# Linear Interpolation Formula: Example, Statistics, uses, history and more

The linear interpolation formula is the most basic approach for predicting a function’s value between any two known values. In addition, the linear interpolation formula is a useful method for fitting curves with linear polynomials. Then, in general, the interpolation approach is used to discover new values for any function based on a set of values. Then, the linear interpolation formula is used to find the unknown values in the table. However, Linear interpolation is a technique for fitting curves with linear polynomials. Then, it aids in the construction of new data points within the range of a discrete set of previously known data points. Also, as a result, linear interpolation is the simplest approach for predicting a channel from a vector of estimates for the given channel. Then, it is extremely valuable for data prediction, forecasting, market research, and a variety of other mathematical and scientific applications.

## Linear Interpolation Formula

However, the linear interpolation method is used for data forecasting, prediction, mathematical and scientific applications, market research, and other purposes. Then, to discover the unknown values in the table, apply the linear interpolation formula. So, the formula for linear interpolation is as follows:

Linear Interpolation(y) = y1 + (x- x1) (y2 – y1)/ (x2 – x1)

Moreover, this formula finds the best fit curve as a straight line by utilising the coordinates of two provided values. Then, at a known value of x, this will produce any needed value of y.

So, in this formula, we have words like:

Atfirst, the first coordinates are x1 and y1.

Then, the second coordinates are x2 and y2.

Then, x is the interpolation point, and y is the interpolated value.

## Linear Interpolation Formula examples

### Example 1

1. Find the value of y at x = 4 given some set of values (2, 4), (6, 7).

Solution: So, we know the values of x and y.

(x = 4, x1 = 2, x2 = 6, y1 = 4, y2 = 7)

So, we know the interpolation formula, which is:

Linear Interpolation(y) = y1 + (x- x1) (y2 – y1)/ (x2 – x1)

y= 4 + (4 – 2)(7 – 4)/ (6 – 2)

Then, y= 112

Hence, the value of y is 112.

### Example 2

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

Solution: So, we know the values of x and y.

(x = 6, x1 = 3, x2 = 6, y1 = 4, y2 = 8)

So, We know the interpolation formula, which is:

Linear Interpolation(y) = y1 + (x- x1) (y2 – y1)/ (x2 – x1)

y= 4 + (6 – 3)(8 – 4)/ (6 – 3)

Then, y= 8

Hence, the value of y is 8.

### Example 3

1. 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: So, we know the values of x and y.

(x = 4, x1 = 3, x2 = 5, y1 = 5, y2 = 6)

So, we know the interpolation formula, which is:

Linear Interpolation(y) = y1 + (x- x1) (y2 – y1)/ (x2 – x1)

y= 5 + (4 – 3)(6 – 5)/ (5 – 3)

Then, y= 5.5

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

### Example 4

1. Consider the following table of data:

 Day 1 3 5 7 9 Height 0 4 8 12 16

Solution: So, we know the values of x and y.

(x1 = 3, x2 = 5, y1 = 4, y2 = 8)

So, we know the interpolation formula, which is:

Linear Interpolation(y) = y1 + (x- x1) (y2 – y1)/ (x2 – x1)

y= 4 + (x – 3)(8 – 4)/ (5 – 3)

Then, y= 2x – 2

So, for the fourth day x = 4.

Then, y=2×4–2

y = 6

Atlast, on the fourth day the height will be 6 units.

## Linear Interpolation Formula statistics

Moreover, the technique of determining a value between two points on a line or curve is known as interpolation. Then, to assist us remember what it implies, consider the initial half of the term, ‘inter,’ as meaning ‘enter’. Also, it reminds us to examine ‘within’ the data we started with. Then, this tool, interpolation, is important not just in statistics, but also in science, commerce, and any other situation where it is necessary to estimate values that lie between two current data points. Moreover, here’s an example to help you understand the notion of interpolation. Every other day, a gardener measured and tracked the growth of a tomato plant. This gardener is inquisitive, and she wants to know how tall her plant was on the fourth day.

1. Day: 1,3,5,7,9 and Height: 0,4,8,12,16

Based on the graph, it’s not difficult to conclude that the plant was probably 6 mm tall on the fourth day. Then, this is because the tomato plant grew in a straight line; there was a straight line link between the number of days measured and the plant’s height development. Then, a linear pattern is formed when the points form a straight line. So, we could even make an educated guess by putting the data on a graph.

## Linear Interpolation Formula in finance

First, Linear Interpolation(y) = y1 + (x- x1) (y2 – y1)/ (x2 – x1)

As we learnt in the preceding definition, it aids in determining a value based on other sets of values, as seen in the following formula: –

X and Y are unknown figures that will be determined based on the other variables supplied.

