Complete Body Mass Frequency Table: A Step-by-Step Guide

Have you ever encountered a frequency table with missing information and felt a bit lost on how to complete it? Don't worry; you're not alone! Frequency tables are a fundamental tool in statistics for organizing and summarizing data, but sometimes they come with gaps that need filling. This comprehensive guide will walk you through the process of completing a body mass frequency table, step by step, ensuring you understand the underlying concepts and can confidently tackle similar problems in the future. We'll break down the different components of a frequency table, explain the relationships between them, and provide practical tips and examples to solidify your understanding. So, grab your calculator and let's dive in!

Understanding Frequency Tables

Before we jump into completing the table, let's first make sure we're all on the same page about what a frequency table is and the different elements it contains. At its core, a frequency table is a way to organize raw data into a more understandable format. It groups data into intervals and then counts how many data points fall within each interval. This allows us to quickly see the distribution of the data and identify patterns or trends. In the context of body mass, a frequency table might group individuals into weight ranges (e.g., 40-45 kg, 45-50 kg) and then show how many people fall into each range.

• Key Components of a Frequency Table:

  • Intervals (or Classes): These are the ranges into which the data is grouped. In our case, the intervals represent weight ranges in kilograms (e.g., [40-45[). The square bracket indicates that the lower limit (40) is included in the interval, while the parenthesis indicates that the upper limit (45) is excluded. This is important for ensuring that each data point falls into only one interval.
  • Frequency (fᵢ): This represents the number of data points (in our case, the number of young people) that fall within each interval. For example, if the frequency for the interval [40-45[ is 10, it means that 10 young people have a body mass between 40 kg (inclusive) and 45 kg (exclusive).
  • Cumulative Frequency (Fᵢ): This is the running total of frequencies. For each interval, the cumulative frequency represents the total number of data points that fall within that interval and all preceding intervals. It helps us understand the overall distribution of the data and how many data points are below a certain value.
  • Relative Frequency (hᵢ%): This expresses the frequency of each interval as a percentage of the total number of data points. It's calculated by dividing the frequency of the interval by the total number of data points and multiplying by 100. The relative frequency allows us to compare the proportion of data points in different intervals, regardless of the total sample size.

Understanding these components and how they relate to each other is crucial for completing a frequency table accurately. Now, let's move on to the practical steps involved in filling in the missing values.

Step-by-Step Guide to Completing the Table

Now that we have a solid understanding of the components of a frequency table, let's dive into the specific steps involved in completing one. We'll use the example table provided in the prompt as a guide, but these steps can be applied to any frequency table with missing information. Remember, the key is to use the relationships between the different components to deduce the missing values. Let's get started!

• 1. Identify Known Values and Missing Values:

  • The first step is to carefully examine the table and identify which values are already provided and which ones are missing. This will give you a clear picture of what you need to calculate. In our example, we have the interval [40-45[ and its relative frequency (hᵢ%) of 10.0%. We need to find the frequency (fᵢ) and cumulative frequency (Fᵢ) for this interval, and potentially other missing values in the table (which are not provided in the example, but we'll discuss the general approach). If you had additional rows in the table, you would apply these steps iteratively to each row.

• 2. Determine the Total Number of Data Points (if needed):

  • Sometimes, the total number of data points (N) is not explicitly given in the table. If this is the case, and you need it to calculate other values, you might need to use other information provided in the table. For example, if you have the relative frequency of one interval and the frequency of another, you can use the relationship between them to find the total number of data points. In our example, let's assume we know the total number of young people (N) is 50. If N was not provided, but you were given a complete row (fᵢ and hᵢ%), you could calculate N using the formula: N = fᵢ / (hᵢ% / 100).

• 3. Calculate the Frequency (fᵢ) using Relative Frequency (hᵢ%) and Total Data Points (N):

  • The relative frequency (hᵢ%) tells us the percentage of the total data points that fall within a particular interval. We can use this information, along with the total number of data points (N), to calculate the frequency (fᵢ) for that interval. The formula for this is:

    • fᵢ = (hᵢ% / 100) * N

  • In our example, we have hᵢ% = 10.0% and we're assuming N = 50. Plugging these values into the formula, we get:

    • fᵢ = (10.0 / 100) * 50 = 5

  • So, the frequency (fᵢ) for the interval [40-45[ is 5. This means that 5 young people have a body mass between 40 kg and 45 kg.

