MCA645 CU-MCA-SEM-II-Statistical & Numerical Methods

ANOVA Table (two-way) 200 CU IDOL SELF LEARNING MATERIAL (SLM)

Step 6: Critical values f (3, 6),0.05 = 4.7571 (for treatments) f (2, 6),0.05 = 5.1456 (for blocks) Step 7: Decision (i) Calculated F0t = 3.40 < f (3, 6),0.05 = 4.7571, the null hypothesis is not rejected and we conclude that there is significant difference in the mean sales among the three programs. (ii) Calculate F0b = 5.39 > f (2, 6),0.05 = 5.1456, the null hypothesis is rejected and conclude that there does not exist significant difference in the mean sales among the four salesmen. Example 2: The illness caused by a virus in a city concerning some restaurant inspectors is not consistent with their evaluations of cleanliness of restaurants. In order to investigate this possibility, the director has five restaurant inspectors to grade the cleanliness of three restaurants. The results are shown below. 201 CU IDOL SELF LEARNING MATERIAL (SLM)

Carry out two-way ANOVA at 5% level of significance. Solution: Step 1: Null hypotheses H0I: µ1 = µ 2 = µ 3 = µ 4 = µ5 (For inspectors - Treatments) That is, there is no significant difference among the five inspectors over their mean cleanliness scores H0R : μI = μII = μIII (For restaurants - Blocks) That is, there is no significant difference among the three restaurants over their mean cleanliness scores Alternative Hypotheses H1I: At least one mean is different from the other among the Inspectors H1R: At least one mean is different from the other among the Restaurants. Step 2: Data Step 3: Level of significance α = 5% 202 Step 4: Test Statistic For inspectors: F0a (treatments) = = MST / MSE For restaurants: F0b (blocks) = MSE / MSB CU IDOL SELF LEARNING MATERIAL (SLM)

Step-5: Calculation of the Test Statistic Squares 203 CU IDOL SELF LEARNING MATERIAL (SLM)

ANOVA Table (two-way) 204 CU IDOL SELF LEARNING MATERIAL (SLM)

Step 6: Critical values f(4, 8),0.05 = 3.838 (for inspectors) f(2, 8),0.05 = 4.459 (for restaurants) Step 7: Decision (i) As F0I = 0.377 < f (4, 8),0.05 = 3.838, the null hypothesis is not rejected and we conclude that there is no significant difference among the mean cleanliness scores of inspectors. (ii) As F0R = 10.49 > f (2, 8),0.05 = 4.459, the null hypothesis is rejected and we conclude that there exists significant difference in at least one pair of restaurants over their mean cleanliness scores. 9.7 SUMMARY Analysis of variance, or ANOVA, is a statistical method that separates observed variance data into different components to use for additional tests. A one-way ANOVA is used for three or more groups of data, to gain information about the relationship between the dependent and independent variables. ANOVA is used when we want to test the equality of means of more than two populations. For example, through ANOVA, one may compare the average yield of several varieties of a crop or average mileages of different brands of cars. ANOVA cannot be used in all situations and for all types of variables. It is based on certain assumptions, and they are listed below: 1. The observations follow normal distribution. 2. The samples are independent. 3. The population variances are equal and unknown. 205 CU IDOL SELF LEARNING MATERIAL (SLM)

9.8 KEYWORDS  Between-group variability. It refers to variations between the distributions of individual groups (or levels) as the values within each group are different.  Mean is a simple or arithmetic average of a range of values.  Hypothesis: A hypothesis is an educated guess about something in the world around us. It should be testable either by experiment or observation.  Variation: a change or slight difference in condition, amount, or level, typically within certain limits.  Interaction effects occur when the effect of one variable depends on the value of another variable 9.9 LEARNING ACTIVITY 1. Discuss Between Group Variability --------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------- ---- 2. Explain important terms in ANOVA table --------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------- 9.1UNIT END QUESTIONS A. Descriptive type Questions 1. What is Hypothesis? 2. Explain One way ANOVA 3. State the limitation of one way ANOVA 4. What is ANOVA table? 5. Brief about two way ANOVA B. Multiple Choice Questions 1. For an experiment comparing more than two treatment conditions you should use analysis of variance rather than separate t tests because: a. Conducting several t tests would inflate the risk of a Type I error. b. Separate t tests would require substantially more computations. c. a test based on variances is more sensitive than a test based on means. 206 CU IDOL SELF LEARNING MATERIAL (SLM)

