Find 'x' If Mode Is 55: Frequency Distribution

Alright, let's dive into this frequency distribution problem where we need to find the value of x given that the mode is 55. Sounds like a fun little mathematical adventure, right? Basically, we've got a table that shows how often certain values appear in different classes, and our mission is to figure out one of the missing frequencies. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into calculations, let's make sure we understand what's going on. We have a frequency distribution table that looks like this:

Class	0-15	15-30	30-45	45-60	60-75	75-90
Frequency	10	7	x	15	10	12

The mode is the value that appears most frequently in a dataset. In a grouped frequency distribution, the mode is within the class that has the highest frequency. We're told that the mode is 55, which means it falls within the 45-60 class. This is super important because it tells us that the frequency of the 45-60 class (which is 15) must be the highest frequency in the entire distribution (or at least equal to the highest). Understanding this concept is crucial before we proceed any further, guys!

Given the mode is 55, the modal class is 45-60. The frequency of the modal class is 15. This implies that x (the frequency of the class 30-45) must be less than or equal to 15. This is because the modal class has the highest frequency.

Mode Formula and Its Components

To solve this problem, we're going to use the formula for the mode of a grouped frequency distribution:

$$Mode = l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h$$

Where:

• $l$ is the lower limit of the modal class.
• $f_1$ is the frequency of the modal class.
• $f_0$ is the frequency of the class preceding the modal class.
• $f_2$ is the frequency of the class succeeding the modal class.
• $h$ is the class width.

Let's break down each of these components in the context of our problem:

• $l$: The lower limit of the modal class (45-60) is 45.
• $f_1$: The frequency of the modal class is 15.
• $f_0$: The frequency of the class preceding the modal class (30-45) is x.
• $f_2$: The frequency of the class succeeding the modal class (60-75) is 10.
• $h$: The class width is the difference between the upper and lower limits of any class. In this case, it's $15 - 0 = 15$.

Applying the Mode Formula

Now that we have all the components, let's plug them into the formula:

$$55 = 45 + \frac{15 - x}{2(15) - x - 10} \times 15$$

Let's simplify this equation step by step. First, subtract 45 from both sides:

$$10 = \frac{15 - x}{30 - x - 10} \times 15$$

$$10 = \frac{15 - x}{20 - x} \times 15$$

Now, let's get rid of the fraction by multiplying both sides by $(20 - x)$:

$$10(20 - x) = 15(15 - x)$$

Expand both sides:

$$200 - 10x = 225 - 15x$$

Solving for x

Now, let's isolate x. Add $15x$ to both sides:

$$200 + 5x = 225$$

Subtract 200 from both sides:

$$5x = 25$$

Finally, divide by 5:

$$x = 5$$

So, we've found that the value of x is 5! Isn't that neat?

Verification

To ensure our answer is correct, let's plug x = 5 back into the mode formula and see if we get 55:

$$Mode = 45 + \frac{15 - 5}{2(15) - 5 - 10} \times 15$$

$$Mode = 45 + \frac{10}{30 - 5 - 10} \times 15$$

$$Mode = 45 + \frac{10}{15} \times 15$$

$$Mode = 45 + 10$$

$$Mode = 55$$

Yep, it checks out! Our calculation is correct. x = 5 indeed gives us a mode of 55.

Final Answer

Therefore, the value of x in the frequency distribution is 5. This means that the frequency of the class 30-45 is 5.

The value of x is 5.

Understanding the mode in a frequency distribution is super important because it helps us quickly identify the most common data point or class in a dataset. Think of it like this: if you're analyzing customer preferences, the modal class could tell you which product category is most popular. Or, in a health study, it could reveal the most common age group affected by a certain condition. The mode gives us a snapshot of what's typical or most frequent, making it a valuable tool for making decisions and understanding trends. It's really like having a finger on the pulse of your data!

Furthermore, the mode can provide insights that other measures of central tendency, like the mean and median, might not capture. For example, the mean can be heavily influenced by extreme values, and the median tells us the middle value but not necessarily the most common one. The mode, on the other hand, is all about what shows up most often. This is particularly useful when dealing with categorical data, where calculating a mean or median might not even make sense. So, knowing how to find and interpret the mode is a key skill for anyone working with data. Guys, it's all about getting that quick, clear picture of what's happening!

Also, when you are trying to understand the distribution of data, the mode acts as a quick reference point. Consider a scenario where you're analyzing the number of hours students spend studying each week. If the modal class is 20-25 hours, you instantly know that the most common study time falls within this range. This information can be used to tailor study programs or allocate resources more effectively. In marketing, knowing the modal income bracket of your customers helps you design targeted campaigns and price your products accordingly. So, whether you're in education, marketing, or any other field, the mode provides that essential, practical insight. This is also especially helpful for identifying outliers in a dataset.

Frequency distributions and the mode aren't just abstract mathematical concepts; they have a ton of practical applications in various fields. For example, in business and marketing, understanding the frequency distribution of customer demographics (like age, income, or location) can help companies tailor their products and marketing campaigns to specific target groups. The mode, in this case, might represent the most common age group that buys a particular product, guiding marketing efforts and product development. This way, companies are not just throwing marketing money around, but actually making informed decisions.

In healthcare, frequency distributions are used to analyze the prevalence of diseases or health conditions within a population. The mode can highlight the most common age group or demographic affected by a particular illness, which helps healthcare providers allocate resources and develop targeted prevention programs. For example, if the mode indicates that a certain disease is most prevalent among older adults, public health campaigns can be tailored to that specific age group. Guys, this is where math turns into real-world solutions!

Education is another area where frequency distributions and the mode play a significant role. Teachers and administrators can use these tools to analyze student performance on tests and assignments. The mode can reveal the most common score or grade achieved by students, which can help teachers identify areas where students are struggling and adjust their teaching methods accordingly. If the mode of scores on a particular topic is low, it signals that the teacher needs to revisit that topic and provide additional support to the students. Also understanding grade distributions is super important when it comes to assessing the overall effectiveness of teaching methodologies.

Frequency distribution and mode are used in environmental science to monitor and analyze environmental data, such as air and water quality. The mode can identify the most common level of a pollutant, which can help scientists and policymakers assess the effectiveness of environmental regulations and develop strategies to reduce pollution. For example, if the mode of air pollution levels is consistently high in a particular area, it may indicate the need for stricter emissions controls or the implementation of green infrastructure. It is clear that the practical applications of frequency distribution and mode are broad and varied.

When you're calculating the mode, there are a few common pitfalls that you'll want to steer clear of to make sure you get the right answer. One of the biggest mistakes is confusing the mode with other measures of central tendency like the mean or median. Remember, the mode is all about the most frequent value, while the mean is the average, and the median is the middle value. Mixing these up can lead to some seriously misleading results. So, always double-check what you're actually trying to find!

Another common error is misidentifying the modal class in a grouped frequency distribution. The modal class is the one with the highest frequency, but sometimes it's not immediately obvious, especially if the frequencies are close together. Make sure you carefully examine the frequency distribution table and correctly identify the class with the highest frequency before you start applying the mode formula. It's easy to glance over the data and make a mistake, so take your time and be thorough.

Lastly, people sometimes forget to verify their answer after calculating the mode. It's always a good idea to plug your calculated mode back into the formula or check it against the original data to make sure it makes sense. This is especially important when dealing with grouped frequency distributions, where the mode is an estimate based on the modal class. Verifying your answer can help you catch any errors in your calculations and ensure that your result is accurate. Always remember this verification step as it is super important. By avoiding these common mistakes, you can confidently calculate the mode and use it to gain valuable insights from your data. It is super helpful to avoid this errors.