Then, sets of variables are assigned to Y1, Y2, X1, and X2 to aid in the determination of an unknown value.

For example, a farmer involved in mango tree farming observes and gathers the following data on the height of the tree on specific days, as shown below: –

 No. of Days 1 2 3 4 Height of tree (mm) 10 20 30 40

Farmers can predict the height of trees for any number of days based on the data provided until the tree achieves its usual height. The farmer wishes to know the height of the tree on the seventh day based on the facts above. So, he can figure it out by interpolating the values shown above. Then, on the seventh day, the tree will be 70 MM tall.

## Linear Interpolation Formula excel

However, Excel does not have a linear interpolation function. But the FORECAST function may be used to do linear interpolation when there are just two pairs of x- and y-values.

So, the syntax is as follows:

=FORECAST(x,known ys,known xs)

where:

x is the input value, while known ys and known xs are the known y-values and known xs, respectively.

Moreover, the FORECAST function estimates the value of y that corresponds to the input value x using linear regression. When there are just two data points, linear regression produces the same result as linear interpolation. When the number of x-values and y-values is larger than 2, the FORECAST function will NOT provide an interpolated y-value. Then, to interpolate between x- and y-values in a big data set (more than two pairs of values), use XLOOKUP or INDEX and MATCH to find a pair of x- and y-values to interpolate between.

However, Linear interpolation is based on the assumption that the change in y for a given change in x is linear. Then, in most circumstances, linear interpolation in Excel will produce suitably accurate results. So, if you want even higher precision, you may want to use a more complex approach such as cubic splines.

## Linear Interpolation Formula from a table

However, Linear interpolation is useful when looking for a value between two or more data points. As a result, mathematicians refer to it as “filling in the gaps” for a set of data values in tabular format. Then, the linear interpolation approach is to employ a straight line to link the available data points on both the positive and negative sides of the unknown location.

Moreover, Linear interpolation is frequently inaccurate when dealing with non-linear data. If the data points in the set vary by a considerable amount, linear interpolation may not provide a decent approximation. So, it also entails calculating a new value by connecting two neighbouring known values using a straight line.

### Example

1. Day: 1,3,5,7,9 and Height: 0,4,8,12,16

Solution: So, we know the values of x and y.

(x1 = 3, x2 = 5, y1 = 4, y2 = 8)

We know the interpolation formula, which is:

Linear Interpolation(y) = y1 + (x- x1) (y2 – y1)/ (x2 – x1)

y= 4 + (x – 3)(8 – 4)/ (5 – 3)

Then, y= 2x – 2

So, for the fourth day x = 4.

Then, y=2×4–2

Then, y = 6

Atlast, on the fourth day the height will be 6 units.

## Linear Interpolation Formula history

Linear interpolation has been used to fill gaps in tables since antiquity. Then, assume you have a table with the population of a nation in 1970, 1980, 1990, and 2000, and you wish to estimate the population in 1994. Also, Linear interpolation is a simple method for accomplishing this. Then, it is said to have been employed throughout the Seleucid Empire (final three centuries BC) and by the Greek astronomer and mathematician Hipparchus (second century BC). However, Linear interpolation is described in the ancient Chinese mathematical treatise The Nine Chapters on the Mathematical Art. However, it dates from 200 BC to AD 100, as well as Ptolemy’s Almagest (2nd century AD). Then, in computer graphics, the basic technique of linear interpolation between two variables is extensively utilised. Then, it is frequently referred to as a lerp in that field’s parlance (from linear interpolation). Moreover, for the operation, the phrase can be used as a verb or a noun. “Bresenham’s algorithm lerps progressively between the line’s two endpoints,” for example.

Also, all current computer graphics processors have Lerp functions in their circuitry. Then, they are frequently used as building pieces for more sophisticated processes, such as bilinear interpolation, which may be completed in three lerps. Because this procedure is inexpensive, it is also a suitable approach to build accurate lookup tables with speedy lookup for smooth operations that do not need a large number of table entries.

## Linear Interpolation formula multivariate

However, Linear interpolation in this context refers to data points in a single spatial dimension. Then, the expansion of linear interpolation to two spatial dimensions is known as bilinear interpolation, and to three dimensions, trilinear interpolation. However, these interpolants are no longer linear functions of the spatial coordinates. But rather products of linear functions; this is exemplified in the figure below by the plainly non-linear example of bilinear interpolation. Then, different linear interpolation extensions may be applied to other types of meshes, such as triangular and tetrahedral meshes, as well as Bézier surfaces. So, these are, in fact, higher-dimensional piecewise linear functions.

