Previous Year Question Papers - BNM.pptx

Business Numerical Methods

1. Previous Year Question Papers - Business Numerical Methods Dr. Nimisha Nandan

2. Matrix Matrix is a rectangular array of numbers arranged in rows and columns . An array is a collection of items arranged at different locations.  A matrix is identified by its order . The order of the matrices is given in the form of number of rows ⨯ number of columns The items contained in a matrix are called Elements of the Matrix . The location of each element is given by the row and column it belongs to.

3. A matrix is represented as [P] m⨯n where P is the matrix, m is the number of rows and n is the number of columns . Let’s say a matrix has 3 rows and 2 columns then the order of the matrix is given as 3⨯2.

4. Transpose of Matrix Transpose of matrix is basically the rearrangement of row elements in column and column elements in a row to yield an equivalent matrix. A matrix in which the elements of the row of the original matrix are arranged in columns or vice versa is called Transpose Matrix. The transpose matrix is represented as A T . It is especially useful in applications where inverse and adjoint of matrices are to be obtained.

5. Trace of a Matrix Trace of a Matrix is the sum of the principal diagonal elements of a square matrix ( that is, matrices with an equal number of rows and columns). It is usually represented as tr (A), where A is any square matrix of order “n × n.” tr (A) = a 11 + a 22 + a 33 + …+ a nn  

6. Order of Transpose Matrix The order of a matrix represents the number of rows and columns in a given matrix. The horizontal lines of the elements are all called the rows of the matrix which is denoted by n, and the vertical lines of the elements are called the columns of the matrix which is denoted by m. Together, they represent the order of a matrix, which is written as n × m. And the order of the transpose of the given matrix is written as m x n.

7. Types of Matrices Based on the number of rows and columns present and the special characteristics shown, matrices are classified into various types . Row Matrix : A Matrix in which there is only one row and no column is called Row Matrix . The elements are arranged in a horizontal manner, and the order of a row matrix is 1 x n. A row matrix, A = [a, b, c, d] has only one row and can have numerous columns, which are equal to the number of elements in the row.

8. Column Matrix: A Matrix in which there is only one column and now row is called a Column Matrix . The elements are arranged in a vertical manner, and the order of a column matrix is n x 1. A column matrix has only one column and can have numerous rows, which are equal to the number of elements in the column.

9. Horizontal Matrix : The horizontal matrix is of the order m×n with the condition that m×n . The number of rows in a horizontal matrix is smaller than the number of columns.

10. Vertical Matrix: A Matrix in which the number of columns is less than the number of rows is called a Vertical Matrix . The basic property of a vertical matrix is that the number of rows in the vertical matrix is more than the number of columns present. A vertical matrix will always be a rectangular matrix. As the condition for a vertical matrix is that the number of rows is greater than the number of columns, a horizontal matrix could never be a square matrix. The transpose of a vertical matrix is a horizontal matrix. While calculating the transports of a matrix, the number of rows is interchanged with the number of columns, the number of rows will decrease, and the number of columns will increase in the transposition of the matrix. Hence, the transpose will be a horizontal matrix. As the number of rows and the number of columns of a vertical matrix will always differ, a vertical matrix will not have properties like a determinant of diagonal elements.

11. Rectangular Matrix: A rectangular matrix is a matrix that is rectangular in shape. If the number of rows in a matrix is not equal to the number of columns in it then the matrix is known as the rectangular matrix Square Matrix A matrix in which the number of rows and columns are the same is called a Square Matrix.

12. Upper Triangular Matrix: A square matrix in which all the elements below the diagonal are zero is known as the upper triangular matrix. Lower Triangular Matrix: A square matrix in which all the elements above the diagonal are zero is known as the lower triangular matrix. Identity Matrix is the matrix which is n × n square matrix where the diagonal consist of ones and the other elements are all zeros. It is also called as a Unit Matrix or Elementary matrix. It is represented as I n  or just by I, where n represents the size of the square matrix.

