Calculating Median: Worked Example & Data Analysis

Hey guys! Let's dive into calculating the median from grouped data. This is a common statistical task, and we'll break it down step-by-step. We'll use a specific example to make it crystal clear. So, buckle up and let's get started!

Understanding the Median

Before we jump into the calculation, let's quickly recap what the median actually is. In simple terms, the median is the middle value in a dataset when the data is arranged in ascending or descending order. It's a measure of central tendency, just like the mean (average), but it's less sensitive to extreme values or outliers. This makes the median a robust measure in many situations.

For example, if we have the numbers 2, 4, 6, 8, and 10, the median is 6 because it's the middle number. If we have an even number of data points, like 2, 4, 6, and 8, the median is the average of the two middle numbers, which is (4 + 6) / 2 = 5. Got it? Great!

Why is the Median Important?

You might be wondering, why bother with the median when we have the average? Well, the median gives us a better idea of the "typical" value when the data has some really high or really low values that could skew the average. Think about income distribution, for example. A few billionaires can significantly raise the average income, but the median income will give a more realistic picture of what most people are earning. That's why understanding how to calculate the median is super important.

The Problem: Working Days and Number of Workers

Okay, let's get to our specific problem. We have data on the number of working days and the corresponding number of workers. Here's the data we're working with:

Our goal is to find the median number of working days. Since the data is grouped (we have a range of working days and the number of workers for each range), we'll need to use a specific formula to calculate the median. Don't worry, it's not as scary as it sounds!

Calculating the Median for Grouped Data

Here's the formula we'll use to calculate the median for grouped data:

Median = L + [ (N/2 - cf) / f ] * h

Let's break down what each of these letters means:

• L = Lower limit of the median class (the class containing the median)
• N = Total number of observations (in our case, the total number of workers)
• cf = Cumulative frequency of the class preceding the median class
• f = Frequency of the median class
• h = Class size (the width of the class interval)

Sounds like a mouthful, right? But trust me, it'll become clear as we work through the example.

Step 1: Find the Total Number of Workers (N)

First, we need to find the total number of workers (N). We simply add up the number of workers in each group:

N = 7 + 10 + 15 + 20 + 25 + 30 = 107

So, we have a total of 107 workers.

Step 2: Determine N/2

Next, we calculate N/2. This will help us find the median class:

N/2 = 107 / 2 = 53.5

This means the median will be the value that corresponds to the 53.5th worker when we line them up according to the number of working days.

Step 3: Find the Cumulative Frequencies

Now, let's calculate the cumulative frequencies. This means we'll keep adding up the number of workers as we go down the table:

The cumulative frequency tells us how many workers are included up to a certain point in the data.

Step 4: Identify the Median Class

Now comes the crucial step: finding the median class. Remember, the median is the value corresponding to the 53.5th worker. Looking at our cumulative frequencies, we see that:

• Up to 52 workers have worked for 13 days or less.
• Up to 77 workers have worked for 16 days or less.

This means the 53.5th worker falls into the class where the number of working days is 16. So, the median class is the one corresponding to 16 working days.

Step 5: Determine the Values for the Formula

Okay, we're almost there! Now we need to identify the values for our formula:

• L (Lower limit of the median class): 16
• N/2: 53.5 (we calculated this earlier)
• cf (Cumulative frequency of the class preceding the median class): 52 (the cumulative frequency for 13 working days)
• f (Frequency of the median class): 25 (the number of workers for 16 working days)
• h (Class size): 3 (the difference between the working days, e.g., 7 - 4 = 3, 10 - 7 = 3, and so on)

Step 6: Plug the Values into the Formula and Calculate

Finally, we plug these values into our formula:

Median = L + [ (N/2 - cf) / f ] * h Median = 16 + [ (53.5 - 52) / 25 ] * 3 Median = 16 + [ (1.5) / 25 ] * 3 Median = 16 + [ 0.06 ] * 3 Median = 16 + 0.18 Median = 16.18

So, the median number of working days is 16.18 days. 🎉

Conclusion

And there you have it! We've successfully calculated the median number of working days from our grouped data. Remember, the key is to break down the formula step-by-step and carefully identify each value. Don't be intimidated by the formula – practice makes perfect! Understanding how to calculate the median is a valuable skill in statistics, helping us get a clearer picture of central tendencies in data.

I hope this example was helpful. If you have any questions, feel free to ask! Keep practicing, and you'll master this in no time. Happy calculating! 📊✨