d. There are no differences between the two tests, you can use either one. 2. Does an ANOVA test identify specifically where the differences exist between the groups? a. No b. Yes c. Yes, provided the data are normally distributed d. Yes, provided the data are not normally distributed 3. An analysis of variance comparing three treatment conditions produced dftotal = 24. For this ANOVA, what is the value of dfwithin? a. 21 b. 3 c. 22 d. 2 4. What is the definition of 'mean square'? a. A sum of squares divided by its degrees of freedom b. The square root of the mean c. The square of the mean d. A table of means with four cells 5. When sample size is large, which of the following assumptions can be ignored? 207 a. That each sample will have an equal variance b. That samples have been drawn from a normal population c. Both d. Neither Answers: CU IDOL SELF LEARNING MATERIAL (SLM)

1 – a; 2 – a; 3 – c; 4 – a; 5 - b 9.11 REFERENCES  Rajaraman V., Computer Oriented Numerical Method. New Delhi: Prentice Hall.  Salaria R.S.A, Textbook of Statistical and Numerical Methods in Engineering, Delhi: Khanna Book Publishing Company.  Gupta S.P. and Kapoor, V.K. (2014). Fundamentals of Mathematical Statistics. Delhi: Sultan Chand and Sons.  Sujatha Sinha, Sushma Pradhan, Numerical Analysis and Statistical Methods, Academic Publishers.  J. H. Wilkinson , The Algebraic Eigenvalue Problem (Numerical Mathematics and Scientific Computation), Clarendon Press  Kendall E. Atkinson , An Introduction to Numerical Analysis, Wiley India Private Limited  Gupta Dey , Numerical Methods , McGraw Hill Education;  Numerical Methods & Analysis – Engineering App – Google Play store  https://en.wikibooks.org/wiki/Statistics/Numerical_Methods/Numerical_Comparison_of_Stati stical_Software 208 CU IDOL SELF LEARNING MATERIAL (SLM)

UNIT 10- TIME SERIES ANALYSIS: Structure: 10.0 Learning Objective 10.1 Introduction 10.2 Components and Analysis of Time Series 10.2.1 Components for Time Series Analysis 10.2.2 Trend 10.2.3 Linear and Non-Linear Trend 10.3 Periodic Fluctuations 10.3.1 Seasonal Variations 10.3.2 Cyclic Variations 10.4 Random or Irregular Movements 10.4.1 Analysis of Time Series 10.4.2 Additive Model for Time Series Analysis 10.4.3 Multiplicative Model for Time Series Analysis 10.4.4 Mixed models 10.5 Measurement of Trend. 10.5.1 Method of the Free-Hand Curve 10.5.2 Semi-average method 10.5.3 Method of Moving Averages 10.6 Summary 10.7 Keywords 10.8 Learning Activity 10.9 Unit End Questions 10.10 References 10.0 LEARNING OBJECTIVE After studying this unit, you will be able to:  Explain the Components and Analysis of Time Series,  Enumerate the Measurement of Trend. 10.1 INTRODUCTION Time Series How do people get to know that the price of a commodity has increased over a period of time? They can do so by comparing the prices of the commodity for a set of a time period. A set of observations ordered with respect to the successive time periods is a time series. 209 CU IDOL SELF LEARNING MATERIAL (SLM)