13. Diagonal Matrix: A square matrix in which every element except the principal diagonal elements is zero is called a Diagonal Matrix. A square matrix D = [ d ij ] n x n will be called a diagonal matrix if d ij = 0, whenever i is not equal to j. Idempotent Matrix: An idempotent matrix is one that when multiplied by itself produces the same matrix . A = A. A is a n × n square matrix. As a result, an idempotent matrix is one that does not change when multiplied by itself . Zero or Null Matrix : A zero matrix is a matrix that has all its elements equal to zero . Since a zero matrix contains only zeros as its elements, therefore, it is also called a null matrix. A zero matrix can be a square matrix. A zero matrix is denoted by 'O'.

14. Arithmetic Mean In statistics, the Arithmetic Mean (AM) or called average is the ratio of the sum of all observations to the total number of observations. The sum of deviations of the items from their arithmetic mean is always zero, i.e. ∑(x – X) = 0. The sum of the squared deviations of the items from Arithmetic Mean (A.M) is minimum, which is less than the sum of the squared deviations of the items from any other values. If each item in the arithmetic series is substituted by the mean, then the sum of these replacements will be equal to the sum of the specific items.

15. Symmetric matrix: A square matrix is said to be symmetric if the transpose of the original matrix is equal to its original matrix. A square matrix A is said to be symmetric if  a ij  =  a ji  for all i and j, where  a ij is an element present at ( i,j ) th position ( i th row and j th  column in matrix A) and a ji is an element present at ( j,i ) th position ( j th row and i th   column in matrix A). In other words, we can say that matrix A is said to be symmetric if the transpose of matrix A is equal to matrix A itself (A T =A).

16. Skew-symmetric Matrix: In mathematics, a skew symmetric matrix is defined as the square matrix that is equal to the negative of its transpose matrix. For any square matrix, A, the transpose matrix is given as A T . A skew-symmetric or antisymmetric matrix A can therefore be represented as, A = -A T . A skew-symmetric matrix finds application in various fields, such as, in machine learning and in statistical analysis.

17. Continuous Series In a continuous series, data-items are kept in certain definite classes. When kept in classes the data-items lose their individual identity, and individual items merge in one or the other class group. The classes thus created, have continuity, since where one class ends, the next class begins from there. Due to this class continuity, such a series is called a continuous series . Continuous series is used the most when number of data-items is more and their magnitude is also more .

18. Singular Matrix A square matrix is said to be a singular matrix if its determinant is zero i.e. |A|= The inverse of a singular matrix is NOT defined and hence it is non-invertible. A null matrix of any order is a singular matrix. The rank of a singular matrix is definitely less than the order of the matrix. For example, the rank of a 3x3 matrix is less than 3. All rows and columns of a singular matrix are NOT linearly independent.

19. Non-singular matrix A non-singular matrix is a square one whose determinant is not zero. The rank of a matrix [A] is equal to the order of the largest non-singular submatrix of [A]. It follows that a non-singular square matrix of n × n has a rank of n. Thus, a non-singular matrix is also known as a full rank matrix.

20. Singular Matrix Non Singular Matrix A square matrix is said to be a singular matrix if its determinant is zero, i.e., det A = 0. A square matrix is said to be a non-singular matrix if its determinant is zero, i.e., det A ≠ 0. If a matrix is singular, then its inverse is not defined. If a matrix is non-singular, then its inverse is defined. The rank of a singular matrix will be less than the order of the matrix, i.e., Rank (A) < Order of A. The rank of a non-singular matrix will be equal to the order of the matrix, i.e., Rank (A) = Order of A. In a singular matrix, some rows and columns are linearly dependent. In a non-singular matrix, all the rows and columns are linearly independent.

21. Geometric Mean The Geometric Mean (GM) is the average value or mean which signifies the central tendency of the set of numbers by finding the product of their values. In other words, the geometric mean is defined as the nth root of the product of n numbers It can be found by multiplying all the numbers in the given data set and take the nth root for the obtained result.

22. Application of Geometric Mean It is used in stock indexes. Because many of the value line indexes which is used by financial departments use G.M. It is used to calculate the annual return on the portfolio. It is used in finance to find the average growth rates which are also referred to the compounded annual growth rate. It is also used in studies like cell division and bacterial growth etc.

23. Harmonic Mean The harmonic mean is a numerical average calculated by dividing the number of observations, or entries in the series, by the reciprocal of each number in the series. Thus, the harmonic mean is the reciprocal of the arithmetic mean of the reciprocals . The harmonic mean helps to find multiplicative or divisor relationships between fractions without worrying about common denominators. Harmonic means are often used in averaging things like rates.

