Solving Data Transformation: Median, Range & The Value Of 4p - Q

by TextBrain Team 65 views

Hey guys! Let's dive into a cool math problem involving data transformation. We've got a dataset, and we're going to play around with its median and range. This is a fun one, so buckle up!

Understanding the Problem: Median, Range, and Transformations

Alright, so the problem throws some terms at us: median and range. Let's quickly recap what they mean, just to make sure we're all on the same page. The median is simply the middle value in a dataset when the values are arranged in order. If you have an even number of values, it's the average of the two middle numbers. The range, on the other hand, is the difference between the highest and lowest values in the dataset. Easy peasy, right? We're also dealing with a transformation here – multiplying each data point by a number (p) and then subtracting another number (q). The challenge is, how do these operations affect the median and range?

We're told that we start with a dataset with a median of 16 and a range of 6. This is our original dataset. Now, we're going to apply a transformation: Every value in the original data is multiplied by p, and then q is subtracted from the result. This creates a new dataset, and we know the new median is 20 and the new range is 9. Our ultimate goal is to find the value of 4p - q. Sounds exciting, doesn't it? This problem is all about understanding how these transformations impact our two key statistics: median and range. This is a classic example of how a simple change in each data point can lead to significant changes in summary statistics. So, let's get down to the specifics of how to solve this type of problem. It's all about keeping track of those transformations.

Now, the first part of this challenge involves grasping what happens to the median and the range when each data point undergoes a linear transformation (multiplication and subtraction). Knowing the impact of each operation on these two key statistical measurements is the critical component of solving these types of math problems. This step is important because it lays the foundation for solving the problem and helps ensure that we avoid errors or get misled during problem-solving. The main point here is to create a good understanding so that we can tackle the problem by converting the given values and applying appropriate formulas.

The Impact of Transformations on Median and Range

Okay, so here's the lowdown on how these transformations affect the median and range. When you multiply each value in a dataset by a constant (p), the median also gets multiplied by that constant. Makes sense, right? If every number gets scaled up or down, the middle number will too. But what happens when we subtract a constant (q)? Well, the median gets shifted by that amount as well. So, if the original median was 16, and we multiply by p and subtract q, the new median will be 16p - q. Easy enough! The impact on the range is a little simpler. Multiplying each value by p means the range also gets multiplied by the absolute value of p. Why the absolute value? Because the range is a measure of spread, and we only care about the magnitude of the change, not whether the numbers get larger or smaller. Subtracting q doesn't change the range at all; it just shifts the entire dataset up or down. We only need to focus on what happens when each value is multiplied by a factor. The process helps us understand the impact of transformations on the range, and this understanding is key to finding the answer. The application of the concepts above to derive the formula to find the missing values helps to develop critical thinking.

With the understanding of the impact on the median and range, it is now easier to solve the problem. The core concepts and understanding of this part are very vital in the process of solving this problem. In short, when you multiply the data by a factor, the median is multiplied by the same factor, and the range is multiplied by the absolute value of that factor. Meanwhile, subtracting a constant will shift the median, but the range will stay the same.

Setting Up the Equations

Now let's get those equations rolling! We know the original median is 16, and the transformation gives us a new median of 20. Using our formula, we get:

16p - q = 20

This equation encapsulates the effect of the transformation on the median. Next, let's look at the range. The original range is 6, and the new range is 9. The range changes only because of the multiplication by p, hence:

6 * |p| = 9

From this equation, we can derive the absolute value of p. So, let's solve for p. But before we jump into the calculation, let's pause to review how the steps above are crucial in solving the math problem. The two equations we constructed from the relationships between the original and transformed medians and ranges lay the groundwork for us to solve for our unknowns p and q. The crucial step here is to translate the information we have into the right mathematical formulas that we can use to get the answers we need. We're just a step away from unlocking the value of the expression we are looking for, 4p - q. Note that the absolute value of p plays a crucial role in this type of problem, because we need to know how the range is impacted by the transformation.

Solving for p and q

Let's solve for p first. From the range equation, we have 6 * |p| = 9. Dividing both sides by 6, we get |p| = 9/6 = 3/2 = 1.5. Therefore, p can be either 1.5 or -1.5. Now we need to figure out which value of p is correct by substituting it into the median equation. Let's consider both cases:

  • If p = 1.5, then 16 * 1.5 - q = 20, which simplifies to 24 - q = 20. Solving for q, we get q = 4.
  • If p = -1.5, then 16 * -1.5 - q = 20, which simplifies to -24 - q = 20. Solving for q, we get q = -44.

Now that we have possible values of p and q, we can calculate 4p - q for both cases:

  • If p = 1.5 and q = 4, then 4p - q = 4(1.5) - 4 = 6 - 4 = 2.
  • If p = -1.5 and q = -44, then 4p - q = 4(-1.5) - (-44) = -6 + 44 = 38.

Since we are looking for one unique value, and the answer must be among the options, we can discard the second case where we arrived at the value of 38.

Finding the Answer: 4p - q

We've found that when p = 1.5 and q = 4, then 4p - q = 2. Let's check our answers against the multiple-choice options. We have successfully navigated this math problem. It is an excellent example that showcases how easy it is to solve the problem by using a systematic approach. This methodical approach allows you to handle complex problems with ease. The process also teaches us how to break down the problem into smaller parts and solve each part step by step. The solution requires only basic algebraic operations and understanding of how transformations affect medians and ranges. The answer is (A) 2. Great job, guys!

Final Thoughts

So there you have it! We've successfully solved a data transformation problem by understanding the behavior of the median and range. Remember, when multiplying data by a constant, the median and range change proportionally. Subtracting a constant affects only the median. Keep practicing, and you'll become a data transformation master! Feel free to ask any questions you might have. Until next time, keep exploring the fascinating world of math!