In other words, the arrangement of data in accordance with their time of occurrence is a time series. It is the chronological arrangement of data. Here, time is just a way in which one can relate the entire phenomenon to suitable reference points. Time can be hours, days, months or years. 10.2 COMPONENTS AND ANALYSIS OF TIME SERIES A time series depicts the relationship between two variables. Time is one of those variables and the second is any quantitative variable. It is not necessary that the relationship always shows increment in the change of the variable with reference to time. The relation is not always decreasing too. It may be increasing for some and decreasing for some points in time. Can you think of any such example? The temperature of a particular city in a particular week or a month is one of those examples. Uses of Time Series  The most important use of studying time series is that it helps us to predict the future behaviour of the variable based on past experience  It is helpful for business planning as it helps in comparing the actual current performance with the expected one  From time series, we get to study the past behaviour of the phenomenon or the variable under consideration  We can compare the changes in the values of different variables at different times or places, etc. 10.2.1 Components for Time Series Analysis The various reasons or the forces which affect the values of an observation in a time series are the components of a time series. The four categories of the components of time series are  Trend  Seasonal Variations  Cyclic Variations  Random or Irregular movements . 210 CU IDOL SELF LEARNING MATERIAL (SLM)

Fig 10.1: Seasonal and Cyclic Variations are the periodic changes or short-term fluctuations. 10.2.2 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. It is observable that the tendencies may increase, decrease or are stable in different sections of time. But the overall trend must be upward, downward or stable. The population, agricultural production, items manufactured, number of births and deaths, number of industry or any factory, number of schools or colleges are some of its example showing some kind of tendencies of movement. 10.2.3 Linear and Non-Linear Trend If we plot the time series values on a graph in accordance with time t. The pattern of the data clustering shows the type of trend. If the set of data cluster more or less round a straight line, then the trend is linear otherwise it is non-linear (Curvilinear). 10.3 PERIODIC FLUCTUATIONS There are some components in a time series which tend to repeat themselves over a certain period of time. They act in a regular spasmodic manner. 10.3.1 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. These variations come into play either because of the natural forces or man-made conventions. The various seasons or climatic conditions play an important role in seasonal variations. Such as production of crops depends on seasons, the sale of umbrella and raincoats in the rainy season, and the sale of electric fans and A.C. shoots up in summer seasons. The effect of man-made conventions such as some festivals, customs, habits, fashions, and some occasions like marriage is easily noticeable. They recur themselves year after year. An upswing in a season should not be taken as an indicator of better business conditions. 10.3.2 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’. 211 CU IDOL SELF LEARNING MATERIAL (SLM)

It is a four-phase cycle comprising of the phases of prosperity, recession, depression, and recovery. The cyclic variation may be regular are not periodic. The upswings and the downswings in business depend upon the joint nature of the economic forces and the interaction between them. 10.4 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. 10.4.1 Analysis of Time Series Mathematical Model for Time Series Analysis Mathematically, a time series is given as yt = f (t) Here, yt is the value of the variable under study at time t. If the population is the variable under study at the various time period t1, t2, t3, … , tn. Then the time series is t: t1, t2, t3, … , tn yt: yt1, yt2, yt3, …, ytn or, t: t1, t2, t3, … , tn yt: y1, y2, y3, … , yn 10.4.2 Additive Model for Time Series Analysis If yt is the time series value at time t. Tt, St, Ct, and Rt are the trend value, seasonal, cyclic and random fluctuations at time t respectively. According to the Additive Model, a time series can be expressed as yt = Tt + St + Ct + Rt. This model assumes that all four components of the time series act independently of each other. 10.4.3 Multiplicative Model for Time Series Analysis The multiplicative model assumes that the various components in a time series operate proportionately to each other. According to this model yt = Tt × St × Ct × Rt 10.4.4 Mixed models Different assumptions lead to different combinations of additive and multiplicative models as yt = Tt + St + Ct Rt. The time series analysis can also be done using the model yt = Tt + St × Ct × Rt or yt = Tt × Ct + St × Rt etc. 212 CU IDOL SELF LEARNING MATERIAL (SLM)