24. Range Range is a fundamental statistical concept to understand the spread or variability of data within a dataset. Range in Statistics provides valuable insights into the extent of variation among the values in a dataset. Range in statistics is the difference between the highest and lowest values in a dataset. The range statistic is simple and straightforward to calculate, but it has limitations because it only takes into consideration the maximum and minimum values and ignores the distribution of values across the dataset . Range = Maximum Value – Minimum Value

25. Coefficient of Range The coefficient of range on the other hand is the ratio of difference between the highest and lowest value of frequency to the sum of highest and lowest value of frequency. The coefficient of the range is a relative measure of dispersion.

26. Quartiles A median divides a given dataset (which is already sorted) into two equal halves similarly, the quartiles are used to divide a given dataset into four equal halves.

27. Quartile deviation Quartile deviation is a statistic that measures the deviation. It measures the deviation of the data from the average value. Quartile deviation is a statistic that measures the deviation in the middle of the data. Quartile deviation is also referred to as the semi interquartile range and is half of the difference between the third quartile and the first quartile value. The formula for quartile deviation of the data is Q.D = (Q3 - Q1)/2 .

28. Minor of a matrix Minor of a matrix for an element is given by the determinant of a matrix obtained after deleting the row and column to which the particular element belongs to. Minor of Matrix is represented by M ij

29. Cofactor of a matrix Cofactor of a matrix is found by multiplying the minor of the matrix for a given element by (-1) i+j . Cofactor of a Matrix is represented as C ij . Hence, the relation between the minor and cofactor of a matrix is given as C ij = (- 1) i+j M ij .

30. Time value of money The time value of money (TVM) is the concept that a sum of money is worth more now than the same sum will be at a future date due to its earnings potential in the interim. The time value of money is a core principle of finance. A sum of money in the hand has greater value than the same sum to be paid in the future. The time value of money is also referred to as the present discounted value. The formula for computing the time value of money considers the amount of money, its future value, the amount it can earn, and the time frame. For savings accounts, the number of compounding periods is an important determinant as well.

31. Future value (FV) Future value (FV) is the value of a current asset at a future date based on an assumed rate of growth. The future value is important to investors and financial planners, as they use it to estimate how much an investment made today will be worth in the future.

32. C o-efficient of variation The co-efficient of variation shows the extent of variability of data in a sample in relation to the mean of the population. The co-efficient of variation (CV) is a statistical measure of the dispersion of data points in a data series around the mean. The co-efficient of variation represents the ratio of the standard deviation to the mean, and it is a useful statistic for comparing the degree of variation from one data series to another, even if the means are drastically different from one another . Coefficient of variation is a type of relative measure of dispersion. It is expressed as the ratio of the standard deviation to the mean.

33. A ssumed mean In statistics, the assumed mean method is used to calculate mean or arithmetic mean of a grouped data. If the given data is large, then this method is recommended rather than a direct method for calculating mean. This method helps in reducing the calculations and results in small numerical values. This method depends on estimating the mean and rounding to an easy value to calculate with. Again this value is subtracted from all the sample values. When the samples are converted into equal size ranges or class intervals, a central class is chosen and the computations are performed.

34. The co-efficient of variation (CV) is a statistical measure of the relative dispersion of data points in a data series around the mean. It represents the ratio of the standard deviation to the mean. The CV is useful for comparing the degree of variation from one data series to another, even if the means are drastically different from one another.

35. Skewness “Skewness essentially is a commonly used measure in descriptive statistics that characterizes the asymmetry of a data distribution, while kurtosis determines the heaviness of the distribution tails.” Understanding the shape of data is crucial while practicing data science. It helps to understand where the most information lies and analyze the outliers in a given data. In this article, we’ll learn about the shape of data, the importance of skewness, and kurtosis in statistics.

36. A distribution is positively skewed when its tail is more pronounced on the right side than it is on the left . Since the distribution is positive, the assumption is that its value is positive. As such, most of the values end up being left of the mean. This means that the most extreme values are on the right side.