## Linear Interpolation formula for x

y = y1 + ((x – x1) / (x2 – x1)) * (y2 – y1), where x is the known value, y is the unknown value, x1 and y1 are coordinates below the known x value, and x2 and y2 are positions above the x value.

Linear Interpolation(y) = y1 + (x- x1) (y2 – y1)/ (x2 – x1)

This formula finds the best fit curve as a straight line by utilising the coordinates of two provided values. Then, at a known value of x, this will produce any needed value of y. In this formula, we have words like:

Atfirst, the first coordinates are x1 and y1.

Then, the second coordinates are x2 and y2.

Then, x is the interpolation point, and y is the interpolated value.

## Linear Interpolation formula a level maths

Because the median is Q2, divide by n/2 and round up if the result is not a whole number. Then apply interpolation (where n = total frequency).

Let’s look at some real-world examples.

class | frequency

10  a  20 | 20

20  a  30 | 30

30  a  40 | 40

Total frequency = n = 20+30+40=70

35 = n/2

The 35th value is what we’re looking for. This value obviously falls between 20 and 30, indicating that we are seeking for the (35-20 =) 15th value in this class. Determine the class width. In this situation, the answer is 30-20 = 10. (For example, if the classes were 10-19 and 20-29, the class width is 29.5-19.5)

Do class width/freq = 10/30 = 1/3 now.

Multiply this by the number of students in the class.

15/3 = 15/3 * 15/3 = 5

Do lower bound + 5 = 20+5 = 25 now.

## Linear Interpolation formula thermodynamics

Linear interpolation is a type of bracketing procedure. This approach, like the bisection method, finds the estimated position of the root through a defined interval. The root of the equation is determined using linear interpolation between two points of function at either end of the interval in this approach. In most circumstances, this approach outperforms the bisection method in terms of convergence speed. In summary, Figure 1 depicts the linear interpolation approach.

Figure 1 depicts the linear interpolation approach.

The linear interpolation method’s algorithm is as follows:

Step 1: Select a and b such that f (a).

f(b)<0.

Step 2: Assume c=a – f. (a).

(b-a) / (f(b) – f(a)

Step 3: If f is positive (a).

If f(c)0, then let b=c; otherwise, let a=c.

Step 4: If |a-b|e, then root = a – f. (a).

(b-a)/(f(b) – f(a)), else on to step 2.

Step 5: Finish.

## Linear Interpolation formula python

Linear interpolation is a technique for calculating the values of functions at any intermediate point given the values of two nearby places. However, Linear interpolation is the process of estimating an unknown value that lies between two known values. Moreover, Linear interpolation is utilised in a variety of fields, including statistics, economics, and pricing determination. Then, it is used to fill gaps in statistical data for the benefit of information continuity. We may Linearly interpolate the supplied data point using the formula below.

Linear Interpolation: y(x) = y1 + (x – x1) frac(y2 – y1) frac(x2 – x1)

The coordinates of the first data point are (x1, y1). And (x2,y2) are the coordinates of the second data point, where x is the interpolation point and y is the interpolated value.

## Example

Let’s look at an example to help you understand. We have the following data values, where x is a number and y is a function of x’s square root. Our job is to calculate the square root of 5.5. (x).

 X 1 2 3 4 5 6 y ( f(x) = √x ) 1 1.4142 1.732 2 2.236 2.4494

In this case, we may apply Linear Interpolation.

1. Determine the two adjacent (x1, y1), (x2, y2) from the x, i.e. (5,2.2360) and (6,2.4494).

In this case, x1 = 5, x2 = 6, y1 = 2.2360, y2 = 2.4494, and we interpolate at x = 5.5.

1. Applying the formula y(x) = y1 + (x – x1) frac(y2 – y1) (x2 – x1)

y(x)  =  y1  +  (x – x1)  \frac{(y2 – y1) }{ (x2 – x1)}

1. After plugging the data into the preceding equation.

y = 2.2360 plus (5.5-5)(2.4494-2.2360)/(6 – 5)

Then, y = 2.3427

The value of Y at x = 5.5 is 2.3427. So, we can quickly find the value of a function between two intervals by using linear interpolation.

### What is Meant by Linear Interpolation Formula?

However, the linear interpolation formula is a handy approach for fitting curves with linear polynomials. Then, in general, the interpolation approach is used to discover new values for any function based on a set of values. Then, the linear interpolation formula is used to find the unknown values in the table. However, the linear interpolation method is used for data forecasting, prediction, mathematical and scientific applications, market research, and other purposes.

(y) = y1+(x−x1)(y2−y1)x2−x1y1+(x−x1)(y2−y1)x2−x1

### What is the Formula to Calculate Linear Interpolation Formula?

Linear Interpolation(y) = y1 + (x- x1) (y2 – y1)/ (x2 – x1)

This formula finds the best fit curve as a straight line by utilising the coordinates of two provided values. Then, at a known value of x, this will produce any needed value of y.

