FORECASTING

1. FORECASTING

2. INTRODUCTION • Forecasting is the estimation of the value of a variable (or set of variables) at some future point in time. A forecasting is usually carried out in order to provide an aid to decision-making and in planning the future. Typically all such exercises work on the premise that if we can predict what the future will be like we can modify our behavior now to be in a better position, than we otherwise would have been, when the future arrives. Applications for forecasting include: • Inventory control/production planning - forecasting the demand for a product enables us to control the stock of raw materials and finished goods, plan the production schedule, etc • Investment policy - forecasting financial information such as interest rates, exchange rates, share prices, the price of gold, etc. This is an area in which no one has yet developed a reliable (consistently accurate) forecasting technique (or at least if they have they haven't told anybody!) • Economic policy - forecasting economic information such as the growth in the economy, unemployment, the inflation rate, etc is vital both to government and business in planning for the future.

3. WHY FORECAST? • “Forecasting is an attempt to foresee the future by examining the past.” • Forecasts require judgment. • Lead times require that decisions be made in advance of uncertain events. • Forecasting is an important for all strategic and planning decisions in a supply chain. • Forecasts of product demand, materials, labor, financing are an important inputs to scheduling, acquiring resources, and determining resource requirements. • Most estimates obtained in quality forecasting are derived in an objective and systematic fashion and do not depend solely on subjective guesses and hunches of the analyst. • Thus, Statistical forecasting concentrates on using the past to predict the future by identifying trends, patterns and business drives within the data to develop a forecast. This forecast is referred to as a statistical forecast because it uses mathematical formulas to identify the patterns and trends while testing the results for mathematical reasonableness and confidence. In many Forecasting Processes, statistical forecasting forms the baseline that is adjusted throughout the process.

4. MEANING OF FORECASTRING • Forecasting is the process of estimation in unknown situations. Prediction is a similar, but more general term. Both can refer to estimation of time series, cross-sectional or longitudinal data. Usage can differ between areas of application: for example in hydrology, the terms "forecast" and "forecasting" are sometimes reserved for estimates of values at certain specific future times, while the term "prediction" is used for more general estimates, such as the number of times floods will occur over a long period. Risk and uncertainty are central to forecasting and prediction. Forecasting is used in the practice of Customer Demand Planning in every day business forecasting for manufacturing companies. The discipline of demand planning, also sometimes referred to as supply chain forecasting, embraces both statistical forecasting and a consensus process. • Forecasting is apart of human conduct. Whatever an individual does at present is in the expectation of that certain events will take place in future. This expectation is generally based on the past experience. Forecast made in this fashion may or may not be true always. In a world, where the future is not taken (known) with certainty, virtually every business and economic decision rests upon a forecast of future condition.

5. MEANING OF FORECASTRING(cont) • Thus, planning which is the backbone of any business activity requires the forecasting of future events, whereas forecasting helps in viewing those events in their proper perspective .Objectivity is the corer stone of forecasting. It thus helps in reducing risks associated with uncertain future events. Thus forecasting reduces the areas of uncertainty that surround management decision- making with respect to costs, profits, sales, production pricing capital investment and so forth. • In Statistics, “the term (forecasting) refers to extending or projecting time-series into the future based on the past behaviour of the quantitative data”. In Other words “business forecasting refers to the statistical analysis of the past and current movements in a given time series, so as to obtain clues about the future pattern of movement.”

6. MEANING OF FORECASTRING(cont) • Business forecasting involves a wide range of tools, including simple electronic spreadsheets; enterprise resource planning (ERP) and electronic data interchange (EDI) networks, advanced supply chain management systems, and other Web-enabled technologies. The practice attempts to pinpoint key factors in business production and extrapolate from given data sets to produce accurate projections for future costs, revenues, and opportunities. This normally is done with an eye toward adjusting current and near-future business practices to take maximum advantage of expectations. • Business forecasting systems often work hand-in-hand with supply chain management systems. In such systems, all partners in the supply chain can electronically oversee all movement of components within that supply chain and gear the chain toward maximum efficiency. With business relationships and supply chains growing increasingly complex particularly in the world of e-commerce, with heavy reliance on logistics outsourcing and just-in-time delivery such forecasting systems become crucial for companies and networks to remain efficient.

