Interpolation Formula

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Linear Interpolation Formula

Interpolation Formula: The method of finding new values for any function using the set of values is done by interpolation. The unknown value on a point is found out using this formula. If the linear interpolation formula is concerned then it should be used to find the new value from the two given points. If compared to Lagrange’s interpolation formula, the “n” set of numbers should be available and Lagrange’s method is to be used to find the new value.

Interpolation is the process of finding a value between two points on a line or curve. To help us remember what it means, we should think of the first part of the word, ‘inter,’ as meaning ‘enter,’ which reminds us to look ‘inside’ the data we originally had. This tool, interpolation, is not only useful in statistics but is also useful in science, business, or any time there is a need to predict values that fall within two existing data points.

Interpolation Formula Calculator

Solved ExamplesQuestion 1: Using the interpolation formula, find the value of y at x = 8 given some set of values (2, 6), (5, 9) ?
Solution:

The known values are,x0=8,x1=2,x2=5,y1=6,y2=9y=y1+(x−x1)(x2−x1)×(y2−y1)

y=6+((8−2)(5−2)×(9−6) y = 6 + 6

y = 12

What is linear interpolation method?

Linear interpolation is the simplest method of getting values at positions in between the data points. The points are simply joined by straight line segments.

How do you find the interpolation between two numbers?

Know the formula for the linear interpolation process. The formula is y = y1 + ((x – x1) / (x2 – x1)) * (y2 – y1), where x is the known value, y is the unknown value, x1 and y1 are the coordinates that are below the known x value, and x2 and y2 are the coordinates that are above the x value.

What is the interpolation method?

In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points. … A few data points from the original function can be interpolated to produce a simpler function that is still fairly close to the original.

Interpolation Formula Excel

Here’s an example that will illustrate the concept of interpolation. A gardener planted a tomato plant and she measured and kept track of its growth every other day. This gardener is a curious person, and she would like to estimate how tall her plant was on the fourth day.

Her table of observations looked like this:

Based on the chart, it’s not too difficult to figure out that the plant was probably 6 mm tall on the fourth day. This is because this disciplined tomato plant grew in a linear pattern; there was a linear relationship between the number of days measured and the plant’s height growth. The linear pattern means the points created a straight line. We could even estimate by plotting the data on a graph.

But what if the plant was not growing with a convenient linear pattern? What if its growth looked more like this?

What would the gardener do in order to make an estimation based on the above curve? Well, that is where the interpolation formula would come in handy.

Interpolation Formula Thermo

Linear interpolation has been used since antiquity for filling the gaps in tables. Suppose that one has a table listing the population of some country in 1970, 1980, 1990 and 2000, and that one wanted to estimate the population in 1994. Linear interpolation is an easy way to do this. The technique of using linear interpolation for tabulation was believed to be used by Babylonian astronomers and mathematicians in Seleucid Mesopotamia (last three centuries BC), and by the Greek astronomer and mathematician, Hipparchus (2nd century BC). A description of linear interpolation can be found in the Almagest (2nd century AD) by Ptolemy.

The basic operation of linear interpolation between two values is commonly used in computer graphics. In that field’s jargon, it is sometimes called a lerp. The term can be used as a verb or noun for the operation. e.g. “Bresenham’s algorithm lerps incrementally between the two endpoints of the line.”

Lerp operations are built into the hardware of all modern computer graphics processors. They are often used as building blocks for more complex operations: for example, bilinear interpolation can be accomplished in three lerps. Because this operation is cheap, it’s also a good way to implement accurate lookup tables with a quick lookup for smooth functions without having too many table entries.

Formula For Interpolation

Let us say that we have two known points x1,y1x1,y1 and x2,y2x2,y2.

Now we want to estimate what yy value we would get for some xx value that is between x1x1 and x2x2. Call this yy value estimate — an interpolated value.

Two simple methods for choosing yy come to mind. The first is to see whether xx is closer to x1x1 or to x2x2. If xx is closer to x1x1 then we use y1y1 as the estimate, otherwise, we use y2y2. This is called nearest-neighbor interpolation.

The second is to draw a straight line between x1,y1x1,y1 and x2,y2x2,y2. We look to see the yy value on the line for our chosen xx. This is a linear interpolation.

It is possible to show that the formula of the line between x1,y1x1,y1 and x2,y2x2,y2 is: y=y1+(x−x1)y2−y1x2−x1y=y1+(x−x1)y2−y1x2−x1

Double Interpolation Formula

In order to perform a linear interpolation in Excel, we’ll use the equation below, where x is the independent variable and y is the value we want to look up:

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.

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 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 –

  • Y= Y1 + (Y2-Y1)/(X2-X1) * (X-X1)
  • =80 + (120-80)/(70-50) * (60-50)
  • =80 + 40/20 *10
  • = 80+ 2*10
  • =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:

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 –

  • 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 recession 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 cost 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

  • 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

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.

Linear Interpolation Formula 

The linear interpolation formula is the simplest method that is used for estimating the value of a function between any two known values. Also, the linear interpolation formula is a method that is useful for curve fitting using linear polynomials. Basically, the interpolation method is used for finding new values for any function using the set of values. The unknown values in the table are found using the linear interpolation formula. Let us learn more about the linear interpolation formula in this section. 

What is Linear Interpolation Formula?