10.5 MEASUREMENT OF TREND. A number of different methods are available to estimate the trend; however, the suitability of these methods largely depends on the nature of the data and the purpose of the analysis. To measure a trend which can be represented as a straight line or some type of smooth curve, the following are the commonly employed methods: (1) Freehand smooth curves (2) Semi-average method (3) Moving average method (4) Mathematical curve fitting Generally speaking, when the time series is available for a short span of time in which seasonal variation might be important, the freehand and semi-average methods are employed. If the available series is spread over a long time span and has annual data where long term cycles might be important, the moving average method and the mathematical curve fitting are generally employed. 10.5.1 Method of the Free-Hand Curve This is a familiar concept, and is briefly described for drawing frequency curves. In case of a time series a scatter diagram of the given observations is plotted against time on the horizontal axis and a freehand smooth curve is drawn through the plotted points. The curve is so drawn that most of the points concentrate around the curve, however, smoothness should not be scarified in trying to let the points fall exactly on the curve. It is better to draw a straight line through the plotted points instead of a curve, if possible. The curve fitted by this method eliminates the short term and long term oscillations and the irregular movements from the time series, and elevates the general trend. After having drawn such a curve or line, the trend values or the estimated YY values, which may be denoted by YY, can be read from the graph corresponding to each time period. One of the major disadvantages of this method is that different individuals draw curves or lines that differ in slope and intercept, and hence no two conclusions are identical. However, it is the most simple and quickest method of isolating the trend. This method is generally employed in situations where the scatter diagram of the original data conforms to some well define trends. Example: Measure the trend using the method of the freehand curve from the given data of production of wheat in a particular area of the world. 213 CU IDOL SELF LEARNING MATERIAL (SLM)

We observe that the graph of the original data does not show any closeness to any type of curve. It looks like it increases very slowly in a straight (linear) manner. Thus we draw a line ABAB as an approximation to the original graph. The line ABAB represents the trend line, and from this we read the trend values for the given years. Advantages This method is very simple and easy to understand. It is applicable to linear and non-linear trends. It gives us an idea about the rise and fall of the time series. For every long time series, the graph of the original data enables us to decide on the application of more mathematical models for the measurement of a trend. Monthly data from 5 years has 60 values. A graph of these values may suggest that the trend is linear for the first two years (24 values) and for the next 3 years, it is non- linear. We accordingly apply the linear approach to the first 24 values and the curvilinear technique to the next 36 values. Disadvantages 214 CU IDOL SELF LEARNING MATERIAL (SLM)

This method is not mathematical in nature, so different people may draw a different trend. The method does not appeal to the common man because it seems rough and crude. 10.5.2 Semi-average method This method is as simple and relatively objective as the free hand method. The data is divided in two equal halves and the arithmetic mean of the two sets of values of Y is plotted against the centre of the relative time span. If the number of observations is even the division into halves will be straightforward; however, if the number of observations is odd, then the middle most item, i.e., (n+1/2) th, is dropped. The two points so obtained are joined through a straight line which shows the trend. The trend values of Y, i.e., , can then be read from the graph corresponding to each time period. Since the arithmetic mean is greatly affected by extreme values, it is subjected to misleading values, and hence the trend obtained by plotting by means might be distorted. However, if extreme values are not apparent, this method may be successfully employed. To understand the estimation of trends, using the above noted two methods, consider the following working example. Example: Measure the trend by the method of semi-averages by using the table given below. Also write the equation of the trend line with origin at 1984-85. 215 CU IDOL SELF LEARNING MATERIAL (SLM)

Trend for 1991 – 92 = 50.60 Trend for 1986 – 87 = 32.32 Increase in trend in 5 years = 18.28 Increase in trend in 1 year = 3.656 The trend for one year is 3.656. This is called the slope of the trend line and is denoted by b. Thus, b = 3.656. The trend for 1987 – 88 is calculated by adding 3.656 to 32.32 and similar calculations are done for the subsequent years. The trend for 1985 – 86 is less than the trend for 1986 – 87. Thus the trend for 1985 – 86 is 32.32 – 3.656 = 28.664. The trend for the year 1984 – 85 = 25.008. This is called the intercept because 1984 – 85 is the origin. The intercept is the value of Y when X = 0. 216 CU IDOL SELF LEARNING MATERIAL (SLM)

