# 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.