7. Need For Forecasting • Business Forecasting is an estimate or prediction of future developments in business such as sales, expenditures, and profits. Given the wide swings in economic activity and the drastic effects these fluctuations can have on profit margins, it is not surprising that business forecasting has emerged as one of the most important aspects of corporate planning. Forecasting has become an invaluable tool for businesspeople to anticipate economic trends and prepare themselves either to benefit from or to counteract them. If, for instance, businesspeople envision an economic downturn, they can cut back on their inventories, production quotas, and hiring’s. If, on the contrary, an economic boom seems probable, those same businesspeople can take necessary measures to attain the maximum benefit from it. Good business forecasts can help business owners and managers adapt to a changing economy.

8. Need For Forecasting (cont) Some of the important needs of forecasting are listed below: • Helps in Production Planning • Helps in Financial Planning: • Helps in Economic Planning: • Helps in Workforce Scheduling: • Helps in Decisions Making: • Helps in Controlling Business Cycles:

9. Need For Forecasting (cont) • Helps in Production Planning: • The rate of producing the products must be matched with the demand which may be fluctuating over the time period in the future. Since its time consuming to change the rate of output of the production processes, so production manager needs medium range demand forecasts to enable them to arrange for the production capacities to meet the monthly demands which are varying.

10. Need For Forecasting (cont) • Helps in Financial Planning: • Sales forecasts are driving force in budgeting. Sales forecasts provide the timing of cash inflows and also provide a basis for budging the requirements of cash outflows for purchasing materials, payments to employees and to meet other expenses of power and utilize etc. Hence forecasting helps finance manager to prepare budgets taking into consideration the cash inflow and cash out flows.

11. Need For Forecasting (cont) • Helps in Economic Planning: • Forecasting helps in the study of macroeconomic variables like population, total income, employment, savings, investment, general price-level, public revenue, public expenditure, balance of trade, balance of payments and a host of other macro aspects at national or regional levels. The forecasts of these variables are generally for a long period of time ranging between one year to ten or twenty years ahead. Much would depend on the perspective of planning, longer the perspective longer would be period of forecasting. Such forecasts are often called as projections. These are helpful not only for planning and public policy making, they also include likely economic environment and aid formulation of business policies as well.

12. Need For Forecasting (cont) • Helps in Workforce Scheduling: • The forecast of monthly demand may further be broken down to weekly demands and the workforce may have to be adjusted to meet these weekly demands. Hence, forecasts are needed to enable managers to get tuned with the workforce changes to meet the weekly production demands.

13. Need For Forecasting (cont) • Helps in Decisions Making: • The goal of the forecaster is to provide information for decision making. The purpose is to reduce the range of uncertainty about the future. Businessmen make forecasts for the purpose of making profits. In business forecast has to be done at every stage. A business man may dislike statistics or statistical theories of forecasting, but he can not do without making forecasts. Business plans of production, sales and investment requires predictions regarding demand for the product, price at which the product can be soled and the availability of inputs. The forecast about demand is the most crucial. Operating budgets of various departments of a company have to be based upon the expected sales. Efficient production schedules, minimization of operating cost and investment in fixed assets is when accurate forecasts recording sales and availability of inputs are available.

14. Need For Forecasting (cont) • Helps in Controlling Business Cycles: • It is commonly believed that business cycles are always very harmful in their effects. Abrupt rise and fall in the price level injurious not only to businessmen, but to all types of persons, industries, trade, agriculture. All suffer from the painful effects of depression. Trade cycle increase the risk f business; create unemployment; induce speculation and discourage capital formation. Their effects are not confined to one country only. Business forecasting reduces the risk associated with business cycles. Prior knowledge of a phase of a trade cycle with its intensity and expected period of happening may help businessmen, industrialist, and economists to plan accordingly to reduce the harmful effects of trade cycle’s .statistics is thus needed for the purpose of controlling the business-cycles.

15. Objectives Of Forecasting • Forecasting has few objectives. Some the few important objectives of forecasting are as follows: • To estimate the amount of error in forecast by using probability theory. • To assist in managerial decision making in uncertainties. • To acquaint businessmen and economists about the future probable circumstances. • To clarify the differences of actual data of the future by comparing it with pre-forecasted data by using theory of probable error. • To provide basis for determination of future policy. • To indicate the probability of happenings in the future. • The forecast provides a warning system of critical functions to be monitored regularly because they might drastically affect the performance of the plan.

