always more appealing to eye than mere numerical data. It is easy for making comparisons and contrasts when more than one diagram is involved. It is easy to understand even for a layman.

Disadvantages: The main disadvantage of this representation is that it only gives rough idea of the variable but not the exact value. Also whenever the number of items are more it is difficult to depict on the diagrams since they require more space, time and un weildy for comparison.

3.3 GRAPHIC REPRESENTATION

Just as in the case of diagrammatic representation, here different methods of graphic representation are presented.

3.3.1 Histogram

It consists of rectangles erected with bases equal to class intervals of frequency distribution and heights of rectangles are proportional to the frequencies of the respective classes in such a way that the areas of rectangles are directly proportional to the corresponding frequencies.

EXAMPLE: Represent the following frequency distribution of farms according to area in a particular village by a histogram given in Table 3.8

From Fig. 3.5 one can infer that the maximum number of farms are lying in the group (2-4) and the minimum number in the group (10-12). The total area under the histogram is equal to the total frequency.

TABLE 3.8

Fig. 3.5 Histogram.

3.3.2 Frequency Polygon

If the points are plotted with midvalues of the class intervals on the X-axis and the corresponding frequencies on the Y-axis, the figure obtained by joining these points with the help of a scale is known as frequency polygon.

EXAMPLE: The frequency polygon for the data given in Table 3.8 is as follows.

Fig. 3.6 Frequency polygon.

The frequency polygon in Fig. 3.6 is drawn with the assumption that the frequencies are concentrated at the midvalues of the corresponding classes. It may be noted that the area under histogram is equal to the area under frequency polygon.

3.3.3 Frequency Curve

If the points are plotted with midvalues of the class intervals on X-axis and the corresponding frequencies on Y-axis, the figure formed by joining these points with a smooth hand is known as frequency curve.

EXAMPLE: The frequency curve for the example given in Table 3.8 is given in Fig. 3.7.

Fig. 3.7 Frequency curve.

3.3.4 Cumulative Frequency Curve (ogive)

If the points are plotted with upper limits of classes on X-axis and the, corresponding cumulative frequencies (less than) on Y-axis, the figure formed by joining these points with a smooth hand is known as cumulative frequency curve (less than). If the lower limits of classes are taken on X-axis and the corresponding cumulative frequencies (greater than) on Y-axis, the curve so obtained is called cumulative frequency curve (greater than).

EXAMPLE: Represent the distribution of rainfall on different days from July to September months in a particular locality and in a particular year by cumulative frequency curves.

TABLE 3.9

Fig. 3.8

The X-co-ordinate of the point of intersection of two cumulative frequency curves is the median value. The Y-co-ordinate will correspond to median no. i.e., where N is the total frequency. From the Fig. 3.8 first quartile and third quartile can be obtained from X-axis for the corresponding values (N+1)/4 and respectively on Y-axis. The reader is advised to refer Sections 4.2 and 5.2 respectively for definitions of median and quartiles.

3.3.5 Lorenz Curve

It is the curve drawn between two variates which are expressed in percentage cumulative frequencies. This curve is useful to depict the income distribution’ of individuals where cumulative percentage of individuals are taken on the X-axis and the corresponding cumulative percentage of incomes are taken on the Y-axis. This is commonly used in graphic representation of the inequality aspect of the income distribution. This is due to Italian statisticians. Gini and Lorenz. This curve can also be used for the distribution of any non-negative variate, with a continuous type of distribution as for example, for the distribution of factories by capital size, (or number of employees), etc. The equality of the income distribution is depicted as a straight line drawn with 45°connecting the two diagonal points, and which is known as ‘egalitarian line’. If the income distribution is not even then the egalitarian line will take a curve shape. This curve is called ‘Lorenz curve’, If Lorenz curve is closer towards ‘egalitarian line’ there is less of inequality of income distribution. If the Lorenz curve is away from the ‘egalitarian line’ there is more of inequality of income distribution.