The intercept is denoted by a. The equation of the trend line is =a+bX = 25.008+3.656 X (1984 – 85 = 0) where shows the trend values. This equation can be used to calculate the trend values of the time series. It can also be used for forecasting the future values of the variable. Advantages This method is very simple and easy to understand, and also it does not require many calculations Disadvantages The method is used only when the trend is linear or almost linear. For non-linear trends this method is not applicable. It is used for the calculation of averages, and averages are affected by extreme values. Thus if there is some very large value or very small value in the time series, that extreme value should either be omitted or this method should not be applied. We can also write the equation of the trend line. 10.5.3 Method of Moving Averages Suppose that there are nn time periods denoted by t1,t2,t3,…,tn and the corresponding values of the Y variable are Y1,Y2,Y3,…,Yn. First of all we have to decide the period of the moving averages. For a short time series we use a period of 3 or 4 values, and for a long time series the period may be 7, 10 or more. For a quarterly time series we always calculate averages taking 4-quarters at a time, and in a monthly time series, 12-monthly moving averages are calculated. Suppose the given time series is in years and we have decided to calculate 3-year moving averages. The moving averages denoted by a1,a2,a3,…,an–2 are calculated as below: 217 CU IDOL SELF LEARNING MATERIAL (SLM)

The average of the first 3 values is Y1+Y2+Y3 / 3 and is denoted by a1. It is written against the middle year t2t2. We leave the first value Y1 and calculates the average for the next three values. The average is Y2+Y3+Y4 / 3 = a2 and is written against the middle yearst3t3. The process is carried out to calculate the remaining moving averages. 4-year moving averages are calculated as: The first average is a1a1 which is calculated as 218 CU IDOL SELF LEARNING MATERIAL (SLM)

It is written against the middle of t3t3 and t4t4. The two averages a1a1 and a2a2 are further averaged to get an average of a1+a2 /2=A1, which refers to the centre of t3t3 and is written against t3t3. This is called centering the 4-year moving averages. The process continues until the end of the series to get 4-years moving averages centered. The moving averages of some proper period smooth out the short term fluctuations and the trend is measured by the moving averages. Advantages Moving averages can be used for measuring the trend of any series. This method is applicable to linear as well as non-linear trends Disadvantages The trend obtained by moving averages generally is neither a straight line nor a standard curve. For this reason the trend cannot be extended for forecasting future values. Trend values are not available for some periods at the start and some values at the end of the time series. This method is not applicable to short time series. Seasonal Component: Using the multiplicative model, i.e., Y = T×S×R the ratio detrended series may be obtained by dividing the actual observations by the corresponding trend values: Y/ T = S×R The remainder now consists of the seasonal and the residual components. The seasonal component may be isolated from the ratio-detrended series by averaging the detrended ratios for each month or quarter. The adjustment seasonal totals are, however, obtained by multiplying the seasonal totals by the following adjustment factor. These adjustment seasonal totals are then averaged over the number of detrended ratios in each quarter or month. The obtained averages represent the seasonal component. After having determined the seasonal component S, the de-seasonalised series may be obtained by dividing the actual observations Y by the corresponding seasonal component. The de-seasonalised series so obtained determines the trend and the residual, for 219 CU IDOL SELF LEARNING MATERIAL (SLM)

The residual component may now be separated by a further division of the de-seasonalised series by the trend, for The entire analysis described above may be briefly summarized in the following steps: 1. Write Y and T 2. Obtained the detrended ratios, i.e., YT 3. Determine he seasonal component S as explained above 4. Obtain the de-seasonalised series by finding YS 5. Further divide the ratios obtained in step 4 by T to separate out the residual R 6. As a final check of the calculations, select any quarter or month and multiply out T, S and R. This product would be equal to Y Seasonal component additive model Using the additive model, i.e. Y=T+S+R the de-trended series may be obtained by subtracting the trend values from the actual observations: Y–T=S+R The remainder now consists of the seasonal and the residual components. The trend values that are subtracted might have been obtained by any of the methods described earlier, however, the moving trend or the least square trend are preferable. The residual component may be eliminated from the de-trended series by averaging the de-trended values for each month or quarter separately. For an efficient trend the sum of these averages must have been zero, however, generally it will not be so. The seasonal component, therefore, needs adjustment. To do this the seasonal totals are averaged, for example in a quarterly time series the four quarterly totals are added and divided by twelve. This average, which may also be called the “adjustment factor,” is subtracted from each quarterly or monthly total. The adjusted totals are then averaged by dividing by the number of quarterly or monthly observations used to arrive at these totals. The averages which are finally obtained represent the seasonal component. The four seasonal components in the case of quarterly data, or the twelve seasonal components in case of monthly data, repeat during the subsequent years. After having determined the seasonal component SS, the de-seasonalised series may be obtained by subtracting the seasonal component SS from the actual observations Y. The de-seasonalised series so obtained also represents a series which jointly determines the trend and the residual, for 220 CU IDOL SELF LEARNING MATERIAL (SLM)