37. Skewness is a statistical measure that assesses the asymmetry of a probability distribution. It quantifies the extent to which the data is skewed or shifted to one side. Positive skewness indicates a longer tail on the right side of the distribution, while negative skewness indicates a longer tail on the left side. Skewness helps in understanding the shape and outliers in a dataset. If the values of a specific independent variable (feature) are skewed, depending on the model, skewness may violate model assumptions or may reduce the interpretation of feature importance.

38. Geometric Progression Geometric Progression (GP) is a type of sequence where each succeeding term is produced by multiplying each preceding term by a fixed number, which is called a common ratio. This progression is also known as a geometric sequence of numbers that follow a pattern.

39. Linear Equations A linear equation is an equation in which the highest power of the variable is always 1. It is also known as a one-degree equation. The standard form of a linear equation in one variable is of the form Ax + B = 0. Here, x is a variable, A is a coefficient and B is constant. The standard form of a linear equation in two variables is of the form Ax + By = C. Here, x and y are variables, A and B are coefficients and C is a constant.

40. Index numbers Index numbers can provide insights into changes in data sets over time. They can be used to identify trends and compare the performance of variables and groups of variables between years or groups of years. The analysis of trends and performance with index numbers is commonly used in fields, e.g., business, economics, and policy making . A statistical measure that helps in finding out the percentage change in the values of different variables, such as the price of different goods, production of different goods, etc., over time is known as the Index Number. The percentage change is determined by taking a base year as a reference. This base year is the year of comparison. When an investigator studies different goods simultaneously, then the percentage change is considered the average for all the goods.

41. " Index number is an economical barometers." Explain. An index number is a number that expresses the relative change in magnitude of a variable or number of variables during a specified period. The variable may be the price of a certain commodity, the quantitative production of certain goods, or the cost of living. It is a statistical device to measure the level of certain phenomena in comparison with a certain period known as the base period, which may be a week month, year, or group of years. The index numbers are known as economic barometers or economic indicators since they help in understanding the changes in economic conditions of the society.

42. T ime series A time series is a collection of observations of well-defined data items obtained through repeated measurements over time. For example, measuring the value of retail sales each month of the year would comprise a time series. This is because sales revenue is well defined, and consistently measured at equally spaced intervals. Data collected irregularly or only once are not time series. An observed time series can be decomposed into three components: the trend (long term direction), the seasonal (systematic, calendar related movements) and the irregular (unsystematic, short term fluctuations).

43. Components for Time Series Analysis Trend : The trend shows the general tendency of the data to increase or decrease during a long period of time. A trend is a smooth, general, long-term, average tendency. It is not always necessary that the increase or decrease is in the same direction throughout the given period of time. Seasonal Variations : These are the rhythmic forces which operate in a regular and periodic manner over a span of less than a year. They have the same or almost the same pattern during a period of 12 months. This variation will be present in a time series if the data are recorded hourly, daily, weekly, quarterly, or monthly.

44. Cyclic Variations : The variations in a time series which operate themselves over a span of more than one year are the cyclic variations. This oscillatory movement has a period of oscillation of more than a year. One complete period is a cycle. This cyclic movement is sometimes called the ‘Business Cycle’. Random or Irregular movements : There is another factor which causes the variation in the variable under study. They are not regular variations and are purely random or irregular. These fluctuations are unforeseen, uncontrollable, unpredictable, and are erratic. These forces are earthquakes, wars, flood, famines, and any other disasters.

45. Mathematically, a time series is given as yt = f (t ) Additive Model for Time Series Analysis : y t = T t + S t + C t + R t . Multiplicative Model for Time Series Analysis : y t = T t × S t × C t × R t

46. Statistics Statistics is a branch that deals with every aspect of the data. Statistical knowledge helps to choose the proper method of collecting the data and employ those samples in the correct analysis process in order to effectively produce the results. In short, statistics is a crucial process which helps to make the decision based on the data.

47. Types of Statistics The two main branches of statistics are: Descriptive Statistics Inferential Statistics Descriptive Statistics – Through graphs or tables, or numerical calculations, descriptive statistics uses the data to provide descriptions of the population. Inferential Statistics – Based on the data sample taken from the population, inferential statistics makes the predictions and inferences.

