MCom I Semester Statistical Analysis Interpolation Extrapolation Study Material Notes

MCom I Semester Statistical Analysis Interpolation Extrapolation Study Material Notes: Meaning and Definition of Interpolation and Extrapolations Need and Importance Interpolation Extrapolation Accuracy Interpolation Extrapolation Methods of Interpolation Parabolic Curve Method Binomial Easier Method Present Pascal Triangle InterpolationOne Missing Value When Extrapolation is to done :

CTET Paper Level 2 Previous Year ( 2011 ) Model paper in Hindi

Interpolation and Extrapolation

In the proper management of any business or administration, data are collected regularly. But it may not be possible to collect figures for every time point. Van OS, we come across situations where we have to estimate a value that is not available in the given series of data or predict a future value. In this situation stead of depending on some guesswork, the techniques of interpolation and extrapolation are useful. For example, the census of the population in India takes place every 10 years, i.e., we have the census figures for 1951, 1961, 1971, 1981, 1991, and 2001. With the help of these available data if one wants to know the census for the year 1996 or 2007 then the technique of interpolation and extrapolation should be applied. The need for interpolating missing observations or making forecasts or projections arises in a number of disciplines like economics, business, social sciences, actuarial work, population studies, etc. Hence, the technique of interpolation and extrapolation are extremely helpful in estimating the missing values or projecting the future values. In the present chapter, a focus is made on Interpolation and Extrapolation.

Meaning and Definition of Interpolation and Extrapolation

In simple words interpolation relates to the estimation of a value within the given range of the series while extrapolation deals with obtaining the forecast or projections in the past or future) beyond the given range of the series. Interpolation, thus, refers to the insertion of an intermediate value in a series of items whereas extrapolation refers to projecting a value for the future. Whenever the method of interpolation or extrapolation is applied it is based on the assumption that the variable whose value is to be estimated is the function of the other variable. A variable is said to be the function of the other if for any values of the independent variable (say x) we can always find a definite value of the dependent variable (say y).

Explanation by an example : Let us suppose that we have two variables x and y, x being the independent variable and y the dependent variable. Let the given values of x be xo, X1, 42, ..,X, and let the corresponding values of y be yo. Yı, y, ….y, respectively. If we want to estimate the value of y, for any value of x between the limits, Xo and in this can be done by applying the technique of Interpolation. For example, suppose we are given the population census figures (w.) for the years (x) 1951,1961, 1971, 1981, and 1991 and we want to estimate the population for any year between 1971 and 1991, say, 1978, 1985, etc. This is done by the method of interpolation. However, if we have to estimate the population for the period outside the range 1951-1991, say, for 1946 or 1995, the technique is known as extrapolation.

Some important definitions are as follows:

“Interpolation consists in reading a value which lies between two extreme points. Extrapolation means reading a value that lies outside the two extreme points.

-W.M. Harper “Interpolation is the estimation of a most likely estimate in given conditions. The Technique of estimating a past figure is termed as interpolation, while that of estimating a probable figure for the future is called extrapolation.”

-Hirach “Interpolation is the art of reading between the lines of the table.” -Theile

Conclusion: Interpolation or extrapolation is a statistical method of estimating the most likely figure of a dependent variable for a given independent variable from the given relevant facts, under certain assumptions.

Interpolation Extrapolation Study Material

Difference between Interpolation and Extrapolation

Both interpolation and extrapolation are statistical methods of finding out the unknown values from the given relevant facts. In fact there is no basic difference between interpolation and extrapolation as far as estimation methods and underlying assumptions are concerned. The only difference between the two is that interpolation relates to estimation of a value within the given range of the series while extrapolation deals with obtaining the forecast or projections in the past or future) beyond the given range of the series.

Frederick Mills says, “There is no difference between interpolation and extrapolation so far as the methods are concerned; but for distinguishing the past from the future, we give them two different names. Interpolation relates to the past, whereas extrapolation gives us the forecast for the future.”

