Mean, Median, Mode: Grouped Frequency Table Calculation

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Calculating Mean, Median, and Mode from a Grouped Frequency Table

Hey guys! Today, we're diving into the world of statistics to tackle a common challenge: calculating the mean, median, and mode from a grouped frequency table. It might sound intimidating, but trust me, we'll break it down step-by-step so it's super easy to understand. We'll use the grouped frequency table you provided as our example, so you can follow along and see exactly how it's done. Let's get started!

Understanding Grouped Frequency Tables

Before we jump into the calculations, let's make sure we're all on the same page about what a grouped frequency table actually is. Think of it as a way to organize data into intervals, or classes, rather than listing every single data point. This is especially helpful when dealing with large datasets, as it makes the information much more manageable. In our case, we have the following table:

Classes 10-20 20-30 30-40
Frequency 4 9 6

This table tells us that we have three classes: 10-20, 20-30, and 30-40. The frequency for each class represents how many data points fall within that interval. So, we have 4 data points between 10 and 20, 9 between 20 and 30, and 6 between 30 and 40. Got it? Great! Now we can move on to the fun part: the calculations.

Calculating the Mean

The mean, also known as the average, is a measure of central tendency that gives us a sense of the "center" of our data. When dealing with a grouped frequency table, we can't simply add up all the data points and divide by the total number of points because we don't know the exact values within each class. Instead, we use a slightly modified formula that takes into account the class midpoints and frequencies. Here's the formula:

Mean = Σ(fᵢ * xᵢ) / Σfᵢ

Where:

  • fᵢ is the frequency of class i
  • xᵢ is the midpoint of class i
  • Σ means "sum of"

Let's break this down step-by-step using our example table.

Step 1: Find the Midpoint of Each Class

The midpoint of a class is simply the average of the lower and upper class limits. For example, the midpoint of the 10-20 class is (10 + 20) / 2 = 15. Let's calculate the midpoints for all our classes:

  • Class 10-20: Midpoint (x₁) = (10 + 20) / 2 = 15
  • Class 20-30: Midpoint (x₂) = (20 + 30) / 2 = 25
  • Class 30-40: Midpoint (x₃) = (30 + 40) / 2 = 35

Step 2: Multiply the Frequency by the Midpoint for Each Class

Now we multiply the frequency (fᵢ) of each class by its midpoint (xᵢ):

  • Class 10-20: f₁ * x₁ = 4 * 15 = 60
  • Class 20-30: f₂ * x₂ = 9 * 25 = 225
  • Class 30-40: f₃ * x₃ = 6 * 35 = 210

Step 3: Sum the Results from Step 2

We add up the values we just calculated: Σ(fᵢ * xᵢ) = 60 + 225 + 210 = 495

Step 4: Sum the Frequencies

Next, we add up all the frequencies: Σfᵢ = 4 + 9 + 6 = 19

Step 5: Divide the Sum from Step 3 by the Sum from Step 4

Finally, we divide the sum of (fᵢ * xᵢ) by the sum of fᵢ: Mean = 495 / 19 ≈ 26.05

So, the mean of our grouped data is approximately 26.05. Not too bad, right? Let's move on to the median.

Finding the Median

The median is another measure of central tendency that represents the middle value in a dataset when it's ordered from least to greatest. For a grouped frequency table, we need to use a slightly different approach to find the median class and then interpolate within that class. Here's the formula for the median of grouped data:

Median = L + [(N/2 - CF) / f] * w

Where:

  • L is the lower class boundary of the median class
  • N is the total frequency (sum of all frequencies)
  • CF is the cumulative frequency of the class before the median class
  • f is the frequency of the median class
  • w is the class width

Let's break this down step-by-step using our example table.

Step 1: Calculate the Total Frequency (N)

We already did this when calculating the mean: N = 4 + 9 + 6 = 19

Step 2: Find the Median Position

The median position is N / 2 = 19 / 2 = 9.5. This means the median lies within the class that contains the 9.5th value.

Step 3: Determine the Median Class

To find the median class, we need to calculate the cumulative frequencies. Cumulative frequency is the running total of frequencies.

  • Class 10-20: Cumulative Frequency = 4
  • Class 20-30: Cumulative Frequency = 4 + 9 = 13
  • Class 30-40: Cumulative Frequency = 13 + 6 = 19

The 9.5th value falls within the class 20-30 (since its cumulative frequency is 13, which is the first to exceed 9.5). So, the median class is 20-30.

Step 4: Apply the Median Formula

Now we have all the pieces to plug into the formula:

  • L (lower class boundary of the median class) = 20
  • N (total frequency) = 19
  • CF (cumulative frequency of the class before the median class) = 4
  • f (frequency of the median class) = 9
  • w (class width) = 30 - 20 = 10

Median = 20 + [(19/2 - 4) / 9] * 10 Median = 20 + [(9.5 - 4) / 9] * 10 Median = 20 + (5.5 / 9) * 10 Median = 20 + 0.6111 * 10 Median = 20 + 6.111 Median ≈ 26.11

Therefore, the median of our grouped data is approximately 26.11. We're on a roll! Let's tackle the mode next.

Identifying the Mode

The mode is the value that appears most frequently in a dataset. In a grouped frequency table, we're looking for the class with the highest frequency, which we call the modal class. To get a more precise estimate of the mode, we can use another formula:

Mode = L + [(fₘ - f₁) / (2fₘ - f₁ - f₂)] * w

Where:

  • L is the lower class boundary of the modal class
  • fₘ is the frequency of the modal class
  • f₁ is the frequency of the class before the modal class
  • f₂ is the frequency of the class after the modal class
  • w is the class width

Let's find the mode for our example table.

Step 1: Determine the Modal Class

Looking at our table, the class with the highest frequency is 20-30, with a frequency of 9. So, the modal class is 20-30.

Step 2: Apply the Mode Formula

Now we plug in the values:

  • L (lower class boundary of the modal class) = 20
  • fₘ (frequency of the modal class) = 9
  • f₁ (frequency of the class before the modal class) = 4
  • f₂ (frequency of the class after the modal class) = 6
  • w (class width) = 10

Mode = 20 + [(9 - 4) / (2 * 9 - 4 - 6)] * 10 Mode = 20 + [5 / (18 - 10)] * 10 Mode = 20 + (5 / 8) * 10 Mode = 20 + 0.625 * 10 Mode = 20 + 6.25 Mode ≈ 26.25

So, the mode of our grouped data is approximately 26.25.

Wrapping Up

And there you have it! We've successfully calculated the mean, median, and mode for our grouped frequency table. We found that:

  • Mean ≈ 26.05
  • Median ≈ 26.11
  • Mode ≈ 26.25

As you can see, these three measures of central tendency are quite close in this particular example, indicating that the data is relatively symmetrical. Remember, these calculations are essential tools for understanding and interpreting data, and now you've got the skills to tackle grouped frequency tables like a pro. Keep practicing, and you'll become a statistics whiz in no time! If you have any questions, feel free to ask. Happy calculating!