• 4. Calculate the Cumulative Frequency (Fᵢ):

  • The cumulative frequency (Fᵢ) represents the total number of data points that fall within a particular interval and all preceding intervals. To calculate Fᵢ for the first interval, it's simply equal to the frequency (fᵢ) of that interval. For subsequent intervals, you add the frequency of the current interval to the cumulative frequency of the previous interval.

  • In our example, since [40-45[ is the first interval, the cumulative frequency (Fᵢ) is equal to the frequency (fᵢ), which we calculated as 5. So, Fᵢ = 5 for the interval [40-45[. If there were more intervals in the table, you would continue to add the frequencies to get the cumulative frequencies for those intervals.

• 5. Repeat Steps 3 and 4 for Other Intervals (if needed):

  • If your table has more intervals with missing values, you would repeat steps 3 and 4 for each interval, using the information you've already calculated and the relationships between the different components to deduce the remaining values. Remember to always double-check your calculations to ensure accuracy.

Practical Tips and Examples

To further solidify your understanding of completing frequency tables, let's look at some practical tips and examples. These tips will help you avoid common mistakes and approach different types of problems with confidence.

• Double-Check Your Calculations:

  • This might seem obvious, but it's worth emphasizing. Always double-check your calculations, especially when dealing with percentages and decimals. A small error in one calculation can propagate through the rest of the table, leading to incorrect results. Use a calculator and take your time to ensure accuracy.

• Understand the Relationships:

  • The key to completing frequency tables is understanding the relationships between the different components. Remember that the relative frequency is a percentage representation of the frequency, and the cumulative frequency is a running total of the frequencies. By understanding these relationships, you can use the information you have to deduce the missing information.

• Work Step-by-Step:

  • Don't try to fill in the entire table at once. Work step-by-step, focusing on one missing value at a time. This will help you stay organized and avoid making mistakes. Start with the easiest values to calculate and then use those values to find the more difficult ones.

• Example with Multiple Intervals (Hypothetical):

  • Let's say we have the following incomplete frequency table:

    Interval (kg)	fᵢ	Fᵢ	hᵢ%
    [40-45[			10.0%
    [45-50[	8
    [50-55[		25
    Total			100.0%

  • Assume the total number of young people (N) is 50.

  • Step 1: We already calculated fᵢ = 5 and Fᵢ = 5 for the interval [40-45[ in the previous example.

  • Step 2: For the interval [45-50[, we know fᵢ = 8. To find hᵢ%, we use the formula hᵢ% = (fᵢ / N) * 100:

    • hᵢ% = (8 / 50) * 100 = 16.0%

  • To find Fᵢ for the interval [45-50[, we add the frequency of this interval to the cumulative frequency of the previous interval:

    • Fᵢ = 5 + 8 = 13

  • Step 3: For the interval [50-55[, we know Fᵢ = 25. To find fᵢ, we subtract the cumulative frequency of the previous interval from the cumulative frequency of the current interval:

    • fᵢ = 25 - 13 = 12

  • To find hᵢ%, we use the formula hᵢ% = (fᵢ / N) * 100:

    • hᵢ% = (12 / 50) * 100 = 24.0%

  • Step 4: To find the total frequency, we add up the frequencies of all intervals:

    • Total fᵢ = 5 + 8 + 12 = 25

  • Notice that this should be equal to the total number of data points (N), which is 50. This indicates that we have made an error. The total frequency should be 50 since that is our N value. Let's calculate the missing frequency that will add up to 50

  • Step 5: This should be the last interval to complete the frequency table and the value of cumulative frequency should be 50

    • 50 - 25 = 25

    • hᵢ% = (25 / 50) * 100 = 50.0%

  • The completed table would look like this:

    Interval (kg)	fᵢ	Fᵢ	hᵢ%
    [40-45[	5	5	10.0%
    [45-50[	8	13	16.0%
    [50-55[	12	25	24.0%
    [55-60[	25	50	50.0%
    Total	50		100.0%

Conclusion

Completing frequency tables is a fundamental skill in statistics. By understanding the components of a frequency table and the relationships between them, you can confidently fill in missing values and analyze data effectively. Remember to work step-by-step, double-check your calculations, and practice with different examples to solidify your understanding. With a little practice, you'll be a pro at completing frequency tables in no time!

For further reading and resources on statistics and frequency tables, you can visit Khan Academy's Statistics and Probability section.