Draw a trend line by semi average method using the following data.. Find the trend from the following series using a three year weighted moving average with weight 1, 2 and 1.. Represent
Trang 1PROBLEM SET 10.1
1
1 Fit a trend line to the following data by the free hand method
Sales
(in million Rs.)
2
2 Draw a trend line by semi average method using the following data
Production
(in tons)
[Ans Semi Average 39.30] 3
3 Obtain the 5 yearly moving averages for the following series of observations
Annual Sales
(Rs’0000)
[Ans 5 yearly moving averages are 4, 4.36, 4.18 and 3.80] 4
4 Find the trend from the following series using a three year weighted moving average with weight 1, 2 and 1
[Ans Trend values 3.75, 5.25, 6.75, 8.25 and 10.25] 5
5 For the following series of observations, verify that 4 year centred moving average
is equivalent to a 5 year weighted moving average with weight 1, 2, 2, 2, 1 respectively
Year 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994
6
6 Represent the following data graphically and show the trend of the series on the basis of three year moving averages
Birthrate 30.9 30.2 29.1 31.4 33.4 30.2 30.4 31.0 29.0
Birthrate 27.9 27.7 26.4 24.7 24.1 23.1 27.7 22.6 23.6
Birthrate 23.0 22.0 22.6 [Ans Trend values are: 30.7, 30.2, 31.3, 31.7, 30.5, 30.1, 29.3, 28.2, 27.3, 26.3, 25.6,
24.0, 25.0, 23.8, 24.6, 23.6, 22.9, and 22.5.]
Trang 27 The revenue from sales Tax in U.P during 1948–99 to 1952–53 is shown in the following table Fit a straight-line trend by the method of least square
Years Revenue (Rs Lakhs)
[Ans Trend values are 311.2, 410.1, 509.0, 607.9, and 706.8.] 8
8 Find the seasonal indices by the method of moving averages from the series observations
Sales of Woollen Yarn (‘000 Rs.)
[Ans Seasonal Indices 10.12, 0.13, –14.08, 3.83.] 9
9 Calculate the seasonal index from the following data using the average method
Year 1st Quarter 2nd Quarter 3rd Quarter 4th Quarter
[Ans 96.4, 92.1, 106.9, 100.5] 10
10.Using 4-Quarterly moving averages find seasonal indices using ratio to moving average method from the given data
Quarter
[Ans 110.9, 99.9, 84.9, 104.3]
Trang 310.6 FORECASTING
The method and principles of Time series are used in the important work of the forecasting Forecasting is an art of making an estimate of future conditions on a systematic basis using prior available information On another way we say that the forecasting is the projection of the past data into future and therefore it has varity of applications Forecasting is done on specified assumption and is always made with probability ranges The need for forecasting arises because future is characterized by uncertainty Successful business activity demands a reasonably accurate forecasting of future business conditions upon which decisions regarding production, inventories, price fixation, etc depend To estimate guesswork modern statistical methods are employed as
a very useful tool of forecasting
10.7 FORECASTING MODES
The time series analysis essentially involves decomposition of the time series into its four components for forecasting The main purpose is to estimate and separate the four types of variations and to bring out the relative impact of each on the over all behaviour of the time series For the purpose of forecasting these will be two-model decomposition of time series
10.7.1 Additive Model
This model is used when it is assumed that the four components of time series are independent
of one another Thus, if Mt is taken represent the magnitude of time series then,
M t = T t + S t + C t + I t where T t = Trend Variation at time t
S t = Seasonal Varaition at time t
C t = Cyclical Variation at time t
I t = Irregular or random Variation at time t
When the time series data are recorded against years, the seasonal component of time series vanish and therefore we have
M t = T t + C t + I t
10.7.2 Multiplicative Model
This model is used when it is assumed that the forces giving rise to the four types of variations
of time series are interdependent i.e.
M t = T t × S t × C t × I t Similarly to additive model, if the time series data are recorded against years then S t
vanish and we have
M t = T t × C t × I t
by taking logarithm on both sides,
log Mt = log Tt + log Ct + log It This implies the four components of time series are essentially additive, in additive as well
as multiplicative models
Trang 4Note: The multiplicative model is better than the additive model for forecasting when the trend is increasing or decreasing over time In such circumstances, seasonal variations are likely to be increasing
or decreasing too The additive model simply adds absolute and unchanging seasonal variations to the trend figures where as the multiplicative model, by multiplying increasing or decreasing trend values by
a constant seasonal variation factor, takes account of changing seasonal variations.