Y–S=T+RY–S=T+R The residual component may now be separated by further subtracting the trend from the seasonally adjusted series for, Y–S–T=RY–S–T=R The entire analysis described above may be briefly summarized in the following steps: 1. Write Y and T in adjacent columns 2. Write the de-trended series by taking the differences (Y–T) 3. Determine the seasonal component S as explained above 4. Write the de-seasonalised series by taking the differences (T–S) 5. Subtract the trend T from the de-seasonalised series to separate out the residual component RR 6. As a final check of the calculations, select any quarter or month and add the values for T, S and R, which would be equal to Y 10.6 SUMMARY Time Series Analysis is used for many applications such as:  Economic Forecasting.  Sales Forecasting.  Budgetary Analysis.  Stock Market Analysis.  Yield Projections.  Process and Quality Control.  Inventory Studies.  Workload Projections 10.7 KEYWORDS  A time series: depicts the relationship between two variables  Cyclic Variations: The variations in a time series which operate themselves over a span of more than one year are the cyclic variations.  Seasonal Variations: These are the rhythmic forces which operate in a regular and periodic manner over a span of less than a year  Trend: The trend shows the general tendency of the data to increase or decrease during a long period of time. 221 CU IDOL SELF LEARNING MATERIAL (SLM)

 Scatter Diagram: The scatter diagram graphs pairs of numerical data, with one variable on each axis, to look for a relationship between them. 10.8 LEARNING ACTIVITY 1. Explain Mathematical Model for Time Series Analysis --------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------- 2. What does the term ‘long period of time’ in a trend depict? --------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------- 10.9 UNIT END QUESTIONS A. Descriptive type Questions 1. Explain Components of Time series 2. What is Seasonal and Cyclic Variation? State the difference 3. Explain Method of free hand curve. 4. Discuss Semi-average method 5. Write a note on Seasonal Component B. Multiple Choice Questions: 222 8. The best method for finding out seasonal variation is a. Simple Average method b. Ratio to moving average c. Ratio to trend d. All of the above 9. Seasonal Variations are a. Short term b. Long Term c. Sudden d. Random CU IDOL SELF LEARNING MATERIAL (SLM)

10. The adjustment seasonal totals are, however, obtained by ________the seasonal totals by the following adjustment factor a. adding b. subtracting c. multiplying d. dividing 11. Linear trend of time series indicates towards a. Change in Geometric progression b. Variable rate of growth c. Constant rate of growth d. Constant rate of change 5. The component of time series attached to long term variation is termed as a. Cyclic Variation b. Irregular Variation c. Seasonal Variation d. Secular Variation Answers: 1 – b; 2 – a; 3 – c; 4 – d; 5 - d 10.10 REFERENCES  Rajaraman V., Computer Oriented Numerical Method. New Delhi: Prentice Hall.  Salaria R.S.A, Textbook of Statistical and Numerical Methods in Engineering, Delhi: Khanna Book Publishing Company.  Gupta S.P. and Kapoor, V.K. (2014). Fundamentals of Mathematical Statistics. Delhi: Sultan Chand and Sons.  Sujatha Sinha, Sushma Pradhan, Numerical Analysis and Statistical Methods, Academic Publishers.  Anderson (1990). Statistical Modelling. New York: McGraw Publishing House.  Gupta S.P. and Kapoor, V.K. (2015). Fundamentals of Applied statistics. Delhi: Sultan Chand & Sons.  Graybill (1990). Introduction to Statistics. New York: McGraw Publishing House.  Numerical Methods & Analysis – Engineering App – Google Play store  https://en.wikibooks.org/wiki/Statistics/Numerical_Methods/Numerical_Comparison_of_ Statistical_Software 223 CU IDOL SELF LEARNING MATERIAL (SLM)