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Linear Interpolation

Linear Interpolation

In statistics, linear interpolation is often used to find the estimated median, quartiles or percentiles of a set of data and particularly when the data is presented in a group frequency table with class intervals. In this article we will look at how to do a linear interpolation calculation with the use of a table and graph to find the median, 1st quartile and 3rd quartile.

Linear interpolation formula

The linear interpolation formula is the simplest method used to estimate the value of a function between any two known points. This formula is also useful for curve fitting using linear polynomials. This formula is often used for data forecasting, data prediction and other mathematical and scientific applications. The linear interpolation equation is given by:

where:

x1 and y1 are the first coordinates.

x2 and y2 are the second coordinates.

x is the point to perform the interpolation.

y is the interpolated value.

Solved example for linear interpolation

The best way to understand linear interpolation is through the use of an example.

Find the value of y if x = 5 and some set of value given are (3,2), (7,9).

Step 1: First assign each coordinate the right value

x = 5 (note that this is given)

x1 = 3 and y1 = 2

x2 = 7 and y2 = 9

Step 2: Substitute these values into the equations, then get the answer for y.

How to do linear interpolation

There are a few useful steps that will help you compute the desired value such as the median, 1st quartile and 3rd quartile. We will go through each step with the use of an example so that it is clear.

In this example, we will look at grouped data with class intervals.

Class
Frequency
0-105
11-2010
21-301
31-408
41-50
18
51-60
6
61-70
20

Frequency is how often a value in a specific class appears in the data.

Step 1: Given the class and the frequency, you have to create another column called the cumulative frequency (also known as CF).

Cumulative frequency is therefore defined as the running total of frequencies.

ClassFrequencyCF
0-1055
11-201015
21-30116
31-40824
41-501842
51-60648
61-702068

Step 2: Plot the cumulative frequency graph. To do this, you plot the upper boundary of the class against the cumulative frequency.

Linear Interpolation Cumulative frequency graph StudySmarter

Finding the median

The median is the value in the middle of the data.

The position of the median is at the value, where n is the total cumulative frequency

In this example, n = 68

Step 1: Solve for the position of the median

Step 2: Look for where the 34th position lies in the data using the cumulative frequency.

According to the cumulative frequency, the 34th value lies in the 41-50 class interval.

Step 3: Given the graph, use linear interpolation to find the specific median value.

We treat the segment of the graph where the class interval lies as a straight line and use the gradient formula to assist.

Linear Interpolation Cumulative frequency graph StudySmarter

Gradient =

We can manipulate this formula and substitute the value of the median (m) as the upper bound and the position of the median as the median cf which is also equal to the gradient.

Gradient =

So it follows that,

So the median is 46.

Finding the first quartile

The 1st quartile is also known as the lower quartile. This is where the first 25% of the data lies.

The position of the 1st quartile is the value.

The steps to find the 1st quartile are very similar to the steps to find the median.

Step 1: solve for the position of the 1st quartile

Step 2: Look for where the 17th position lies in the data using the cumulative frequency.

According to the cumulative frequency, the 17th value lies in the 31-40 class interval.

Step 3: Given the graph, use linear interpolation to find the specific 1st quartile value.

We treat the segment of the graph where the class interval lies as a straight line and use the gradient formula to assist.

Linear Interpolation Cumulative frequency graph StudySmarter

Gradient =

We can manipulate this formula and substitute the value of the 1st quartile (Q1) as the upper bound and the position of the 1st quartile as the 1st quartile cf which is also equal to the gradient.

Gradient =

It follows that,

So the 1st quartile is 32.125.

Finding the third quartile

The 1st quartile is also known as the lower quartile. This is where the first 25% of the data lies.

The position of the 3rd quartile is the value.

Step 1: solve for the position of the 3rd quartile

Step 2: Look for where the 51st position lies in the data using the cumulative frequency.

According to the cumulative frequency, the 51st value lies in the 61-70 class interval.

Step 3: Given the graph, use linear interpolation to find the specific 3rd quartile value.

We treat the segment of the graph where the class interval lies as a straight line and use the gradient formula to assist.

Linear Interpolation Cumulative frequency graph StudySmarter

Gradient =

We can manipulate this formula and substitute the value of the 3rd quartile (Q3) as the upper bound and the position of the 3rd quartile as the 3rd quartile cf which is also equal to the gradient.

Gradient =

It follows that,

So the 3rd quartile is 32.125.

Linear Interpolation - Key takeaways

  • Linear interpolation is used to find an unknown value of a function between any two known points.
  • The formula for linear interpolation is
  • Linear interpolation can also be used to find the median, 1st quartile and 3rd quartile
  • The position of the median is
  • The position of the 1st quartile is
  • The position of the 3rd quartile
  • A graph of the upper bounds in each class interval plotted against the cumulative frequency can be used to locate the median, 1st quartile and 3rd quartile.
  • The gradient formula can be used to find the specific value of the median, 1st quartile and 3rd quartile

Frequently Asked Questions about Linear Interpolation

Linear interpolation is a method to fit a curve using linear polynomials.

How to calculate linear interpolation: Linear interpolation can be calculated using the formula

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

where,

x1 and y1 are the first coordinates.

x2 and y2 are the second coordinates.

x is the point to perform the interpolation. 

y is the interpolated value. 

How to use linear interpolation: Linear interpolation can be be used by substituting the values of x1, x2, y1  and y2 in the below formula

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

where,

x1 and y1 are the first coordinates.

x2 and y2 are the second coordinates.

x is the point to perform the interpolation. 

y is the interpolated value. 

Final Linear Interpolation Quiz

Question

What is linear interpolation?

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Answer

Linear interpolation is a statistical method used to find the estimated median, quartiles, or percentiles of a set of data and particularly when the data is presented in a group frequency table with class intervals. 

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Question

One of the simplest methods used to estimate the value of a function between any two known points is known as ____

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Answer

Linear interpolation

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Question

if a set of values are given as (1, 7), (8, 9), what will be y2 considering the linear interpolation formula?

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Answer

y2 = 9

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Question

How often a value in a specific class appears in data is known as ____


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Answer

Frequency

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Question

What is cummulative frequency?

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Answer

 Cumulative Frequency is defined as the running total of frequencies

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Question

Where the lower 25% of a data lies is called the____

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Answer

Lower quartile or !st quartile

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Question

What will be the mean given given the data points of (23, 78, 43, 22, 12)

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Answer

35.6

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Question

  • A graph of the upper bounds in each class interval plotted against the cumulative frequency can be used to locate the median, 1st quartile and 3rd quartile. Is this statement true or false?

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Answer

True

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Question

The gradient formula can be used to find the specific value of the median, 1st quartile and___


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Answer

3rd quartile

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