10.8 TYPES OF FORECASTING AND FORECASTING METHODS
Forecasting are of two types:
(a) Qualitative Forecasting: Qualitative forecasting is used when past data is not available
(b) Quantitative Forecasting: Quantitative forecasting is used if historical or past data are available
Quantitative forecasting are two types One is Time Series Forecasting and another is Casual Forecasting In casual forecasting methods, factors relating to the variable whose values are to be predicted are determined and in time series forecasting method, projection of the future values of a variable is indicated depending on the past and the present movements of the variable Different forecasting methods using time series are given in the following
1 Mean Forecast: It is the simplest forecasting method According to this method the
mean y– of the time series is taken as a forecast or predicted value for the value of y t
of the series for the time period t i.e., yˆt = y
2 Naive Forecast: In this method, recent past is considered for the predication of immediate future If there exist high correlation between the pair of values in the time series then
the value y t for the time period t is the forecast of the value y t+1 for the time period
(t + 1) i.e., yˆt+ 1 = y t.
3 Linear Trend Forecast: In this method, the equation of the trend line y= +a bx for the given time series is first determined by the method of least squares Then the forecast
for the period t is found from the relation y– t = a + bx, where x is obtained from the value of t.
4 Non Linear Trend Forecast: In this method a parabolic or non-linear relationship between the time and the response value (time series observation) is first determined
by the method of least squares Then the forecast for the period t is found from the relation yˆt = a + bx + cx2, where x is obtained from value of t.
10.9 SMOOTHING OF CURVE
Smoothing techniques are used to reduce irregularities (random fluctuations) in time series data They provide a clearer view of the underlying behaviour of the series In some, time series, seasonal variation is so strong it obscures any trends or cycles, which are very important for the understanding of the process being observed Smoothing can remove seasonality and makes long-term fluctuations in the series stand out more clearly The most common type of smoothing technique is moving average smoothing although others do exist Since the type of seasonality will vary from series to series, so must the type of smoothing
Trang 5(=) Exponential Smoothing: Exponential smoothing is a smoothing technique used to reduce irregularities (random fluctuations) in time series data, thus providing a clearer view
of the true underlying behaviour of the series It also provides an effective means of predicting future values of the time series (forecasting)
(>) Moving Average Smoothing: A moving average is a form of average, which has been adjusted to allow for seasonal or cyclical components of a time series Moving average smoothing
is a smoothing technique used to make the long-term trends of a time series clearer When a variable, like the number of unemployed, or the cost of strawberries, is graphed against time, there are likely to be considerable seasonal or cyclical components in the variation These may make it difficult to see the underlying trend These components can be eliminated by taking a suitable moving averages By reducing random fluctuations, moving average smoothing makes long term trends clearer
(?) Running Medians Smoothing: Running medians smoothing is a smoothing technique analogous to that used for moving averages The purpose of the technique is the same, to make
a trend clearer by reducing the effects of other fluctuations
GGG
Trang 6CHAPTER 11
Statistical Quality Control
11.1 INTRODUCTION
The important, appealing and easily understood method of presenting the statistical data is the use of diagrams and graphs They are nothing but geometrical figures like points, lines, bars, squares, rectangles, circles, cubes etc., pictures, maps or charts Diagrammatic and graphic representation has a number of advantages Some of them are given below:
1 Diagrams are generally more attractive and impressive than the set of numerical data They are more appealing to the eye and leave a much lasting impression on the mind
as compared to the uninteresting statistical figures
2 Diagrams and graphs are visuals aids, which give a bird’s eye view of a given set of numerical data They present the data in simple, readily comprehensible form
3 They register a meaning impression on the mind almost before we think They also save lot of time, as very little effort is required to grasp them and draw meaningful inferences from them
4 The technique of diagrammatic representation is made use of only for purpose of comparison It is not to be used when comparison is either not possible or is not necessary
5 When properly constructed, diagrams and graphs readily show information that might otherwise be lost a mid the detail of numerical tabulations They highlight the salient features of the collected data, facilitate comparisons among two or more sets of data and enable use to study the relationship between them more readily
11.1.1 Difference between Diagrams and Graphs
There are no certain method to distinguish between diagrams and graphs but some points of difference may be observed
1 Generally graph paper is used in the construction of the graph, which helps us to study the mathematically relationship between the two variables, whereas diagrams are generally constructed on a plain paper and used for comparison only not for studying the relationship between two variables
2 In graphic mode of representation points or lines (dashes, dot, dot-dashes) of different kinds are used to represent the data while in diagrammatic representation data are presented by bars, rectangles, circles, squares, cubes, etc
451
Trang 73 Diagrams funish only approximate information They do not add anything to the meaning of the data and therefore, are not of much use to a statistician or researcher for further statistical analysis On the other hand graphs are more obvious, precise and accurate than the diagrams and are quite helpful to the mathematician for the study of
slopes, rate of change and estimation i.e., interpolation and extrapolation, whenever
possible
4 Construction of graphs is easier as compared to the construction of diagrams Diagrams are useful in depicting categorical and geographical data but it fails to present data relating to frequency distributions and time series
11.1.2 Types of Diagrams
A variety of diagrammatic devices are used commonly to present statistical data
(a) One Dimensional Diagrams i.e., line diagrams and bar diagrams.
