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Practical Manual Statistical Methods

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Practical Manual Statistical Methods PRACTICAL MANUAL STATISTICAL METHODS ( UG COURSE) Compiled by DEPARTMENT OF MATHEMATICS AND STATISTICS Jawaharlal Nehru Krishi Vishwa Vidyalaya, JABALPUR 482 004 1 2 Contents S. Chapter Name Description Page No. No. 1 Graphical Representation of 1. Construction of Discrete and continuous 1-8 data frequency distribution 2. Construction of Bar Diagram, Histogram, Pie Diagram, Frequency curve and Frequency polygon 2 Measures of Central tendency 1. Definition, Formula and Calculation of Mean, 9-21 Median , Mode, Geometric Mean and Harmonic Mean for grouped and ungrouped data 2. Definition, Formula and Calculation of Quartiles, Deciles and Percentiles for grouped and ungrouped data 3 Measures of Dispersion 1. Definition, Formula and Calculation of 22-29 absolute measures of Dispersion, Range, Quartile Deviation, Mean Deviation, Standard Deviation 2. Definition, Formula and Calculation of relative measures of Dispersion, CD and CV for grouped and ungrouped data 4 Moments, Skewness and 1. Definition and types of moments, skewness 30 -40 Kurtosis and Kurtosis 2. Formula and calculation of raw moments, moments about origin, central moments and different types of coefficient of skewness and kurtosis 5 Correlation and Regression 1. Definition and types of Correlation and 41-49 Regression. 2. Calculation of Correlation and regression coefficient along with their test of significance. 6 Test of Significance 1. Definition of Null and Alternative Hypothesis 50-59 and different tests of significance 2. Application of t test for single mean, t-test for independent samples, paired t test, F-test, Chi-square test 7 Analysis of Variance( One way 1. Definition and steps of analysis of one way 60-79 and Two way classification) and two way classification. 2. Analysis of CRD and RBD as an example of one way and two way ANOVA 8. Sampling Methods 1. Definition of SRS, SRSWR and SRSWOR and 80-86 difference between census and sampling 2. Procedures of selecting a simple random sample 3 4 1. Graphical Representation of data Mujahida Sayyed Asst. professor (Maths & Stat.), College of Agriculture, JNKVV, Ganjbasoda, 464221(M.P.), India Email id : [email protected] Frequency Distribution: A tabular presentation of the data in which the frequencies of values of a variable are given along with class is called a frequency distribution. Two types of frequency distribution are available 1. Discrete Frequency Distribution: A frequency distribution which is formed by distinct values of a discrete variable eg. 1,2,5 etc. 2. Continuous Frequency Distribution: A frequency distribution which is formed by distinct values of a continuous variable eg. 0-10, 10-20, 20-30 etc. Process: For construction of Discrete Frequency Distribution Step I. Set the data in ascending order. Step II. Make a blank table consisting of three columns with the title: Variable, Tally Marks and Frequency. Step III. Read off the observations one by one from the data given and for each one record a tally mark against each observation. In tally marks for each variable fifth frequency is denoted by cutting the first four frequency from top left to bottom right and then sixth frequency is again by a straight tally marks and so on. Step IV. In the end, count all the tally marks in a row and write their number in the frequency column. Step V. Write down the total frequency in the last row at the bottom. Objective : Prepare a discrete frequency distribution from the following data Kinds of data: 5 5 2 6 1 5 2 9 5 4 3 4 11 7 2 5 12 6 Solution : First arrange the data in ascending order 1 2 2 2 3 4 4 5 5 5 5 5 6 6 7 9 11 12 Prepare a table in the format described above in the process. Count the numbers by tally method we get the required discrete frequency distribution: No. of Letters, Variable (X) Tally Marks No. of Words, Frequency(f) 1 │ 1 2 │││ 3 3 │ 1 4 ││ 2 5 ││││ 5 6 ││ 2 7 │ 1 9 │ 1 11 │ 1 12 │ 1 Total 18 1 Continuous Frequency Distribution: A continuous frequency distribution i.e. a frequency distribution which obtained by dividing the entire range of the given observations on a continuous variable into groups and distributing the frequencies over these groups . It can be done by two methods 1. Inclusive method of class intervals : When lower and upper limit of a class interval are included in the class intervals. 