Interpolation

What is Interpolation?

Interpolation can be described as the mathematical procedure applied in order to derive value in between two points having a prescribed value. In simple words, we can describe it as a process of approximating the value of a given function at a given set of discrete points. It can be applied in estimating varied concepts of cost, mathematics, statistics, etc.

Interpolation can be said as the method of determining the unknown value for any given set of functions with known values. The unknown value is found out. If the given sets of values work on a linear trend, then we can apply linear Interpolation in excel to determine the unknown value from the two known points.

Interpolation Formula

The formula is as follows: –

Y = Y1 + (Y2 – Y1)/(X2 – X1) * (X * X1)

As we have learned in the definition stated above, it helps to ascertain a value based on other sets of value, in the above formula: –

  • X and Y are unknown figures which will be ascertained on the basis of other values given.
  • Y1, Y2, X1, and X2 are given sets of variables that will help in determining unknown value.

For example, A farmer engaged in farming of mango trees observes and collects the following data regarding the height of the tree on particular days shown as follows: –

No. of DaysHeight of Tree (MM)
110
220
330
440

Based on the given set of data, farmers can estimate the height of trees for any number of days until the tree reaches its normal height. Based on the above data, the farmer wants to know the height of the tree on the 7th day.

He can find it out by interpolating the above values. The height of the tree on the 7th day will 70 MM.

Interpolation-Formula

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Examples of Interpolation

Now, let us understand the concept with the help of some simple and practical examples.

You can download this Interpolation Formula Excel Template here – Interpolation Formula Excel Template

Example #1

Calculate the unknown value using the interpolation formula from the given set of data. Calculate the value of Y when X value is 60.

XY
100
3040
5080
70120
90160

Solution:

The value of Y can be derived when X is 60 with the help of Interpolation as follows: –

Here X is 60, Y needs to be determined. Also,

X150
Y180
X270
Y2120

So, the Calculation of Interpolation will be –

Interpolation Formula - Example 1.3
  • Y= Y1 + (Y2-Y1)/(X2-X1) * (X-X1)
  • =80 + (120-80)/(70-50) * (60-50)
  • =80 + 40/20 *10
  • = 80+ 2*10
  • =80+20
Interpolation Formula - Example 1.4
  • Y = 100

Example #2

Mr. Harry shares details of Sales and profits. He is eager to know the profits of his business when the sales figure reaches $75,00,000. You are required to calculate profits based on the given data:

SalesProfits
1,000,000200,000
2,000,000300,000
3,000,000400,000
4,000,000500,000
5,000,000600,000

Solution:

Based on the above data, we can estimate the profits of Mr. Harry using the interpolation formula as follows:

Here

X7,500,000
Y1500,000
Y2600,000
X14,000,000
X25,000,000

So, the Calculation of Interpolation will be –

2.3 - Calculation of Y
  • Y= Y1 + (Y2-Y1)/(X2-X1) * (X-X1)
  • = $ 5,00,000 + ($6,00,000 – $5,00,000)/($50,00,000 – $40,00,000) * ($75,00,000 – $40,00,000)
  • = $ 5,00,000 + $1,00,000 / $10,00,000 * $ 35,00,000
  • = $5,00,000 + $ 3,50,000
2.4 - Value of Y
  • Y = $8,50,000

Example #3

Mr. Lark shares details of production and costs. In this era of global recessionGlobal RecessionGlobal recession refers to a condition when the countries across the globe experience an economic downturn for an extended period. It is a form of synchronized economic downfall encountered by the various interrelated economies throughout the world.read more fears, Mr. Lark is also having a fear of decreasing the demands of his product and eager to know the optimum production level to cover the total cost of his business. You are required to calculate the optimum quantity level of production based on the given data. Lark wants to determine the quantity of production required to cover the estimated costEstimated CostCost estimate is the preliminary stage for any project, operation, or program in which a reasonable calculation of all project costs is performed and thus requires precise judgement, experience, and accuracy.read more of $90,00,000.

QuantityCost ($)
10,0004,000,000
200,0004,500,000
300,0005,000,000
400,0005,500,000
500,0006,000,000

Solution:

Based on the above data, we can estimate the quantity required to cover the cost of $90,00,00 using interpolation formula as follows:

Here,

Y9,000,000
Y15,500,000
Y26,000,000
X1400,000
X2500,000

Y= Y1 + (Y2-Y1)/(X2-X1) * (X-X1)

To get the quantity of production required we have modified the above formula as follows

X = (Y – Y1) /[(Y2-Y1)/(X2-X1)] + X1

Quantity of Production (X)
  • X =(9,000,000 – 5,500,000) /[(6,000,000 – 5,500,000) / (500,000 – 400,000)] + 400,000
  • = 3,500,000 /(5,00,000/1,00,000) + 400,000
  • = 3,500,000 /5 + 400,000
  • = 7,00,000 + 400,000
  • = 11,00,000 Units

Interpolation Calculator

You can use the following calculator.

X
X1
X2
Y1
Y2
Interpolation Formula
 

Interpolation Formula = Y1 + (Y2 - Y1)/(X2 - X1) * (X - X1)
0 + (0 - 0)/(0 - 0) * (0 - 0) = 0

Relevance And Use

In the era where data analysis plays an important role in each and every business, an organization can make varied use of interpolation to estimate different values from the known set of values. Mentioned below are some of the relevance and uses of interpolation.

  • Interpolation can be used by data scientists to analyze and derive meaningful results from a given set of raw values.
  • It can be applied by an organization to determine any financial information which is based on a given set of function like the cost of goods sold; profits earned, etc.
  • Interpolation is being used in numerous statistical operations to derive meaningful information.
  • This is being used by scientists to determine possible results out of numerous estimates.
  • This concept can also be used by a photographer to determine useful information out of raw collected data.

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This has been a guide Interpolation and its definition. Here we discuss the formula for the calculation of Interpolation along with examples and downloadable excel sheets. You may also have a look at the following articles –

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