16. Requirements Of Good Forecast. • Agood forecast should satisfy the following criteria: • Time frame • Pattern of the data • Cost of forecasting/ Economy • Accuracy • Availability of data • Durability • Plausibility/Ease of operation and understanding • Flexibility

17. Requirements Of Good Forecast. (cont) Time frame: • The first factor that can influence the choice of forecasting is the time frame of the forecasting situation. Forecasts are generally for points in time that may be a number of days, weeks, months, quarters, or years in the future. This length of time is called the time frame or time horizon. The length of the time frame is usually categorized as follows: • Immediate: less than one month • Short term: one to three months • Medium: more than three months to less than two years. • Long term: two years or more • In general, the length of the time frame will influence the choice of the forecasting technique. Typically a longer time frame makes accurate forecasting more difficult with qualitative forecasting techniques becoming more useful as the time frame lengthens.

18. Requirements Of Good Forecast. (cont) Pattern of the data: • The pattern of the data must also be considered when choosing a forecasting model. The components present i.e.’ trend, cycle, seasonal or some combination of these will help determine the model that will be used. Thus it is extremely important to identify the existing data pattern. Cost of forecasting/ Economy: • Though the firm is interested in accurate forecasts, the benefits of accurate results must be weighed against the cost of the method. While choosing a forecasting technique, several costs are relevant. First, the cost of developing the model must be considered. Second the cost of storing the necessary data must be considered. Some forecasting methods require the storage of a relatively small amount of data, while other methods require the storage of large amounts of data. Last, the cost of the actual operation of the forecasting technique is obviously very important. Some forecasting methods are operationally simple, while others are very complex. The degree of complexity can have a definite influence on the total cost of forecasting.

19. Requirements Of Good Forecast. (cont) Accuracy desired: • Accuracy in forecasting is very important. The previous method must be checked for want of accuracy by observing that the predictions made in the past are accurate or not. The accuracy of past forecasting can be checked against present performance and of present forecasts against future performance. In some situations a forecast that is in error by as much as 20% may be acceptable. In other situations a forecast that is in error by 1% might be disastrous. The accuracy that can be obtained using any particular forecasting method is always an important consideration. Availability of data: • Immediate availability of data is an important requirement and the method employed should be able to produce good results quickly. The technique which takes much time to produce useful information is of no use. Historical data on the variable of interest are used when quantitative forecasting methods are employed. The availability of this information is a factor that may determine the forecasting method to be used. Since various forecasting methods require different amounts of historical data, the quantity of data available is important. Beyond this, the accuracy and the timeliness of the data that are available must be examined, since the use of inaccurate or outdated historical data will obviously yield inaccurate predictions. If the needed historical data are not available, special data-collection procedures may be necessary.

20. Requirements Of Good Forecast. (cont) Plausibility/Ease of operation and understanding: • The ease with the forecasting method is operated and understood is important. Management must be able to understand and have confidence in the technique used. It has to understand clearly how the estimate was made. Mathematical and statistical techniques should be avoided if the management cannot understand what the forecaster does. • Managers are held responsible for the decisions they make and if they are to be expected to base their decisions on predictions, they must be able to understand the techniques used to obtain these predictions. A manager simply will not have confidence in the predictions obtained from a forecasting technique he or she does not understand, and if the manager does not have confidence in these predictions, they will not be used in the decision-making process. Thus, the managers understanding of the forecasting system is of crucial importance. Durability: • The forecast should be durable and should not be changed frequently. The durability of the forecasts depends on the simplicity and ease of comprehension as well as on continuous link between the past and the present and between present and the future. Flexibility: • The technique used in forecasting must be able to accommodate and absorb frequent changes occurring in the economy.

21. Types Of Forecasting • A forecast is a prediction or an estimation of a future situation .The objectives of an organization are facilitated by a number of different types of forecast. These may be related to cash flows, operating budgets, personnel requirement, inventory levels, and so on. However a broad classification of the types of forecasts is as follows. • Demand forecasts • Environmental forecasts • Technological forecasts

22. Types Of Forecasting (cont) • Demand forecasts: • Demand forecasting means an estimation of the level of demand that might be realized in future under given circumstances. These are concerned with the predictions of demand for products or services to minimize the uncertainties of the unknown future. These forecasts facilitate in formulating material and capacity plans and serves as inputs to financial, marketing and personnel planning. The forecast itself may be generated in a number of ways, many of which depend heavily upon sales and marketing information.