48. Importance of Statistics The important functions of statistics are: Statistics helps in gathering information about the appropriate quantitative data It depicts the complex data in graphical form, tabular form and in diagrammatic representation to understand it easily It provides the exact description and a better understanding It helps in designing the effective and proper planning of the statistical inquiry in any field It gives valid inferences with the reliability measures about the population parameters from the sample data It helps to understand the variability pattern through the quantitative observations

49. Functions of Statistics Statistics simplifies complexity Statistics presents fact in a definite form Statistics facilities comparison To help in formulation of policies Statistics helps in forecasting Statistics helps in formulating and testing hypothesis

50. Limitations of Statistics Statistics does not deal with individuals Statistics does not study qualitative phenomena Statistical laws are not exact Statistics is only a means Statistics is liable to be misused

51. Sequence A sequence is an arrangement of any objects or a set of numbers in a particular order followed by some rule. If a1, a2, a3, a4,……… etc. denote the terms of a sequence, then 1,2,3,4,…..denotes the position of the term. A sequence can be defined based on the number of terms i.e. either finite sequence or infinite sequence . An itemized collection of elements in which repetitions of any sort are allowed is known as a sequence, whereas a series is the sum of all elements. An arithmetic progression is one of the common examples of sequence and series. In short, a sequence is a list of items/objects which have been arranged in a sequential way.

52. Types of Sequence Some of the most common examples of sequences are: Arithmetic Sequences : A sequence in which every term is created by adding or subtracting a definite number to the preceding number is an arithmetic sequence . If “a” is the first term and “d” is the common difference of an arithmetic sequence, then it is represented by a, a+d , a+2d, a+3d, …

53. Geometric Sequences : If “a” is the first term and “r” is the common ratio of a geometric sequence, then the geometric sequence is represented by a, ar , ar 2 , ar 3 , …., ar n-1 , .. In a Geometric Sequence each term is found by multiplying the previous term by a constant . A geometric progression or a geometric sequence is the sequence, in which each term is varied by another by a common ratio . The next term of the sequence is produced when we multiply a constant (which is non-zero) to the preceding term.

54. Series A series is defined as the sum of the elements of a sequence . A series may contain a number of terms in the form of numerical, functions, quantities, etc. When the series is given, it indicates the symbolised sum, not the sum itself. For example, 2 + 4 + 6 + 8 + 10 + 12 is a series with six terms. To find the sum of these numbers, we use the phrase “sum of a series”, which means the number that results from adding the terms of the series, the sum of the series is 42 . A series with an infinite number of terms is called an infinite series.

55. A series is called convergent or summable if this limit exists, which means the sequence is summable . Otherwise, the series is called divergent series. In the above representation, the limit is called the sum of the series . The arithmetic series is represented by a + ( a+d ) + (a+2d) + (a+3d) + … The geometric series is represented by a + ar + ar2 + ar3 + ….+ arn-1+ ..

56. H armonic sequence In algebra, a harmonic sequence, sometimes called a harmonic progression, is a sequence of numbers such that the difference between the reciprocals of any two consecutive terms is constant. In other words, a harmonic sequence is formed by taking the reciprocals of every term in an arithmetic sequence . The harmonic sequence in mathematics can be defined as the reciprocal of the arithmetic sequence with numbers  other than 0. The sum of harmonic sequences is known as harmonic series. It is an infinite series that never converges to a limit. For example, let’s take an arithmetic sequence as 5, 10, 15, 20, 25,... with the common difference of 5. Then its harmonic sequence is: 1/5, 1/10, 1/15,1/20,1/25….

57. Difference Between Sequences and Series Sequences Series Set of elements that follow a pattern Sum of elements of the sequence Order of elements is important Order of elements is not so important Finite sequence: 1,2,3,4,5 Finite series: 1+2+3+4+5 Infinite sequence: 1,2,3,4,…… Infinite Series: 1+2+3+4+……

58. EMI An equated monthly installment (EMI) is a fixed payment amount made by a borrower to a lender at a specified date each calendar month. Equated monthly installments are applied to both interest and principal each month so that over a specified number of years, the loan is paid off in full. In the most common types of loans—such as real estate mortgages, auto loans, and student loans—the borrower makes fixed periodic payments to the lender over several years to retire the loan.

