Calculating Missing Frequencies: A Median Deep Dive

by SLV Team 52 views

Hey math enthusiasts! Let's dive into a cool problem where we get to flex our statistical muscles. We're given a frequency distribution, and the median is already handed to us, at 35. Our mission, should we choose to accept it, is to figure out the missing frequencies, denoted by a and b. Sounds like a fun challenge, right?

Understanding the Median and Frequency Distributions

Alright, before we get our hands dirty with calculations, let's make sure we're all on the same page. Remember what the median is? It's the middle value in a dataset when the data is arranged in ascending order. If we have an odd number of data points, it's the exact middle number. If we have an even number, it's the average of the two middle numbers. In the context of a frequency distribution, the median tells us the value that separates the data into two equal halves. The table is designed to show the class intervals and their corresponding frequency.

Frequency distributions are tables that organize data into intervals (also known as classes) and show how many data points fall into each interval. It's like sorting your socks – you put all the black ones in one pile, all the blue ones in another, and so on. Each pile is a class interval, and the number of socks in each pile is the frequency. A frequency distribution helps us understand the shape, center, and spread of our data. It's an awesome tool for making sense of large amounts of data. It helps you quickly see where the bulk of your data lies and identify any patterns or trends. If you're working with a large dataset, creating a frequency distribution is often the first step towards uncovering valuable insights. Now, knowing the median gives us a key piece of information. Since the median is 35, it falls within the class interval of 30-40. This is our median class, and it's where a big chunk of our work will happen. The median class is the interval that contains the median value. The boundaries of this class will be important for our calculations. This lets us know where our median lies within the distribution and gives us our starting point. This is where the magic happens. Since we know the median is 35, we know it falls within the 30-40 class. We can use this to set up our calculations to find a and b.

Setting up the Calculations

Let's get down to business, shall we? We will need to use the formula for calculating the median in a grouped frequency distribution. The formula looks like this:

Median = L + [(N/2 - CF) / F] * W

Where:

  • L = Lower limit of the median class
  • N = Total frequency
  • CF = Cumulative frequency of the class preceding the median class
  • F = Frequency of the median class
  • W = Width of the median class

From the table, we know that the median is 35, and the median class is 30-40. So, let's identify the known values. L (lower limit of the median class) is 30. Since the median is 35, it is definitely within the 30-40 class. The total frequency, which we are given, we can express it as 10 + 20 + a + 15 + b + 10 + 5 = 60 + a + b. The cumulative frequency of the class preceding the median class is 10 + 20 + a = 30 + a. The frequency of the median class (F) is 20. The width of the median class (W) is 40 - 30 = 10.

Let’s plug these values into the median formula, and we get:

35 = 30 + [( (60 + a + b) / 2 - (30 + a) ) / 20] * 10

Now, let's simplify and solve this equation step by step. First, subtract 30 from both sides:

5 = [( (60 + a + b) / 2 - (30 + a) ) / 20] * 10

Next, divide both sides by 10:

0.5 = [( (60 + a + b) / 2 - (30 + a) ) / 20]

Multiply both sides by 20:

10 = (60 + a + b) / 2 - (30 + a)

Then, multiply all terms by 2:

20 = 60 + a + b - 60 - 2a

Combine like terms:

20 = -a + b

Therefore, we have our first equation, which is:

-a + b = 20

Unveiling the Values of a and b

Remember that the total frequency is 60 + a + b. Also, the question is missing a total value. We can get the total frequency from the original problem. The total number of data points is 10 + 20 + a + 20 + 15 + b + 10 + 5, but if the table is a complete one, then it would be a total of 100. Thus, to determine the values of a and b, we also need to consider the total frequency. Let's assume the total frequency is 100. So we have:

60 + a + b = 100

Which simplifies to:

a + b = 40

Now we have a system of two equations:

  1. -a + b = 20
  2. a + b = 40

Let's solve this system. We can use the elimination method by adding the two equations together. Adding the two equations, we get:

(-a + b) + (a + b) = 20 + 40

Which simplifies to:

2b = 60

Now, solve for b by dividing both sides by 2:

b = 30

Great, we have found the value of b!

Now, substitute b = 30 into the second equation to find a:

a + 30 = 40

Subtract 30 from both sides:

a = 10

So, we have a = 10 and b = 30.

Checking the Results

Now, let's do a quick check to make sure our answers are correct. We can verify the values of a and b by plugging them back into the frequency table and recalculating the median. If the median comes out to be 35, then we know we've nailed it.

Let's reconstruct the frequency table with our new values:

Class Interval 0-10 10-20 20-30 30-40 40-50 50-60 60-70 Total
Frequency 10 20 10 20 15 30 5 100

Now, let's recalculate the median using the formula. In this table, the median class remains 30-40, L = 30, N = 100, CF = 10 + 20 + 10 = 40, F = 20, and W = 10.

Median = 30 + [(100/2 - 40)/20] * 10

Median = 30 + [(50 - 40)/20] * 10

Median = 30 + [10/20] * 10

Median = 30 + 5

Median = 35

Awesome! The calculated median is 35, which matches the given value. This confirms that our values for a and b are correct. Congratulations, guys, you have successfully solved this problem! You've not only found the missing frequencies but also deepened your understanding of the median and frequency distributions. Keep up the amazing work! This problem shows how a little bit of statistical knowledge can go a long way in unraveling complex data scenarios. You can use this to build the skills for your career. Now, go forth and conquer those math problems! Keep practicing, keep learning, and most importantly, keep having fun! Don't be afraid to take on new challenges, and always remember that every mistake is a learning opportunity. Now, go ahead and apply this knowledge. You guys are rockstars.