Interpolation Definition Formula Calculation With Examples

What is Interpolation?

Interpolation is the mathematical procedure applied to derive value 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. Hence, one can apply it in estimating varied cost concepts, 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. If the given sets of values work on a linear trend, then we can apply linear Interpolation in excelLinear Interpolation In ExcelIn Excel, linear interpolation refers to forecasting or guessing the next value of any given variable based on current data. To execute a linear interpolation in Excel, we use the forecast function and the lookup function to create a straight line that connects two values and estimates the future value through it.read more 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 values in the above formula: –

X and Y are unknown figures they will ascertain based on other values.Y1, Y2, X1, and X2 are given variables that will help determine unknown values.

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

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

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

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Examples of Interpolation

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

Example #1

Calculate the unknown value using the interpolation formula from the data set. For example, calculate the value of Y when the X value is 60.

Solution:

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

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

So, the calculation of interpolation will be –

Y= Y1 + (Y2-Y1)/(X2-X1) * (X-X1)=80 + (120-80)/(70-50) * (60-50)=80 + 40/20 10= 80+ 210=80+20
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:

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

Here,

So, the calculation of interpolation will be:

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
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 afraid of decreasing the demand for his product. He is 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.

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

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

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.

Relevance And Use

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

Data scientists can use interpolation to analyze and derive meaningful results from a given set of raw values.Organizations can determine any financial information based on a given set of functions like the cost of goods soldCost Of Goods SoldThe Cost of Goods Sold (COGS) is the cumulative total of direct costs incurred for the goods or services sold, including direct expenses like raw material, direct labour cost and other direct costs. However, it excludes all the indirect expenses incurred by the company.
read more, and profits earned, etc.Interpolation is being used in numerous statistical operations to derive meaningful information.Scientists are using this to determine possible results out of numerous estimates.A photographer can also use this concept to determine useful information from raw collected data.

What is Interpolation?#

Interpolation Formula#

Examples of Interpolation#

Example #1#

Example #2#

Example #3#

Interpolation Calculator#

Relevance And Use#

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