The linear interpolation formula is used for data forecasting, data prediction, mathematical and scientific applications and, market research, etc. The linear interpolation formula can be used for finding the unknown values in the table. The formula for linear interpolation formula is given by:

Linear Interpolation Formula 

The formula to calculate linear interpolation is:

Examples Using Linear Interpolation Formula 

Example 1: Find the value of y if x = 6 and some set of values are given as (3, 4), (6, 8)?

Solution:

x = 6 ;

 = 3 ; = 6 ; = 4 ;

= 8 (given)

Using linear interpolation formula,

Example 2: Calculate the estimated height of the boy in the fourth position.

position(x) x 1  2  3 5
Height in feet (y) y
3
 
4.5 5 6

Solution:

x = 4 ;

= 3 ; = 5 ; = 5 ;

= 6 (given)

Using linear interpolation formula,

y = 5 + 1(1/2)

y= 5 + 0.5
y = 5.5

Therefore, the height of the boy in the fourth position is 5.5 feet.

Example 3: Find the value of y if x = 8 and some set of values are given as (5, 3.5), (10, 6)?

Solution: 

x = 8 ;

 = 5 ; = 10 ; = 3.5 ;

= 6 (given)

Using linear interpolation formula,

y = 5 + 1(1/2)

y= 5 + 0.5
y = 5.5

Therefore, the height of the boy in the fourth position is 5.5 feet.

Example 3: Find the value of y if x = 8 and some set of values are given as (5, 3.5), (10, 6)?

Solution: 

x = 8 ;

 = 5 ; = 10 ; = 3.5 ;

= 6 (given)

Using linear interpolation formula,

y = 3.5 +3(2.5/5)
y = 3.5 + 3(1/2)

y = 3.5 + 1.5

y = 5

Therefore, the value of y is 5

FAQs on Linear Interpolation Formula 

What is Meant by Linear Interpolation Formula?

the linear interpolation formula is a method that is useful for curve fitting using linear polynomials. Basically, the interpolation method is used for finding new values for any function using the set of values. The unknown values in the table are found using the linear interpolation formula. The linear interpolation formula is used for data forecasting, data prediction, mathematical and scientific applications and,

What are the Uses of Linear Interpolation Formula?

The linear interpolation formula is helpful in determining the values between any two given points. Hence, linear interpolation is also considered as a method of filling in the gaps for any value in a table format. The formula helps in creating a straight line along with the given points on both the negative and positive sides. 

What is the Interpolation Formula?

The term “Interpolation” refers to the curve fitting technique that is used in the prediction of intermediate values and patterns on the basis of available historical data along with recent data points. In other words, the interpolation technique can be used to predict the missing data points in-between the available data points.

The formula for interpolation is basically building a function for the unknown variable (y) based on the independent variable and at least two data points – (x1, y1) and (x2, y2). Mathematically, it is represented as,

Formula,

Example of Interpolation Formula (With Excel Template)

Let’s take an example to understand the calculation of the Interpolation Formula in a better manner.

Interpolation Formula – Example #1

Let us take the example of a hot Rod to illustrate the concept of interpolation. Suppose that the Temperature of the Rod was 100°C at 9.30 A.M which gradually came down to 35°C at 10.00 A.M. Find the temperature of the Rod at 9.40 A.M based on the given information.

  • Temperature of the rod (y) = (35 – 100) / (1000 – 930) * (940 – 930) + 100֯C
  • Temperature of the rod (y) = 78.33֯C

Therefore, the temperature of the rod was 78.33֯C at 9.40 A.M.

Interpolation Formula – Example #2

Let us take the curious case of John Doe who has been gaining significant weight in the last couple of months. As such, his doctor decided to monitor his weight and so started tracking his weight every 6 days for the last 60 days. The following information has been collected:

Solution:

The weight of John′s is calculated using the formula given below.

y =  (y2 – y1) / (x2 – x1) * (x – x1) + y1

On 14th day

  • On 14th day = (160 – 154) / (18 – 12) * (14 – 12) + 154
  • On 14th day = 156 lbs

On 33rd day

  • On 33rd day = (188 – 180) / (36 – 30) * (33 – 30) + 180
  • On 33rd day = 184 lbs

On 49th day

  • On 49th day = (216 – 210) / (54 – 48) * (49 – 48) + 210
  • On 49th day = 211 lbs

Therefore, John’s weight on 14th, 33rd and 49th day was 156 lbs, 184 lbs, and 211 lbs respectively.

Explanation

The formula for Interpolation can be calculated by using the following steps:

Step 1: Firstly, identify the independent and dependent variables for the function.

Step 2: Next, gather as many as possible historical and current data points in order to build a function. Make sure that there are at least two data points as it is the minimum data points required.

Step 3: Next, calculate the slope of the available data points by dividing the difference between the ordinates by that of the abscissas of the available data points.

Slope =  (y2 – y1) / (x2 – x1)

Step 4: Finally, the function for interpolation can be derived by multiplying the slope (step 3) with the difference between the independent variable and the abscissa of any one data point and then adding the corresponding ordinate to the result as shown below.

y =  (y2 – y1) / (x2 – x1) * (x – x1) + y1

Relevance and Use of Interpolation Formula

The importance of the interpolation technique can be gauzed from the fact that linear interpolation is believed to be used by Babylonian mathematicians and astronomers in the last three centuries BC, while the Greeks and Hipparchus used it in the 2nd century BC. One of the basic variants of interpolation is the linear interpolation technique which is commonly used by analysts in the field of mathematics, finance and computer programming. Please keep in mind that interpolation is a statistical and mathematical tool that is used to predict the intermediate values between two points.

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