59. An equated monthly instalment (EMI) is a fixed payment made by a borrower to a lender on a specified date of each month. EMIs are applied to both interest and principal each month so that over a specified time period, the loan is paid off in full. EMIs can be calculated in two ways: the flat-rate method or the reducing-balance method. The EMI reducing-balance method generally is more favorable for borrowers, as it results in lower interest payments overall. EMIs allow borrowers the peace of mind of knowing exactly how much money they will need to pay each month toward their loan.

60. The EMI reducing-balance method is calculated using this formula: EMI = P * [( r * (1 + r)^n)) / ((1 + r)^n - 1)] where: P = Princiapl amount borrowed r = Periodic monthly interest rate n = Total number of monthly payments

61. Dispersion The field of statistics is used across every sector and industry to help people better understand, and predict, potential outcomes . In finance, investors often turn to statistics to gain a sense of how returns on certain assets, or groups of assets, could be distributed. This range of possible investment returns is called dispersion. Dispersion is often interpreted as a measure of the degree of uncertainty, and thus risk, associated with a particular security or investment portfolio. Descriptive statistics is a means of using summaries of a data sample to describe features of a larger data set. For example, a population census may include descriptive statistics regarding the ratio of men and women in a specific city.

62. What are the major measures of central tendency ? List out its merit and limitations of each measure ? (ESSAY) A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. The mean, median and mode are all valid measures of central tendency.

63. Mean The mean (or average) is the most popular and well known measure of central tendency . It can be used with both discrete and continuous data, although its use is most often with continuous data. The mean is equal to the sum of all the values in the data set divided by the number of values in the data set . An important property of the mean is that it includes every value in your data set as part of the calculation. In addition, the mean is the only measure of central tendency where the sum of the deviations of each value from the mean is always zero. It is the value that produces the lowest amount of error from all other values in the data set.

64. Merits of Mean It is straightforward to calculate and comprehend. It is for this reason that it is the most widely used central tendency measure. Every item has an impact because it is included in the calculation. The result remains the same since the mathematical formula is rigid. When repeated samples are gathered from the same population, fluctuations are minimal for this measure of central tendency. Unlike other measures like as mode and median, it can be subjected to algebraic treatment. A.M . has an advantage in that it is a calculated quantity that is not depending on the order of terms in a series. Due to its strict definition, it is mostly used to compare issues.

65. Draw backs of arithmetic mean It is very much affected by sampling fluctuation.  It cannot be located graphically . Arithmetic mean cannot be advocated to open end classification . Only if the frequency is regularly distributed will it be useful. If the skewness is greater, the results will be ineffectual . Unlike the mode and median, X cannot be found by inspection.

66. Median The median is a positional number that determines the position of the middle set of data. The value of the median remains unchanged if the size of the largest value increases because it is defined by the position of various values . Median is the middle value of the series when items are arranged either in an ascending or a descending order.

67. Merits of median Easy to calculate and understand Not affected by extreme values Rigidly defined Best average in case of qualitative data Useful in case of an open-ended distribution Represented graphically

68. D emerits of Median Arrangement of data is necessary Not based on all the observations Not a representative of the universe Affected by fluctuations in sampling Lack of further algebraic treatment

69. Mode The mode is the most frequent score in our data set . On a histogram it represents the highest bar in a bar chart or histogram. T herefore , sometimes consider the mode as being the most popular option . It is the point of greatest density. It is that value of the variable which has the highest frequency.

70. Merits of Mode Easy to calculate and simple to understand R epresentative value N ot affected by the value of extreme items N o need of complete data U seful for both quantitative and qualitative data Graphic determination

71. DEMERITS OF MODE N ot based on all the observations of the series S ometimes it is indeterminate or ill defined Not rigidly defined A ffected by the fluctuations of sampling C omplex grouping process N ot capable of algebraic treatment

72. Requisites of a Good Average It should be easy to understand. It should be simple to compute. It should be based on all the items. It should not be unduly affected by extreme observations . It should be rigidly defined. It should be capable of further algebraic treatment. It should have sampling stability.