(b) Two Dimensional Diagrams i.e., rectangle, squares, circles and pie diagrams.
(c) Three Dimensional Diagrams i.e., cubes, spheres, prisms, cyclinders etc.
(d) Pictograms.
(e) Cartograms.
11.1.3 Rules for Drawing Diagrams
1 The first and the most important thing is the selection of a proper scale No definite rules can be laid down as regards the selection of scale But as a guiding principle the scale should be selected consistent with the size of the paper and the size of the observations to be displayed so that the diagram obtained is neither too small nor too large
2 The vertical and horizontal scales should be clearly shown on the diagram itself The former on the left hand side and the latter at the bottom of the diagram
3 Neatness should be strictly being written on the top in bold letter and should be very explanatory If necessary the footnotes may be given at the left hand bottom of the diagram to explain certain points of facts
11.2 LINE DIAGRAM
This is the simplest of all the diagrams It consists in drawing vertical lines, each vertical line being equal to the frequency The variate values are presented on a suitable scale along the X-axis and the corresponding frequencies are presented on a suitable scale along Y-axis
Example 1 Draw line diagram for the following data:
Trang 8Line Diagram
250 200 150 100 50 0
FIG 11.1
11.3 BAR DIAGRAM
The terms ‘bar’ is used for a thick wide line The width of the bar diagram shows merely to make the diagram more explanatory Bar diagrams are one of the easiest and the commonly used diagram of presenting most of the business and economics data They consist of a group
of equidistant rectangles one for each group or category of the data in which the length or height of the rectangles represents the values or the magnitudes, the width of the rectangles being arbitrary There are various types of bar diagrams
(a)Simple Bar Diagram: It is used for comparative study of two or more items or values of a single variable or category of data
Example 2 Birth rate of a few countries of the World during the year 1934.
Country India Germany Irish Free State Soviet Russia New Zealand Swedon
Sol
45 40 35 30 25 20 15 10 5 0
Ind ia
Ge rm a y
Iris
h F
e S ta
S
vie
t R
uss ia
Ne w e
lan
Sw e o
FIG 11.2
Trang 9(b)Subdivided Bar Diagram: If a magnitude is capable of being broken into component parts or if there are independent quantities which form the subdivisions of the total,
in either of these cases, bars may be subdivided into the ratio of the various components
to show the relationship of the parts to the whole
Example 3 Represent the following data by sub-divided bar diagram.
Income Rs 500 Income Rs 300
Sol
600
500
400
300
200
100
0 –100 Income Rs 500 Income Rs 300
Saving or Deficit Miscellaneous Education Clothing Food
Subdivided Bar Diagram
FIG 11.3 (c)Percentage Bar Diagram: Subdivided bar diagrams presented graphically on percentage basis give percentage bar diagrams They are especially useful for the diagrammatic portrayal of the relative changes in the data
Example 4 Draw a bar chart for the following data showing the percentage of the total population in villages and towns.
Percentage of total Population in
Trang 10% Cumulative % % Cumulative %
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
Percentage Diagram Showing Total Population
Cumulative % Villages
Cumulative % Towns
Elderly Persons
Middle aged men and Women
Young men and Women
Boys and Girls
Infants and young children
FIG 11.4 Some other bar diagrams are multiple bar diagram, Deviation bar, Broken bars etc In a multiple bar diagram two or more sets of interrelated data are represented The method of drawing multiple bar diagram is the same as that of simple bar diagram Deviation bars are popularly used for representing net quantities excess or deficit, i.e., net loss, net profit etc Such types of bars have both positive and negative values Obviously positive values are shown above the base line and negative values below the base line
Example 5 Draw a multiple bar diagram from the following data.
Year Sales (‘000 Rs.) Gross Profit (‘000 Rs.) Net Profit (‘000 Rs.)