In this formula, we have words like:

Atfirst, the first coordinates are x1 and y1.

Then, the second coordinates are x2 and y2.

Then, x is the interpolation point, and y is the interpolated value.

### How do you use linear interpolation?

However, Linear interpolation is useful when attempting to find a value between two or more data points. It may be thought of as “filling in the gaps” of a data table. The linear interpolation approach aims to link the known data points on each side of the unknown location with a straight line.

### How do you linearly interpolate in Excel?

Excel does not have a linear interpolation function, but the FORECAST function may be used to do linear interpolation when there are just two pairs of x- and y-values.

The syntax is as follows:

=FORECAST(x,known ys,known xs)

where:

x is the input value, while known ys and known xs are the known y-values and known xs, respectively. The FORECAST function estimates the value of y that corresponds to the input value x using linear regression. When there are just two data points, linear regression produces the same result as linear interpolation.

### What is linear interpolation A level maths?

Linear interpolation is a method of curve fitting in mathematics that uses linear polynomials to create new data points within the range of a discrete set of existing data points.

### What is the interpolation method?

Interpolation is a statistical strategy for estimating an unknown price or possible yield of an asset using related known variables. Interpolation is accomplished by employing other proven values that are positioned in the same sequence as the uncertain value.

### How do you use interpolation to find percentiles?

To compute an interpolated percentile, perform the following steps: Determine the rank to be used for the percentile. Use this formula: rank = p(n+1), where p is the percentile and n is the sample size. In our case, 0.7*(11 + 1) = 8.4 is the rank for the 70th percentile.

### Some more frequently asked questions

#### 1. What is the linear interpolation formula?

Linear interpolation is a method used to estimate a value between two known points on a straight line. The formula for linear interpolation is as follows:

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

Where x1 and x2 are the known x-values, y1 and y2 are the corresponding known y-values, and x is the point at which we want to estimate the y-value (y).

#### 2. How do you perform linear interpolation using Excel?

In Excel, you can perform linear interpolation using the FORECAST.LINEAR function. The syntax is:

=FORECAST.LINEAR(x, known_y’s, known_x’s)

Where x is the point you want to estimate, known_y’s is the range of known y-values, and known_x’s is the range of known x-values.

#### 3. Can you use a calculator for linear interpolation?

Yes, many scientific calculators have built-in linear interpolation functions. Consult your calculator’s manual or search for instructions online to perform linear interpolation on your specific calculator model.

#### 4. How is linear interpolation used in statistics?

In statistics, linear interpolation is used to estimate missing data points, percentiles, and medians from existing data. It provides a simple and efficient method to approximate values within the range of the given data.

#### 5. Can you provide an example of the linear interpolation formula?

Suppose you have two points (3, 4) and (7, 10), and you want to estimate the y-value at x=5. Using the linear interpolation formula:

y = 4 + (5 – 3) * (10 – 4) / (7 – 3) y = 4 + 2 * 6 / 4 y = 4 + 3 y = 7

#### 6. How do you apply the linear interpolation formula to data from a table?

To apply the linear interpolation formula to data from a table, identify the two closest points surrounding the desired x-value, and use their respective x and y values in the formula.

#### 7. How is linear interpolation used in thermodynamics?

In thermodynamics, linear interpolation is often used to estimate properties like temperature, pressure, and specific volume from steam tables or other property tables, given two known data points.

#### 8. How do you use linear interpolation to find the median?

To find the median using linear interpolation, first, arrange the data in ascending order and calculate the position of the median. Next, identify the two data points surrounding the median position, and use linear interpolation to estimate the median value.

#### 9. How do you use linear interpolation for percentiles?

For percentiles, linear interpolation is used similarly to finding the median. Determine the position of the desired percentile, identify the two surrounding data points, and apply the linear interpolation formula to estimate the value.

#### 10. How do you use linear interpolation for quartiles?

Quartiles are a type of percentile. Use linear interpolation as described above for percentiles, with the positions corresponding to the first (25th percentile), second (50th percentile), and third (75th percentile) quartiles.

#### 11. What is log-linear interpolation?

Log-linear interpolation is a variant of linear interpolation that operates on the logarithm of the y-values. It is particularly useful for interpolating data that exhibits exponential growth or decay.

#### 12. How do you perform log-linear interpolation in Excel?

To perform log-linear interpolation in Excel, first, take the logarithm of the known y-values, and use the FORECAST.LINEAR function as described earlier. Finally, exponentiate the result to obtain the interpolated value.