23. Types Of Forecasting (cont) • Environmental forecasts: • Environmental forecasting is attempting to predict the nature and intensity of the micro environmental and macro environmental forces that are likely to affect a firm's decision making and have an impact upon its performance in a given period. Environmental concerns such as pollution control, are much better managed from an anticipatory rather than an after the fact standpoint. Environmental forecasts are concerned with the social,political and economic environment of the state or the country.

24. Types Of Forecasting (cont) • Technological forecasts: • It is forecasting the future characteristics of useful technological machines, procedures or techniques. These are concerned with new developments in existing technologies as well as the development of new technologies. They have become increasingly important to major firms in the computer, aerospace, nuclear and many other technologically advanced industries.

25. QUALITATIVE METHODS OF FORECASTING: • Qualitative forecasting methods consist of collecting the opinions and judgments of individuals who are expected to have the best knowledge of current activities or future plans of the organization. For example, knowledge of demand trends and consumer plans are often known to marketing and sales personnel are presumably familiar with individual customers or retail market segment. Management usually maintains broader market information on trends by product line, geographic area, customer groups, etc. • Qualitative forecasting methods have the advantage that they can incorporate subjective experience as inputs along with objectives data. It is a human brain that permits assimilation of all types of information and the ultimate issuance of a prediction. • Since each human being has different knowledge, experience, and perceptive of reality, intuitive forecasts are likely to differ from one individual to another. Furthermore, the less they are based upon fact and quantified data, the less they lend themselves to analyses and resolution of differences of opinion. The quantification of data gives them a more precise meaning than words which are in exact and are capable of being misunderstood. Also, if the forecast prove to be inaccurate there is an objective bases for improvement the next time around.

26. QUALITATIVE METHODS OF FORECASTING: (cont) A number of approaches fall under qualitative methods and these are as follows: PERSONAL OPINION- In this approach of forecasting, an individual does some forecast of the future based on his or her own judgement or opinion without using a formal quantitative model. Such an assessment can be relatively reliable and accurate. This approach is usually recommended when condition in the past are not likely to hold in the future. For instance, getting an assessment of whether inventory levels are likely to last until the next replenishment; whether a machine will require, repair in the next month and so on.

27. QUALITATIVE METHODS OF FORECASTING: (cont) Advantages: • It is fast and efficient. • It is timely and based on good information content. • It uses the collective knowledge of experts. Disadvantages: • Experts can make mistakes. • Subjectivity and bias of experts and vitiate the forecast. • The group dynamics of the experts could be greatly influenced by the degree of dominance of a particular person. He who could shout loudest might get his way.

28. QUALITATIVE METHODS OF FORECASTING: (cont) MARKET SURVEY: • This method is used to collect data on well defined objectives and assumptions of about the future value of a variable. A carefully designed questionnaire is administered to the selected target audience of customers. Customers are selected independently using a representative random sample. This method is very popular and if carefully implemented will give you good results. • This is the apt technique to use, particularly if you want to forecast sales for a new product or new brand. • This method of forecasting requires the active cooperation of the target audience. • The sample size must be reasonably large. Larger the sample size smaller will be the standard error and sampling error. • Larger the sample size the more time consuming and costly the survey will be. Swo, you have to strike a balance between sample size and cost.

29. QUALITATIVE METHODS OF FORECASTING: (cont) DELPHI METHOD: • It is a quantitative forecasting method that obtains forecasts through group consensus. In the expert opinion method of forecasting a consensus forecast is arrived at after eliciting the opinions and views of experts with diverse background. Certainly this method is subject to group dynamics. At times, judgements may be highly influenced by persuasions of some group members who have strong likes and dislikes. Delphi method attempts to retain the wisdom and accumulated knowledge of a group while simultaneously attempting to reduce the group effects. • In this method, group members are asked to make individual assessment about a forecast. These assessments are complied and then fed back to the members, so that they get the opportunity to compare their judgement with others. They are then given an option to revise their forecasts. After three or four replications, group members reach their final conclusion.

