Find The Median: A Step-by-Step Guide

Alright, guys, let's dive into a fun little statistics problem! We've got a set of numbers and we need to find the median. Then, we're going to tweak one of the numbers and see how that changes things. Finally, we'll average the two medians we found. Sounds like a plan? Let's get started!

Understanding the Median

Before we jump into the calculations, let's make sure we're all on the same page about what the median actually is. The median is simply the middle value in a dataset when the data is arranged in ascending or descending order. It's a measure of central tendency that's particularly useful because it's not as affected by extreme values (outliers) as the mean (average) is. So, if you have a dataset with some really big or really small numbers, the median can give you a better sense of the 'typical' value.

For example, think about house prices in a neighborhood. If there's one mega-mansion that's worth millions while all the other houses are worth a few hundred thousand, the average house price will be skewed upwards by that mansion. The median house price, on the other hand, will give you a more accurate idea of what most houses in the neighborhood are actually worth. So, the median is your friend when you want to understand the center of your data without being misled by outliers. Keep this in mind as we proceed with our calculations.

The median is important because it tells us where the middle of a dataset lies. Unlike the mean, which is the average and can be skewed by extreme values, the median remains stable, offering a more accurate representation of the central tendency, especially in datasets with outliers. Why is this important? Well, imagine you're analyzing income data for a city. A few billionaires can significantly inflate the average income, making it seem like everyone is doing better than they actually are. The median income, however, would give you a clearer picture of the income level of the typical resident. This makes the median an indispensable tool in various fields, from economics to social sciences, where understanding the true center of a distribution is crucial for making informed decisions. So, when you want a reliable measure of what's 'normal' in a dataset, the median is the way to go!

Step 1: Finding the Median of the Original Data Set

First things first, we need to arrange our original data set in ascending order. Our data set is: 40, 50, 30, 20, 80, 70, 90, 50. Let's sort it:

20, 30, 40, 50, 50, 70, 80, 90

Now, since we have an even number of values (8 values), the median will be the average of the two middle numbers. In this case, the two middle numbers are 50 and 50. So, the median is:

(50 + 50) / 2 = 50

So, the median of the original data set is 50.

To reiterate, finding the median involves a couple of key steps: First, you must arrange the data in ascending (or descending) order. This organizes the data in a way that makes it easy to identify the middle value(s). Second, if the dataset contains an odd number of values, the median is simply the middle value. However, if the dataset contains an even number of values, as in our case, you need to find the two middle values and calculate their average. This average represents the median of the dataset. This method ensures that the median accurately reflects the central tendency of the data, regardless of the presence of extreme values. By following these steps carefully, you can confidently determine the median of any dataset, making it a valuable skill for data analysis and interpretation.

Step 2: Finding the Median After Changing 30 to 120

Now, let's change 30 to 120 in our original data set. Our new data set is: 40, 50, 120, 20, 80, 70, 90, 50. Again, we need to sort it in ascending order:

20, 40, 50, 50, 70, 80, 90, 120

We still have 8 values, so the median will be the average of the two middle numbers, which are 50 and 70. So, the median is:

(50 + 70) / 2 = 60

So, the median of the modified data set is 60.

When we change a value in the dataset, it can affect the median, especially if the changed value influences the order of the middle numbers. In this case, replacing 30 with 120 shifted the numbers around, causing the two middle values to become 50 and 70 instead of 50 and 50. As a result, the median changed from 50 to 60. This illustrates how sensitive the median can be to changes in the data, particularly when those changes occur near the center of the distribution. Understanding this sensitivity is crucial for accurately interpreting data and making informed decisions based on statistical analysis. So, always be mindful of how modifications to the dataset can impact the median and other measures of central tendency.

Step 3: Finding the Mean of Both Medians

Finally, we need to find the mean (average) of the two medians we calculated. We found that the median of the original data set was 50, and the median of the modified data set was 60. So, the mean of both medians is:

(50 + 60) / 2 = 55

Therefore, the mean of both medians is 55.

Calculating the mean of the medians provides a way to summarize the overall central tendency across different versions of the dataset. In this case, we started with an original dataset, modified one of its values, and then found the medians of both datasets. By averaging these medians, we obtain a single value that represents the 'average' median. This can be useful for comparing different scenarios or assessing the impact of changes in the data on the central tendency. It's a simple yet effective way to consolidate information and gain a broader understanding of the dataset's characteristics. So, whether you're analyzing financial data, scientific measurements, or any other type of data, calculating the mean of medians can provide valuable insights and support informed decision-making.

Conclusion

So, after walking through all the steps, we found that the mean of both medians is 55. Therefore, the answer is (c) 55.

Hope this helped, guys! Let me know if you have any other questions.