Linear approximation method for interpolation, Mathematics

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Linear Approximation Method

This is a rough and ready method of interpolation and is best used when the series moves in predicted intervals. It can be applied to interpolate values in both ascending and descending series. The method can be best described with the help of illustrations.

Example

The sales in units of a consumer durable are ascertained as follows:

Year	1986	1987	1988	1990
Sales Units (in '000s)	8	16	24	40

The records of sales for the year 1989 were accidentally lost in a fire. The sales in this year could be interpolated by the following procedure:

Use the value of the immediately preceding year as the base value or starting value. We may connote this as base value.

The year immediately preceding 1989, is 1988. Hence 24,000 units sold in 1988 is the base value.

Ascertain whether the series is an ascending one or a descending one.

In the illustration, since demand for the units is steadily increasing, we may conclude that it is an ascending series.

Find out the difference in values of the variable corresponding to the immediately succeeding and preceding years of the year for which the value is to be interpolated. We may connote this as upper limit minus lower limit.

Value corresponding to the immediately succeeding year (1990) = 40,000 units

Value corresponding to the immediately preceding year (1988) = 24,000 units

Upper limit - Lower limit = 40,000 - 24,000

= 16,000

Find out the time interval between the two known values. We may denote this as t_s - t_p.

In the above illustration, the time interval between 1988 and 1990 is 2 years.

Find out the time interval between the immediately preceding year and the year for which the value is to be interpolated. We may denote this as t_i - t_p.

Time interval between 1988 and 1989 is one year.

Interpolate the value as follows:

For the illustration, the sales for the year 1989 will be,

24000 +
x 1

= 24,000 + 8,000 = 32,000

If the series is a descending series, the formula will be,

Base Value -

x (t_i - t_p)

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