Assumptions of Interpolation and Extrapolation

As has been stated above, interpolation or extrapolation is done on certain assumptions. The following assumptions are made while making use of the techniques of interpolation and extrapolation.

(1) No Sudden or Violent Fluctuations in the Intervening Period : While interpolating or extrapolating a value, we always presume that there are no sudden ups and downs in the given data. In other words, the values should relate to periods of normal and stable economic conditions, i.e., the given data should be free from all sorts of abnormalities and all sorts of random and irregular fluctuations like earthquakes, wars, floods, labour strikes, lock outs, economic boom and depression and political disturbances, etc., which may result in violent ups and downs in the values.

If, for example, we are interpolating the figure of population of India in the year 1993 and we are given the figures of Indian population for the years 1961. 1971, 1981, 1991 and 2001 our presumption would be that the population in this country has grown up smoothly and there are no violent ups and downs in these figures.

(2) The Rate of Change of Figures from One Period to Another is Uniform : The second assumption is that the rate of change of the figures is uniform. Thus in the example of census population given above, our assumptions would be that from 1961 to 2001 the growth rate of population has been uniform.

On the basis of the above said assumptions, missing figures can be interpolated with a fair degree of accuracy.

Accuracy of Interpolation and Extrapolation

Since the interpolation techniques are based on certain assumptions which may not hold good in practice, the estimates so obtained, may not always be accurate or reliable, it is not possible to ascertain the error of estimate. According Dr. A.L. Bowley, the accuracy of the interpolated value depends on:

possible fluctuations in the values of the phenomenon under study, which is provided by the available data at our disposal.

(ii) A knowledge

(1) A knowledge of

about the course of events which affect the value of the phenomenon under investigation. If we know that the estimated value of the given phenomenon at a particular period is affected by random factors, like political riots, floods, etc., then the interpolated value should be modified in the light of this information and a better estimate may be obtained.

Interpolation Extrapolation Study Material

Need and Importance of Interpolation and Extrapolation

The techniques of interpolation and extrapolation are of great practical use, because of:

(1) Non-availability of Data : Interpolation may also be required in case the data are insufficient due to gaps in coverage, or are inadequately collected.

(2) Loss of Data : Figures of some of the periods may be erased, destroyed or lost due to certain reasons like improper handling or random and natural causes like fire, floods, etc. Such a figure may be obtained with the help of interpolation technique. Interpolation is thus helpful in filling up the gaps in available data.

(3) To Estimate the Intermediate Values. : Due to certain financial and organisational difficulties, data may not be collected on census basis and sampling techniques may be used to obtain the relevant information. The intermediate gaps are then filled by interpolation methods.

(4) To bring Uniformity in the Data : Sometimes it is found that the data pertaining to a specific phenomena are grouped by different agencies in different types of groups and this makes them meaningless for comparison. To bring uniformity in the groups, interpolation technique is resorted to. If the data are collected for two different dates, for making comparison in them, they have to be brought at one point of time. For example, if in a country the population census is taken in 2000 and in India the census is taken in 2001. For making comparison between the populations of the two countries either India’s population is to be interpolated for 2000 or the other country’s population is to be estimated by extrapolation for 2001.

(5) For Making Forecasts : Prediction of the future figures is a basic need in any policy formulation or economic planning. Extrapolation is thus helpful in making forecasts.

(6) To Determine the Positional Averages in Continuous Frequency Distribution : The interpolation technique has been used to derive the formulae for the computation of median, quartiles, quintiles, cortiles, deciles, percentiles and mode in case of continuous frequency distribution.

Methods of Interpolation and Extrapolation

The methods of interpolation or extrapolation may be broadly classified as follows:

(a) Graphic Method,

(b) Algebraic Method,

(C) Other Methods.