30. QUALITATIVE METHODS OF FORECASTING: (cont) HISTORICAL ANALOGY: • This method is applied when a new product is about to be introduced by a company. Forecasting sales for new products are difficult in view of lack of proper historical data. Historical analogy method attempts to forecast sales for a new product based on the performance of related or similar products in the market place. The database of sales of these products forms the basis for forecasting. Disadvantages: • You cannot precisely say how your new product is similar or related to a particular product. • Suppose you have a number of products that you feel are similar to yours. Which of these will you consider as most similar to yours? • Products that are similar to yours could have failed in the past for a variety of reasons. Let us say a similar product failed in the past because whenever there was an advertisement about this product, it was not available on the shelf. So, the consumers developed a negative perception about this product and became skeptical about its availability. You may not know all these and simply conclude your product will also fail!

31. PANEL CONSENSES: • To reduce the prejudices and ignorance that may arise in the individual judgement , it is possible to develop consensus among group to individuals. Such a panel of individuals is encouraged to share information, opinion, and assumptions to predict future value of some variable. Disadvantages: • It is dependent on group dynamics and frequently requires a facilitator or convenor to coordinate the process of developing a consensus.

32. QUANTITATIVE FORECASTING METHODS: Time series Forecasting Methods: • Time series forecasting methods are based on analysis of historical data (time series: a set of observations measured at successive times or over successive periods). They make the assumption that past patterns in data can be used to forecast future data points. • 1. Moving averages (simple moving average, weighted moving average): forecast is based on arithmetic average of a given number of past data points • 2. Exponential smoothing (single exponential smoothing, double exponential smoothing): a type of weighted moving average that allows inclusion of trends, etc. • 3. Mathematical models (trend lines, log-linear models, Fourier series, etc.): linear or non-linear models fitted to time-series data, usually by regression methods • 4. Box-Jenkins methods: autocorrelation methods used to identify underlying time series and to fit the "best" model

33. QUALITATIVE METHODS OF FORECASTING: (cont) Components of Time Series Demand: • 1. Average: the mean of the observations over time • 2. Trend: a gradual increase or decrease in the average over time • 3. Seasonal Influence: predictable short-term cycling behaviour due to time of day, week, month, season, year, etc. • 4. Cyclical Movement: unpredictable long-term cycling behaviour due to business cycle or product/service life cycle • 5. Random Error: remaining variation that cannot be explained by the other four components

34. QUALITATIVE METHODS OF FORECASTING: (cont) Simple Moving Average: • Moving average techniques forecast demand by calculating an average of actual demands from a specified number of prior periods • each new forecast drops the demand in the oldest period and replaces it with the demand in the most recent period; thus, the data in the calculation "moves" over time • Simple moving average: At = Dt + Dt-1 + Dt-2 + ... + Dt-N+1 • N • Where N = total number of periods in the average • Forecast for period: t+1: Ft+1 = At • Key Decision: N - How many periods should be considered in the forecast • Tradeoff: Higher value of N - greater smoothing, lower responsiveness

35. QUALITATIVE METHODS OF FORECASTING: (cont) Weighted Moving Average: • A weighted moving average is a moving average where each historical demand may be weighted differently • Average: At = W1Dt + W2 Dt-1 + W3 Dt-2 + ... + WN Dt-N+1 • Where: • N = total number of periods in the average • Wt = weight applied to period t's demand • Sum of all the weights = 1 • Forecast: Ft+1 = At = forecast for period t+1

36. QUALITATIVE METHODS OF FORECASTING: (cont) Exponential Smoothing: • Exponential smoothing gives greater weight to demand in more recent periods, and less weight to demand in earlier periods • Average: At = a Dt + (1 - a) At-1 = a Dt + (1 - a) Ft • Forecast for period t+1: Ft+1 = At • Where: • At-1 = "series average" calculated by the exponential smoothing model to period t-1 • a = smoothing parameter between 0 and 1 • The larger the smoothing parameter , the greater the weight given to the most recent demand