2. Exclusive method of class intervals: When the upper limit of a class interval is equal to the lower limit of the next higher class intervals. Process: For construction of Continuous Frequency Distribution Step I. Set the data in ascending order. Step II. Find the range= max value –min value. Step III. Decide the approximate number k of classes by the formula K= 1+3.322 log10N, where N is the total frequency. Round up the answer to the next integer. After dividing the range by number of classes class interval is obtained. Step IV. Classify the data by exclusive and/or inclusive method for the desired width of the class intervals. Step V. Make a blank table consisting of three columns with the title: Variable, Tally Marks and Frequency. Step VI. Read off the observations one by one from the data given and for each one record a tally mark against each observation. Step VII. In the end, count all the tally marks in a row and write their number in the frequency column. Step VIII. Write down the total frequency in the last row at the bottom. ******************************************************************************** Objective : Prepare a continuous grouped frequency distribution from the following data. Kinds of data: 20 students appear in an examination. The marks obtained out of 50 maximum marks are as follows: 5, 16, 17, 17, 20, 21, 22, 22, 22, 25, 25, 26, 26, 30, 31, 31, 34, 35, 42 and 48. Prepare a frequency distribution taking 10 as the width of the class-intervals . Solution: Arrange the data in the ascending order 5 16 17 17 20 21 22 22 22 25 25 26 26 30 31 31 34 35 42 48 Here lower limit is 5 and upper limit is 48. Since it is given that the desired class interval is 10, so frequency distribution for Inclusive Method of Class intervals: Marks Tally Marks No. of students 1-10 │ 1 11-20 ││││ 4 21-30 │││││││ │ 9 31-40 ││││ 4 41-50 ││ 2 Total 20 2 Exclusive Method of Class intervals: Marks Tally Marks No. of students 0-10 │ 1 10-20 │││ 3 20-30 │││││││ │ 9 30-40 │││ │ 5 40-50 ││ 2 Total 20 ******************************************************************************** Conversion of Inclusive series to Exclusive series: To apply any statistical technique (Mean, Median etc.) , first the inclusive classes should be converted to exclusive classes. 푙표푤푒푟 푙푚푡 표푓 푠푒푐표푛푑 푐푙푎푠푠−푢푝푝푒푟 푙푚푡 표푓 푓푟푠푡 푐푙푎푠푠 For this purpose we find the difference of and add 2 this amount to upper limit of first class and subtract it from the lower limit of next higher class. ퟏퟏ−ퟏퟎ In the present example the conversion factor = = 0.5. So we add 0.5 to 10 and subtract 0.5 ퟐ from 11 and finally get the exclusive classes 1-10.5, 10.5-20.5, etc. ******************************************************************************** Graphical Representation of data:- Graphical Representation is a way of analysing numerical data. It exhibits the relation between data, ideas, information and concepts in a diagram. It is easy to understand and it is one of the most important learning strategies. It always depends on the type of information in a particular domain. There are different types of graphical representation. Some of them are as follows • Bar Diagram – Bar Diagram is used to display the category of data and it compares the data using solid bars to represent the quantities. • Histogram – The graph that uses bars to represent the frequency of numerical data that are organised into intervals. Since all the intervals are equal and continuous, all the bars have the same width. • Pie diagram–Shows the relationships of the parts of the whole. The circle is considered with 100% and the categories occupied is represented with that specific percentage like 15%, 56% , etc. • Frequency Polygon – It shows the frequency of data on a given number to curve. • Frequency curve - Frequency curve is a graph of frequency distribution where the line is smooth. Merits of Using Graphs Some of the merits of using graphs are as follows: • The graph is easily understood by everyone without any prior knowledge. • It saves time. • It allows to relate and compare the data for different time periods • It is used in statistics to determine the mean, median and mode for different data, as well as in interpolation and extrapolation of data. 3 1. Simple Bar Diagram: Bar graph is a diagram that uses bars to show comparisons between categories of data. The bars can be either horizontal or vertical. Bar graphs with vertical bars are sometimes called vertical bar graphs. A bar graph will have two axes. One axis will describe the types of categories being compared, and the other will have numerical values that represent the values of the data. It does not matter which axis is which, but it will determine what bar graph is shown. If the descriptions are on the horizontal axis, the bars will be oriented vertically, and if the values are along the horizontal axis, the bars will be oriented horizontally. Objective : Prepare a simple Bar diagram for the given data: Kinds of data: Aggregated figures for merchandise export in India for eight years are as Follows. Years 1971 1972 1973 1974 1975 1976 1977 1978 Exports (million Rs.) 1962 2174 2419 3024 3852 4688 5555 5112 Solution: For Simple Bar Diagram Step I: Draw X and Y axis.
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