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**

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.

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.

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.

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.

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.

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.

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