Methods of Interpolation

Interpolation is the statistical technique used to estimate unknown values within the range of given data. When some values in a series are missing, interpolation helps in finding these values by assuming a smooth and continuous movement in the data.

Among various methods, the Binomial Expansion Method and Newton’s Forward Difference Method are widely used algebraic methods for accurate estimation of missing values.

1. Binomial Expansion Method

The Binomial Expansion Method is an algebraic method of interpolation used when the data values are equally spaced and one or more values in the middle of the series are missing. It assumes that the data follows a polynomial trend and uses binomial coefficients to estimate missing values.

Conditions for Application

• The data must be equally spaced

• Missing values should lie within the given data range

• The number of missing values should be limited

• The series should show a smooth trend

Estimation of One Missing Value Using Binomial Expansion Method

Procedure

• Write the given data in order

• Assume the missing value as a variable (say x)

• Prepare successive differences

• Apply the binomial condition that the sum of alternate differences equals zero

• Solve the equation to find the missing value

Illustrative Problem (One Missing Value)

Second differences:
(x − 7) − 3 = x − 10
(15 − x) − (x − 7) = 22 − 2x
7 − (15 − x) = x − 8

Step 2: Apply binomial condition

For one missing value:
Sum of alternate second differences = 0

(x − 10) + (x − 8) = 22 − 2x

2x − 18 = 22 − 2x

4x = 40

x = 10

Estimated Missing Value = 10

Estimation of Two Missing Values Using Binomial Expansion Method

Procedure

• Assume the two missing values as x and y

• Construct the difference table

• Apply binomial conditions for second and third differences

• Form simultaneous equations

• Solve to obtain missing values

Illustrative Problem (Two Missing Values)

Step 1: Construct difference table

First differences:
x − 5
11 − x
y − 11
23 − y

Second differences:
(11 − x) − (x − 5) = 16 − 2x
(y − 11) − (11 − x) = y − 22 + x
(23 − y) − (y − 11) = 34 − 2y

Step 2: Apply binomial conditions

For two missing values:

Sum of alternate second differences = 0

(16 − 2x) + (34 − 2y) = y − 22 + x

50 − 2x − 2y = x + y − 22

3x + 3y = 72

x + y = 24

(Second equation formed using higher differences or trend assumption)

Solving equations gives:
x = 8
y = 16

Estimated Missing Values: x = 8, y = 16

2. Newton’s Forward Difference Method

Newton’s Forward Difference Method is an algebraic interpolation technique used when the missing value lies near the beginning of the data series and the data is equally spaced. It is based on the principle of finite differences.

Interpolation Problem Using Newton’s Forward Difference Method (One Missing Value)

Illustrative Problem

Step 1: Assume missing value = x

Step 2: Prepare forward difference table

First differences:
x − 40
90 − x
160 − 90 = 70

Second differences:
(90 − x) − (x − 40) = 130 − 2x
70 − (90 − x) = x − 20

Step 3: Apply condition for forward interpolation

Second differences are assumed equal:

130 − 2x = x − 20

3x = 150

x = 50

Interpolated Value = 50