Fig. 3.9 Lorenz curve

From Fig. 3.9, it can be inferred that the distribution of income in year y2has tended towards equality in comparison to the year y1.

Since Ay2 Lorenz curve and egalitarian line and B is the area between y1 and egalitarian line as shown in. Fig. 3.9. The procedure for finding out the area A or B is given in the following subsection of ‘Fitting of Lorenz Curve’. If any curve coincides with the ‘egalitarian line then the area would become zero and the Gini’s concentration’ ratio would be zero.

3.3.6 Fitting of Lorenz Curve

"The approximate procedure of fitting Lorenz curve as will as the method of finding out the area between Lorenz curve and egalitarian line is given here.

TABLE 3.10

Let Δ be the area of the trepezium between Lorenz curve and the X-axis in Fig. 3.10.

Fig. 3.10 Lorenz curve.

The area between the ‘Lorenz curve’ "and ‘egalitarian line’ can be obtained by subtracting Δ from 0.5.

Area of trepezium,

Area between ‘Lorenz curve’ and ‘egalitarian line’ is

It may be noted that the above method is an approximate one for finding out the area of trepezium.

EXAMPLE: The following is the distribution of income of different staff in an educational institution. Represent the data by Lorenz curve and also find the proportionate number of persons having income upto 20 per cent.

TABLE 3.11

Fig. 3.11 Egalitarian line.

From Fig, 3.11, the proportion of persons having income upto 20 per cent is 60 per cent.

3.3.7 Remarks

The graphic representation generally depicts the trend when the number of observations is large. Also it provides intermediary values, though roughly.

EXERCISES

1. The following is the distribution of heights of plants of a particular crop.

Draw the (i) Histogram (ii) Frequency Polygon (iii) Frequency curve and (iv) ogive.

2. Draw the histogram of the following distribution of marriages classified according to the age of the bridegroom, and give your comments.

(B.Sc. Madras, April, 1969)

3. Draw the ‘Ogive’ for the following frequency distribution.

4. The following are the data regarding the area under grape cultivation in different years.

Represent the above data by a suitable diagram.

5. Represent the following data by a bar diagram and comment on their relationship. (B.Sc Madras, Sept., 1969)

6. Represent the following data by sub-divided bar diagram drawn on the percentage basis. (B.Sc. Madras, April, 1969)

7. The data given below relates to the income of workers’ families in an Industrial area

Draw a Lorenz curve to represent the data and determine there from, what percentage of the total income of the working classes is earned by the highly paid 25 per cent of the families.

(B.Sc. Madras, Aprtl.1967)

8. The expenditure pattern of¹ two cultivators on one hectare farm for different items of agricultural inputs and the corresponding sector angles are given in the following table.

CHAPTER 4

MEASURES OF LOCATION

It is always advisable to represent group of data by a single observation provided it does not loose any important information contained in the data and brings out every important information from it. This single value, which represents the group of values, is termed as a ‘measure of central tendency’ (or a measure of location or an ‘average’). This should be a representative value or a typical member of the group. The different measures of location are 1. Arithmetic mean, 2. Median, 3. Mode, 4. Geometric Mean and 5. Harmonic mean.

4.1 ARITHMETIC MEAN

It is defined as the sum of the observations divided by its number.

Let X1, X2 ,Xn be n observations then the Arithmetic mean (A.M), is defined as which can be written as , where ‘Σ’ is the summation which indicates the summing up of the observations from X1 to Xn.

EXAMPLE: Compute the mean daily milk yield of a buffalo given the following milk yields (in kgs) for the consecutivelO days.

15, 18, 16, 9, 13, 20, 16, 17, 21, 19

4.1.1 Linear Transformation Method

If the observation values are large, more in number and the deviation among themselves is small, the linear transformation method will save time in computation.

Let di = Xi - A where A is called arbitrary mean and which is taken as round figure mid way between highest and lowest values.

For the above example, let A = 15

TABLE 4.1