37. QUALITATIVE METHODS OF FORECASTING: (cont) Double Exponential Smoothing: • (TREND-ADJUSTED EXPONENTIAL SMOOTHING) • When a trend exists, the forecasting technique must consider the trend as well as the series average ignoring the trend will cause the forecast to always be below (with an increasing trend) or above (with a decreasing trend) actual demand • Double exponential smoothing smoothes (averages) both the series average and the trend • Forecast for period t+1: Ft+1 = At + Tt • Average: At = aDt + (1 - a) (At-1 + Tt-1) = aDt + (1 - a) Ft • Average trend: Tt = B CTt + (1 - B) Tt-1 • Current trend: CTt = At - At-1 • Forecast for p periods into the future: Ft+p = At + p Tt • Where: • At = exponentially smoothed average of the series in period t • Tt = exponentially smoothed average of the trend in period t • CTt = current estimate of the trend in period t • a = smoothing parameter between 0 and 1 for smoothing the averages • B = smoothing parameter between 0 and 1 for smoothing the trend

38. QUALITATIVE METHODS OF FORECASTING: (cont) Multiplicative Seasonal Method: • What happens when the patterns you are trying to predict display seasonal effects? • What is seasonality? - It can range from true variation between seasons, to variation between months, weeks, days in the week and even variation during a single day or hour. • To deal with seasonal effects in forecasting two tasks must be completed: • A forecast for the entire period (ie year) must be made using whatever forecasting technique is appropriate. This forecast will be developed using whatever • The forecast must be adjust to reflect the seasonal effects in each period (ie month or quarter) • The multiplicative seasonal method adjusts a given forecast by multiplying the forecast by a seasonal factor • Step 1: calculate the average demand y per period for each year (y) of past data by dividing total demand for the year by the number of periods in the year • Step 2: divide the actual demand Dy,t for each period (t) by the average demand y per period (calculated in Step 1) to get a seasonal factor fy,t for each period; repeat for each year of data • Step 3: calculate the average seasonal factor t for each period by summing all the seasonal factors fy,t for that period and dividing by the number of seasonal factors • Step 4: determine the forecast for a given period in a future year by multiplying the average seasonal factor t by the forecasted demand in that future year

39. STEPS IN FORECASTING: • Forecasting business change involves more than analysis of statistical data-it also embodies the prediction of economic change such as secular trend. Seasonal variations. Cyclical variations and a consideration of cause and effect. • Broadly speaking the forecasting of business fluctuations consists of the following steps: Understanding why changes in the past have occurred: One of the basic principles of statistical forecasting-indeed of all forecasting when historical data are available – is that the forecaster should use the data on past performance to get a “speedometer reading” of the current rate (say, of sales) and of how fast this rate is increasing or decreasing. The current rate and changes in the rate- “acceleration” and “deceleration”-constitute the basis of forecasting. Once they are known, various mathematical techniques can develop projections from them. If an attempt is made to forecast business fluctuations without understanding why past changes have taken place, the forecast will be purely mechanical based solely upon the application of mathetical formulae and subject to serious error.

40. STEPS IN FORECASTING: (cont) • Observation and analysis of the past behavior is one of the most vital parts of forecasting. However, it should be carefully noted that though future may be some sort of extension of the past. It may not be an exact replica. Changes in business and economic activity are caused by numerous forces or factors which are often difficult to discover and measure. Not only this, they may appear in all kinds of combinations and may be constantly changing. Hence in making forecasts, we should not assume that history repeats itself. Rather, we should believe that there are certain regularities in the past behavior which can be observed and used as a basis for reducing the uncertainities of the future. It is often said that the past, imperfect indicator of the future though it is, is the best guide we have in attempting to make predictions.

41. STEPS IN FORECASTING: (cont) Determining which phases of business activity must be measured: After it is known why business fluctuations have occurred, or if there is a reasonable supposition, it is necessary to measure certain phases of business activity in order to predict what changes will probably follow the present level of activity. Selecting and compiling data to be used as measuring devices: There is an interdependent relationship between the selection of statistical data and determination of why business fluctuations occur. Statistical data cannot be collected and analysed in an intelligent manner unless there ia a sufficient understanding of business fluctuations; likewise, it is important that reasons for business fluctuations be stated in such a manner that it is possible to secure data that are related to the reasons.

42. STEPS IN FORECASTING: (cont) Analyzing the data: In this last step, the data are analysed in the light of one’s understanding of the reason why change occurs. For example, if it is reasoned that a certain combination of forces will result in a given change, the statistical part of the problem is to measure these forces and from the data available, to draw conclusions on the future course of action. The methods of drawing conclusions may be called forecasting techniques and they represent any one of a large number of analytical devices for summarizing data and drawing inferences from the summaries.