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Linear Interpolation

- Calculus
- Absolute Maxima and Minima
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- Accumulation Function
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- Surface Area of Revolution
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- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
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- Inductive Reasoning
- Inequalities Maths
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- Integers
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- Interest
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- Law of Cosines in Algebra
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- Laws of Logs
- Limits of Accuracy
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- Bias in Experiments
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- Confidence Interval for Population Mean
- Confidence Interval for Population Proportion
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- Confidence Intervals
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- Linear Interpolation
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- Transforming Random Variables
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- Types of Data in Statistics
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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, 1^{st} quartile and 3^{rd} quartile.

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:

x_{1} and y_{1} are the first coordinates.

x_{2} and y_{2} are the second coordinates.

x is the point to perform the interpolation.

y is the interpolated value.

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)

x_{1} = 3 and y_{1} = 2

x_{2} = 7 and y_{2} = 9

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

There are a few useful steps that will help you compute the desired value such as the median, 1^{st} quartile and 3^{rd} 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 |

0-10 | 5 |

11-20 | 10 |

21-30 | 1 |

31-40 | 8 |

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.

Class | Frequency | CF |

0-10 | 5 | 5 |

11-20 | 10 | 15 |

21-30 | 1 | 16 |

31-40 | 8 | 24 |

41-50 | 18 | 42 |

51-60 | 6 | 48 |

61-70 | 20 | 68 |

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

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 34^{th} position lies in the data using the cumulative frequency.

According to the cumulative frequency, the 34^{th} 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.

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.

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

The position of the 1^{st} quartile is the value.

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

Step 1: solve for the position of the 1^{st} quartile

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

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

Step 3: Given the graph, use linear interpolation to find the specific 1^{st} 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.

Gradient =

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

Gradient =

It follows that,

So the 1^{st} quartile is 32.125.

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

The position of the 3^{rd} quartile is the value.

Step 1: solve for the position of the 3^{rd} quartile

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

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

Step 3: Given the graph, use linear interpolation to find the specific 3^{rd} 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.

Gradient =

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

Gradient =

It follows that,

So the 3^{rd} quartile is 32.125.

- 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, 1
^{st}quartile and 3^{rd}quartile - The position of the median is
- The position of the 1
^{st}quartile is - The position of the 3
^{rd}quartile - A graph of the upper bounds in each class interval plotted against the cumulative frequency can be used to locate the median, 1
^{st}quartile and 3^{rd}quartile. - The gradient formula can be used to find the specific value of the median, 1
^{st}quartile and 3^{rd}quartile

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=y_{1}+(x-x_{1})(y_{2}-y_{1})/(x_{2}-x_{1})

where,

x_{1} and y_{1} are the first coordinates.

x_{2} and y_{2} 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 x_{1, }x_{2,} y_{1} and y_{2} in the below formula

y=y_{1}+(x-x_{1})(y_{2}-y_{1})/(x_{2}-x_{1})

where,

x_{1} and y_{1} are the first coordinates.

x_{2} and y_{2} are the second coordinates.

x is the point to perform the interpolation.

y is the interpolated value.

More about Linear Interpolation

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