Growth Curve Method of the Least Square
The growth curves are some special type of curves which are plotted on graph papers for analyzing and estimating the trend values in the business and economic phenomena. Where initially, the growth rate is very slow but gradually it picks up at a faster rate till it reaches a point of stagnation, or saturation. Such situations are quite common in business fields where new products are introduced for marketing. In such a situation, when a new product say the book, “Statistical Methods” is introduced into the market, the growth rate of its sale is quite slow, and if the product proves to be worthwhile, the growth rate of its begins to decline. As such, a curve, representing this type of phenomena continues to grow more and more slowly approaching the upper limit, but never reaching the same. It takes the shape of an elongated f which indicates the pattern of growth in terms of the actual amount as small in the early years increasingly greater in the middle Years, and large and stabilized in the later years. But when such curves are plotted on a semi logarithmic chart, they show a growth at rapidly increasing rates in the earlier periods and at a declining ratio in the later periods of the series.
There are different types of growth curves used in different fields of business and economics, but the most popular among them are the following twos which have come into being since 1920:
- Gompertz Growth Curve.
- Logistic or Pearl-Read Growth Curve.
A brief introduction to these curves is given as below:
Gompertz Growth Curve. The fundamental trend equation of this curve is given by
Yc = Kabx
When put to the logarithmic form, the above equation is modified as under:
Ye = Al of [log K + bX log a]
Where, k represents the constants of the highest point, an intercept of y i.e. the trend value of the origin of X, and b the slope of the line i.e. the rate of growth.
Logistic or Pearl-Read Growth Curve.
This curve is given by the following model
1/yc = k + ab×
The above equation may also be used in the alternative form as under:
It may be noted that the Logistic curve is nothing but a modified exponential curve in terms of the reciprocal of the y variable. This curve, however, is very popular in the demographic studies and in many business and economic analysis.
