Find The Mode: Math Test Scores For Class XI SMA Z

Hey guys! Let's dive into this math problem together. We've got a table showing the scores from a math test for class XI at SMA Z, and our mission is to figure out the mode of this data. No sweat, we can totally do this! Figuring out the mode is super useful because it helps us quickly see which score range was the most common in the class. This gives us a snapshot of how the class performed overall. Let's break down what the mode is all about and then tackle the problem step-by-step.

The mode in statistics is simply the value that appears most frequently in a dataset. Think of it as the most popular choice. When we're dealing with grouped data, like in the table we have, we're looking for the class interval with the highest frequency. This interval is often referred to as the modal class. The actual mode is then calculated using a formula that takes into account the boundaries of the modal class and the frequencies of the classes around it. This gives us a more precise estimate of the most common score.

Before we jump into calculations, let's take a closer look at the table. We need to identify the class interval that has the highest frequency. This will be our modal class, and it's the key to unlocking the mode. Once we've spotted the modal class, we can gather the information we need for the formula. This includes the lower boundary of the modal class, the frequencies of the classes before and after the modal class, and the class width. Each of these pieces plays a role in calculating the mode accurately. Understanding these components will not only help us solve this problem but also give us a solid foundation for tackling similar statistical challenges in the future.

Understanding the Data

Okay, first things first, let's get friendly with the data table. We have two columns here: Nilai (which means "Score") and f (which stands for "frequency"). The "Score" column shows the ranges of scores, like 58-60, 61-63, and so on. The "f" column tells us how many students fall into each of those score ranges. So, a frequency of 9 in the 64-66 range means that 9 students scored somewhere between 64 and 66. Got it? This is how we understand frequency distribution, which is key to finding our mode. The frequency tells us how often each score range pops up, and the one that appears most often is our modal class.

To pinpoint the mode, we need to find the score range with the highest frequency. Scan the 'f' column, guys. Which number jumps out at you? Which score range has the most students? That's our modal class. Once we've nailed that, we're one step closer to calculating the exact mode. Remember, the mode gives us a sense of the most typical score in the class, so it's a really useful piece of information. It helps us understand where the bulk of the students landed in terms of their performance. So, let's keep our eyes peeled and find that highest frequency!

Now, let's zoom in on that frequency column. We're looking for the biggest number, the one that screams, "Hey, I'm the most frequent!" In this case, the highest frequency is 9. That means the score range corresponding to the frequency of 9 is our modal class. This is super important because it's the foundation for our mode calculation. Think of it like finding the tallest building in a city skyline – it stands out, right? The highest frequency does the same thing in our data. It highlights the score range that most students achieved.

Once we've identified the modal class, we can start gathering the specific values we need for the mode formula. We'll need the lower limit of the modal class, the frequencies of the classes before and after it, and the class width. These values are like the ingredients in a recipe – each one plays a specific role in getting the final result. So, let's make sure we're crystal clear on which score range has the highest frequency before we move on. This step is crucial for an accurate mode calculation!

Identifying the Modal Class

Alright, let's get down to business. Looking at the table, we can see the frequencies are: 2, 6, 9, 6, 4, and 3. Which one is the highest? That's right, it's 9! So, the class interval corresponding to the frequency of 9 is our modal class. And that interval is 64-66. Awesome! We've found our modal class. This is like discovering the treasure chest on a treasure map – we're getting closer to our goal.

The modal class is super important because it's where the mode lives. It's the range of scores where the most students landed. Now that we know the modal class is 64-66, we can start thinking about the formula we'll use to pinpoint the exact mode within this range. The formula helps us fine-tune our estimate and get a more precise value. Think of the modal class as the neighborhood, and the mode formula as the street address within that neighborhood. We're zeroing in on the most common score!

So, now that we've identified the modal class, we know the mode is somewhere between 64 and 66. But we need to be more specific. That's where the mode formula comes in. It uses the lower limit of the modal class and the frequencies of the surrounding classes to give us a more accurate estimate. It's like using a GPS to navigate within that neighborhood – we're getting turn-by-turn directions to the exact location. So, let's get ready to put that formula to work!

Calculating the Mode

Now comes the fun part: calculating the mode! The formula for the mode of grouped data is:

Mode = L + [(f1 - f0) / (2f1 - f0 - f2)] * c

Where:

• L is the lower boundary of the modal class
• f1 is the frequency of the modal class
• f0 is the frequency of the class before the modal class
• f2 is the frequency of the class after the modal class
• c is the class width

Don't let the formula intimidate you, guys. It looks complicated, but we'll break it down step by step. Each part of the formula has a specific role, and once we understand what each variable represents, it becomes much easier to use. Think of it like following a recipe – each ingredient has a purpose, and when we combine them in the right way, we get a delicious result. In this case, our delicious result is the mode!

Let's identify each of these values from our table. First up, L, the lower boundary of the modal class. Remember, our modal class is 64-66. So, the lower boundary is 64 - 0.5 = 63.5. Next, f1 is the frequency of the modal class, which we already know is 9. f0 is the frequency of the class before the modal class, which is 6 (from the 61-63 range). And f2 is the frequency of the class after the modal class, which is also 6 (from the 67-69 range). Finally, c is the class width, which is the range of each interval (e.g., 61-58 = 3). Now we have all the pieces we need!

So, now that we have all the values, we can plug them into the formula and calculate the mode. It's like putting the pieces of a puzzle together – each piece fits in a specific spot, and when we connect them all, we see the complete picture. In this case, the pieces are the lower boundary, frequencies, and class width, and the complete picture is the mode. Let's take a deep breath and work through the calculation step by step. We've got this!

Plugging in the Values

Let's plug those values into the formula:

Mode = 63.5 + [(9 - 6) / (2 * 9 - 6 - 6)] * 3

Now, let's simplify this step by step. First, we tackle the numerator inside the brackets: 9 - 6 = 3. Then, the denominator: 2 * 9 = 18, and 18 - 6 - 6 = 6. So, our equation becomes:

Mode = 63.5 + [3 / 6] * 3

We're making great progress! We've simplified the fraction inside the brackets and now we're ready to move on to the next step. It's like climbing a staircase – each step we take gets us closer to the top. In this case, each simplification brings us closer to the mode. So, let's keep going, we're almost there!

Next, we simplify the fraction 3 / 6 to 0.5. Then, we multiply 0.5 by 3, which gives us 1.5. So, our equation now looks like this:

Mode = 63.5 + 1.5

See how much simpler it's getting? We're almost at the finish line! It's like running a race – the final stretch is always the most exciting. And in this case, the final stretch is adding 1.5 to 63.5. So, let's do it!

Final Calculation and Answer

Finally, we add 63.5 and 1.5 to get:

Mode = 65

So, the mode of the data is 65. Bam! We did it! We successfully navigated the data, identified the modal class, and crunched the numbers to find the mode. Give yourselves a pat on the back, guys! This is a fantastic achievement. We've not only solved the problem but also deepened our understanding of statistics.

Finding the mode is like being a detective – we gather clues from the data, follow the trail, and uncover the most frequent value. And in this case, the most frequent score is 65. This tells us that the most common score range among the students in class XI SMA Z was around 65. This is valuable information that can be used to understand the overall performance of the class and identify areas for improvement. So, congratulations on cracking the case!

Therefore, the correct answer is 65.0. You guys nailed it! Understanding how to calculate the mode is a super useful skill, not just for math class, but for understanding data in the real world too. Whether it's figuring out the most popular product, the most common age group, or the most frequent response in a survey, the mode can give you quick insights. So, keep practicing and keep exploring the world of statistics! You're doing great!