73. Empirical Relationship between Mean, Median and Mode In statistics, for a moderately skewed distribution, there exists a relation between mean, median and mode. This mean median and mode relationship is known as the “empirical relationship” which is defined as Mode is equal to the difference between 3 times the median and 2 times the mean . In case of a moderately skewed distribution, the difference between mean and mode is almost equal to three times the difference between the mean and median. Thus, the empirical mean median mode relation is given as : Mean – Mode = 3 (Mean – Median ) Mode = 3 Median – 2 Mean

74. Variance Variance is the measure of how notably a collection of data is spread out. If all the data values are identical, then it indicates the variance is zero. All non-zero variances are considered to be positive . A little variance represents that the data points are close to the mean, and to each other, whereas if the data points are highly spread out from the mean and from one another indicates the high variance. In short, the variance is defined as the average of the squared distance from each point to the mean.

75. Standard Deviation Standard Deviation is a measure which shows how much variation (such as spread, dispersion, spread,) from the mean exists. The standard deviation indicates a “typical” deviation from the mean. It is a popular measure of variability because it returns to the original units of measure of the data set. Standard deviation calculates the extent to which the values differ from the average. Standard Deviation, the most widely used measure of dispersion, is based on all values. Therefore a change in even one value affects the value of standard deviation. It is independent of origin but not of scale. It is also useful in certain advanced statistical problems.

76. Mean Deviation The mean deviation of a given standard distribution is the average of the deviation from the central tendency. Central Tendency can be computed using the Arithmetic Mean, Median, or Mode of the data. It is used to show how far the observations are situated from the central point of the data (the central point can be either mean, median or mode ). Mean deviation measures the arbitrary change in the values of the data set from the centre point of the data set.

77. Merits of Mean Deviation It is simple to understand. It is easy to calculate. It is based on all the observations of a series.  It shown the dispersion, or scatter of the various items of a series from its central value. It is not very much affected by the values of extreme items of a series. It facilitates comparison between different items of a series. It truly represents the average of deviations of the items of a series. It has practical usefulness in the field of business and commerce.

78. Demerits of Mean Deviation It is not rigidly defined in the sense that it is computed from any central value namely Mean, Median, Mode etc. and thereby it can produce different results. It violates the algebraic principle by ignoring the + and – signs while calculating the deviations of the different items from the central value of a series. It is not capable of further algebraic treatment. It is affected much by the fluctuations in sampling. It is difficult to calculate when the actual value of an average comes out in fraction, or recurring figure for that in such a case it requires the use of the shortcut method which involves a cumbersome formula subject to adjustment in different cases. It is not suitable for sociological study.

79. Kurtosis Kurtosis describes the "fatness" of the tails found in probability distributions. Kurtosis is a statistical measure used to describe the degree to which scores cluster in the tails or the peak of a frequency distribution. The peak is the tallest part of the distribution, and the tails are the ends of the distribution. Kurtosis is a statistical measure used to describe a characteristic of a dataset. When normally distributed data is plotted on a graph, it generally takes the form of a bell. This is called the bell curve. The plotted data that are furthest from the mean of the data usually form the tails on each side of the curve. Kurtosis indicates how much data resides in the tails.

80. Distributions with a large kurtosis have more tail data than normally distributed data, which appears to bring the tails in toward the mean . Distributions with low kurtosis have fewer tail data, which appears to push the tails of the bell curve away from the mean.

81. Simple Interest and Compound Interest Simple Interest: Simple interest can be defined as the principal amount of a loan or deposit a person makes into their bank account. Compound Interest: Compound interest is the interest that accumulates and compounds over the principal amount . The major difference between simple interest and compound interest is that simple interest is based on the principal amount. In contrast, compound interest is based on the principal amount and the interest compounded for a cycle of the period.

82. Simple Interest Compound Interest Simple Interest can be defined as the sum paid back for using the borrowed money over a fixed period of time. Compound Interest can be defined as when the sum principal amount exceeds the due date for payment, along with the rate of interest for a period of time. S.I. = (P × T × R) ⁄ 100 C.I. = P(1+R⁄100) t − P The return is much lesser when compared to compound interest. The return is much higher. The principal amount is constant. The principal amount keeps on varying during the entire borrowing period. The growth remains quite uniform in this method. The growth increases quite rapidly in this method. The interest charged on is for the principal amount. The interest charged on it is for the principal and accumulated interest.