# Linear Interpolation

## Mastering Linear Interpolation in Statistics

In statistics, linear interpolation is a powerful tool used to estimate key values in a dataset, such as the median, quartiles, and percentiles. This method is especially useful when working with grouped data and class intervals. Let's take a closer look at the steps involved in calculating linear interpolation and how to find the desired values using a table and graph.

But first, let's review the formula for linear interpolation:

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

Where:

• x1 and y1 represent the coordinates of the first known point
• x2 and y2 represent the coordinates of the second known point
• x is the point at which we want to estimate the value
• y is the estimated value

Example:

Find the value of y when x = 5, given the following coordinates: (3,2) and (7,9).

Step 1: Assign values to each coordinate:

• x = 5 (given)
• x1 = 3 and y1 = 2
• x2 = 7 and y2 = 9

Step 2: Plug the values into the formula and solve for y:

y = 2 + (5 - 3)(9 - 2) / (7 - 3) = 4.5

## Performing Linear Interpolation

To successfully perform linear interpolation and accurately estimate key values within a dataset, follow these essential steps:

Step 1: First, take note of the class intervals and their corresponding frequencies. Next, create a column for the cumulative frequency (CF).

Note: The cumulative frequency is simply the running total of the frequencies.

Step 2: Plot the cumulative frequency graph, with the upper boundary of each class interval on the x-axis and the cumulative frequency on the y-axis.

## Finding the Median

In statistics, the median is the middle value in a dataset. To find the median using linear interpolation, follow these steps:

Step 1: Determine the position of the median:

n = total cumulative frequency

Step 2: Locate the class interval where the n/2th value lies on the cumulative frequency graph.

Step 3: Perform linear interpolation to find the specific value of the median.

## Finding the 1st Quartile

The 1st quartile represents the value at which the first 25% of the data lies. To find this value using linear interpolation, follow the steps below:

Step 1: Calculate the position of the 1st quartile:

n/4 = position of the 1st quartile

Step 2: Locate the class interval where the n/4th position lies on the cumulative frequency graph.

Step 3: Use linear interpolation to find the value of the 1st quartile.

## Finding the 3rd Quartile

The 3rd quartile represents the value at which the first 75% of the data lies. The steps for finding the 3rd quartile are similar to those for finding the 1st quartile. To find the 3rd quartile:

Step 1: Calculate the position of the 3rd quartile:

3n/4 = position of the 3rd quartile

Step 2: Locate the class interval where the 3n/4th position lies on the cumulative frequency graph.

Step 3: Use linear interpolation to find the value of the 3rd quartile.

## Key Takeaways

Understanding linear interpolation and how to perform it is essential for making accurate estimations in statistics. Whether you need to find the median, quartiles, or percentiles of a dataset, linear interpolation is a helpful tool to have in your arsenal. By following the steps outlined in this article, you can confidently calculate linear interpolation and make precise estimates.

## Maximizing Accuracy with Linear Interpolation

The use of linear interpolation has widespread applicability, from predicting data points on a line graph to determining missing values in a data set. It is a straightforward and efficient method for approximating values between known data points.

To utilize linear interpolation, input the values for x1, x2, y1, and y2 into the formula y = y1 + (x - x1)(y2 - y1) / (x2 - x1). Here, x1 and y1 represent the first coordinates, x2 and y2 represent the second coordinates, and x stands for the point where you wish to apply the interpolation. The resulting y value will serve as your estimated value.

In summary, linear interpolation is a valuable approach for making estimations and determining values in various scenarios. By comprehending the formula and its utilization, you can easily incorporate linear interpolation into your problem-solving repertoire.