( Graphic Method : This is the simplest method of interpolation and extrapolation. Under this method, the given data are plotted on a graph paper independent variable is plotted on the x-axis or the horizontal scale and the depended variable is plotted on the y-axis or the vertical scale. The various points so obtains are joined together by a smooth freehand curve. From this curve, which represents the general trend of the relationship between the two variables, we can find out the value of y for any given value of x within the given range of the series. From the point on the x-axis, for which the value of y is to be interpolated, a line parallel to y-axis will be drawn. From the point where this line will cut the curve, a line parallel to r-axis will be drawn. The point where this line cuts y-axis, value of y will be found out.

If the value of y (or x) lies outside the given range, i.e., if it is a case of extrapolation, then the smoothed curve is extended to the required point and then the estimated value is read from the graph.

Graphic method is simple to calculate. But the results obtained by this method are not much accurate. It requires graphic skill. If the figures are large, then the result will be a rough estimate. Different persons will get different smooth curves for the same set of data, depending upon the investigator.

Illustration 1.

Interpolation Extrapolation Study Material

The following table gives the population of states in India :

Solution.

Interpolation and Extrapolation of Population by Graphic Method

It is clear from the graph that the estimated population in 1965 is 5.8 crore and in 1981 will be 7.4 crore.

(b) Algebraic Method : Interpolation and extrapolation can also be done by algebraic method. The algebraic methods may be :

When there are Abnormal Fluctuations in the Known Values

For accurate results of interpolation and extrapolation, it is essential that there should not be sudden ups and downs in the given data. If we see that there is abnormal fluctuations in anyone of the given values of y then such value of y will also be treated as unknown value. Such type of problems may be of two types :

(a) When both the unknown value as well as the abnormally fluctuated value are related to Interpolation : In such a case two equations (A7 = 0 and Aj = 0) will be required to find out both values. It has been explained in detail carlier. It can be understood very well with the help of following example.

Illustration 8.

From the following data of the population of a city in laths, find out population for 1995 :

There is equal interval in x-series. Hence, we will use the binomial expansion method. But we see that there is sudden downfall in the population of 1985. Therefore, the population of 1985 would also be treated as unknown. In this way, two values (1985 and 1995) becomes unknown.

Now the known values would be 5 instead of 6. Since there are two unknowns, yz and ys and both values are related to interpolation. Therefore, two equations (A = 0 and A = 0) will be required to find out the missing values. They are :

Newton’s Method

A number of formulae were given by Newton to be applied in different situations. Some of these formulae are :

(1) Newton’s Advancing Difference Method

(2) Newton’s Gauss (Forward) Method

(3) Newton’s Gauss (Backward) Method

(4) Newton’s Divided Difference Method

Here we are explaining only the Newton’s Advancing Difference Method and other methods have been explained in the latter part of this chapter.

Newton’s Method of Finite or Advancing Differences

This method is applicable when the independent variable (x) advances by equal intervals and gives the best estimate for interpolation near the beginning of the series. It is rememberable that like Binomial expansion method it is not necessary here that the value of x for which y is to be interpolated is one of the class limits of X-series. For example, if the given data are :

X: 5 10 15 – 20 25

y: 40 y.) 55 (1) 85 (2) 100 (3) 120 (2)

We can interpolate the value of y for x = 13 or 22 etc. Similarly, we can extrapolate the value for x = 27.

This method is called finite or advancing differences method because after finding out differences in the values of y, the process is extended further till only one difference remains.

Procedure for Interpolating or Extrapolating by this method :

(i) The given values of independent variable(x-variable) are denoted by X0, X1, X2, X3, X4,…. Similarly, the given values of dependent variable (y-variable) are denoted by yo, Yu, Y2, y3, 94,….

(ii) We will calculate the difference between the various values of y, until the only one difference remains. The algebraic signs + and – are taken into account when differences are calculated. The differences are indicated by the sign A (pronounced as delta). Table given below shows the method of calculating advancing differences:

Interpolation Extrapolation Study Material

Table showing Advancing Differences

In each column of the differences, the first difference is called the leading difference. Al, As A A…. are the 1st, 2nd, 3rd and 4th leading differences

Interpolation Extrapolation Study Material