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**Freehand Curve Method :** The term freehand is used to any non-mathematical curve in statistical analysis even if it is drawn with the aid of drafting instruments. This is the simplest method of studying trend of a time series. The procedure for drawing free hand curve is an follows :

(i) The original data are first plotted on a graph paper.

(ii) The direction of the plotted data is carefully observed.

(iii) A smooth line is drawn through the plotted points.

While fitting a trend line by the freehand method, an attempt should be made that the fitted curve conforms to these conditions.

(i) The curve should be smooth either a straight line or a combination of long gradual curves.

(ii) The trend line or curve should be drawn through the graph of the data in such a way that the areas below and above the trend line are equal to each other.

(iii) The vertical deviations of the data above the trend line must equal to the deviations below the line.

(iv) Sum of the squares of the vertical deviations of the observations from the trend should be minimum.

**Illustration :** Draw a time series graph relating to the following data and fit the trend by freehand method :

**Year Production of Steel**

**(million tonnes)**

1994 20

1995 22

1996 30

1997 28

1998 32

1999 25

2000 29

2001 35

2002 40

2003 32

The trend line drawn by the freehand method can be extended to project future values. However, the freehand curve fitting is too subjective and should not be used as a basis for prediction. Method of Moving Averages : The moving average is a simple and flexible process of trend measurement which is quite accurate under certain conditions. This method establishes a trend by means of a series of averages covering overlapping periods of the data.

The process of successively averaging, say, three years data, and establishing each average as the moving-average value of the central year in the group, should be carried throughout the entire series. For a five-item, seven-item or other moving averages, the same procedure is followed : the average obtained each time being considered as representative of the middle period of the group.

The choice of a 5-year, 7-year, 9-year, or other moving average is determined by the length of period necessary to eliminate the effects of the business cycle and erratic fluctuations. A good trend must be free from such movements, and if there is any definite periodicity to the cycle, it is well to have the moving average to cover one cycle period. Ordinarily, the necessary periods will range between three and ten years for general business series but even longer periods are required for certain industries.

In the preceding discussion, the moving averages of odd number of years were representatives of the middle years. If the moving average covers an even number of years, each average will still be representative of the midpoint of the period covered, but this mid-point will fall halfway between the two middle years. In the case of a four-year moving average, for instance each average represents a point halfway between the second and third years . In such a case, a second moving average may be used to ‘recentre’ the averages.

That is, if the first moving averages gives averages centering half-way between the years, a further two-point moving average will recentre the data exactly on the years.

This method, however, is valuable in approximating trends in a period of transition when the mathematical lines or curves may be inadequate. This method provides a basis for testing other types of trends, even though the data are not such as to justify its use otherwise.

**Illustration :** Calculate 5-yearly moving average trend for the time series given below.

Year : 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000

Quantity : 239 242 238 252 257 250 273 270 268 288 284

Year : 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

Quantity : 282 300 303 298 313 317 309 329 333 327

**Solution :**

**Year Quantity 5-yearly moving total 5-yearly moving average**

1990 239

1991 242

1992 238 1228 245.6

1993 252 1239 247.8

1994 257 1270 254.0

1995 250 1302 260.4

1996 273 1318 263.6

1997 270 1349 269.8

1998 268 1383 276.6

1999 288 1392 278.4

1990 284 1422 284.4

2001 282 1457 291.4

2002 300 1467 293.4

2003 303 1496 299.2

2004 298 1531 306.2

2005 313 1540 308.0

2006 317 1566 313.2

2007 309 1601 320.2

2008 329 1615 323.0

2009 333

2010 327

**To simplify calculation work:** Obtain the total of first five years deta. Find out the difference between the first and sixth term and add to the total to obtain the total of second to sixth term. In this way the difference between the term to be omitted and the term to be included is added to the preceding total in order to obtain the next successive total.

**Illustration :** Fit a trend line by the method of four-yearly moving average to the following time series data.

Year : 1995 1996 1997 1998 1999 2000 2001 2002

Sugar production (lakh tons) : 5 6 7 7 6 8 9 10

Year : 2003 2004 2005 2006

Sugar production (lakh tons) : 9 10 11 11

**Solution :**

**Remark : **Observe carefully the placement of totals, averages between the lines.

**Merits**

1. This is a very simple method.

2. The element of flexibility is always present in this method as all the calculations have not to be altered if same data is added. It only provides additional trend values.

3. If there is a coincidence of the period of moving averages and the period of cyclical fluctuations, the fluctuations automatically disappear.

4. The pattern of moving average is determined in the trend of data and remains unaffected by the choice of method to be employed.

5. It can be put to utmost use in case of series having strikingly irregular trend.

**Limitations**

1. It is not possible to have a trend value for each and every year. As the period of moving average increases, there is always an increase in the number of years for which trend values cannot be calculated and known. For example, in a five yearly moving average, trend value cannot be obtained for the first two years and last two years, in a seven yearly moving average for the first three years and last three years and so on. But usually values of the extreme years are of great interest.

2. There is no hard and fast rule for the selection of a period of moving average.

3. Forecasting is one of the leading objectives of trend analysis. But this objective remains unfulfilled because moving average is not represented by a mathematical function.

4. Theoretically it is claimed that cyclical fluctuations are ironed out if period of moving average coincide with period of cycle, but in practice cycles are not perfectly periodic.

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