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13.3.2 - Example 1

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Introduction to Frequency Tables

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Teacher
Teacher

Today, we're discussing frequency tables and how they can help us organize survey data. Can anyone explain what a frequency table represents?

Student 1
Student 1

It's a table that shows how often each value occurs.

Teacher
Teacher

That's right! For example, if we surveyed households about family sizes, we might list different family size categories and how many families fit into each. Can anyone give me an example of a family size category?

Student 2
Student 2

Like 1 to 3 family members?

Teacher
Teacher

Exactly! Now, how would we represent our findings in a frequency table using those categories?

Student 3
Student 3

We list the categories in one column and the number of families for each in another.

Teacher
Teacher

Great job! Remember, this organization helps us identify patterns in the data easily.

Identifying the Modal Class

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Teacher
Teacher

Now that we have our frequency table, how do we find the mode from it?

Student 4
Student 4

We look for the class with the highest frequency!

Teacher
Teacher

Correct! In our data, which class has the highest frequency?

Student 1
Student 1

The 3-5 family members category has 8 families.

Teacher
Teacher

Right, so our modal class is 3-5. Why is it important to identify the modal class?

Student 2
Student 2

Because it tells us which family size is most common!

Calculating the Mode

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Teacher
Teacher

Now let's calculate the mode using the formula. Do you remember what the formula is for finding the mode?

Student 3
Student 3

It involves the lower limit of the modal class, the frequencies, and the class size.

Teacher
Teacher

Excellent! Let’s substitute in the values from our table. What are the values we'll use?

Student 4
Student 4

Lower limit is 3, class size is 2, and we have frequencies of 8 for modal, 7 for previous, and 2 for the next.

Teacher
Teacher

Perfect! Let's plug those values into the formula together.

Interpreting the Mode Result

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Teacher
Teacher

After plugging in the numbers, we found the mode is 3.286. What does this number tell us?

Student 1
Student 1

It means that the most common family size is just over 3 members.

Teacher
Teacher

Exactly! It provides insight into family sizes in our locality. Why do you think this information is useful?

Student 2
Student 2

It can help in planning resources or services for families.

Teacher
Teacher

Exactly! Understanding our community's needs is important.

Introduction & Overview

Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.

Quick Overview

This section demonstrates how to find the mode in a frequency table derived from a household survey.

Standard

The section provides a detailed example of calculating the mode using a frequency table from a survey of family sizes in households. It explains the identification of the modal class and shows the application of the formula for calculating the mode.

Detailed

In this section, we explore the calculation of mode, a statistical measure that represents the most common value in a data set. A survey of 20 households generated a frequency table showing the distribution of family sizes. The table indicated that the highest frequency occurs in the class range of 3-5 family members, thereby designating this as the modal class. To find the precise mode, we apply the mode formula, substituting in values for class limits and frequencies. This process not only highlights the importance of the modal class but also underscores practical statistical applications in real-life scenarios.

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Mean median mode range
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Audio Book

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Introduction to the Survey

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A survey conducted on 20 households in a locality by a group of students resulted in the following frequency table for the number of family members in a household:

Detailed Explanation

A group of students conducted a survey where they analyzed the number of family members living in 20 different households. This process helped them to collect data that is essential for understanding family size in that locality. The result of this survey was summarized in a frequency table, which is a useful way to organize the data in a clear manner.

Examples & Analogies

Imagine you and your friends are curious about how many pets people have in your neighborhood. You decide to ask 20 houses. The responses you gather can be organized in a similar table, making it easier to see how many households have one pet, two pets, and so on.

Understanding the Frequency Table

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Family size 1 - 3 3 - 5 5 - 7 7 - 9 9 - 11
Number of families 7 8 2 2 1

Detailed Explanation

The frequency table breaks down the number of families according to their family size ranges. For example, 7 families have between 1 to 3 members, while 8 families have between 3 to 5 members. This allows us to see how many families fall into each size range, which is critical for determining the 'mode' of the data, or the most common family size range.

Examples & Analogies

Think of it like categorizing your favorite movies based on their genres. You might find that you have more comedies than any other genre. The frequency table works in a similar way by showing us which family size range is the most common, much like you discovering that your collection has the most comedies.

Identifying the Modal Class

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Here the maximum class frequency is 8, and the class corresponding to this frequency is 3 – 5. So, the modal class is 3 – 5.

Detailed Explanation

To find the mode, we look for the frequency (or the number of occurrences) that is highest. In this case, the highest frequency is 8 and it corresponds to the family size range of 3 to 5 members. This class is referred to as the 'modal class' because it represents the most frequently occurring family size in the survey.

Examples & Analogies

Imagine in a classroom, the teacher asks how many books each student has. After counting, it turns out that most students have between 3 to 5 books. Just like identifying the most common family size in the survey, identifying how many books are most commonly owned by the students highlights the 'modal' category.

Calculating the Mode

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Now
modal class = 3 – 5, lower limit (l) of modal class = 3, class size (h) = 2 frequency (f) of the modal class = 8, frequency (f) of class preceding the modal class = 7, frequency (f) of class succeeding the modal class = 2.

Detailed Explanation

To calculate the mode more precisely, we will use a formula that takes into account the frequencies of both the modal class and the classes adjacent to it. Here, we identify the modal class as 3–5, with a lower limit of 3, a class size of 2, and the frequencies of the classes before and after it.

Examples & Analogies

Envision that you are measuring how tall groups of plants are in a garden. If the most common height is around 3.5 to 5 feet, you want to find out the average height precisely by looking at the heights of the other nearby plants too, so you can make adjustments for better growth in the future.

Using the Mode Formula

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Now, let us substitute these values in the formula : Mode = l + (f1 - f0) / (2f1 - f0 - f2) × h = 3 + (8 - 7) / (2 × 8 - 7 - 2) × 2 = 3 + 1 / 7 × 2 = 3 + 0.2857 = 3.286

Detailed Explanation

The formula used helps in deriving the mode more accurately. The values are substituted into the formula, resulting in a calculated mode of approximately 3.29. This gives us a more precise representation of the most common family size rather than just the range it falls within.

Examples & Analogies

If you were refining your favorite pizza topping to find out exactly how many people prefer pepperoni or vegetarian, you would analyze the responses in detail, adjusting for various choices to find the exact most favored topping.

Final Result

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Therefore, the mode of the data above is 3.286.

Detailed Explanation

In conclusion, after all calculations and observations, we have determined that the mode, which indicates the most common number of family members among the surveyed households, is approximately 3.286. This means that the average family size tends to hover around this value based on the data analyzed.

Examples & Analogies

This is similar to concluding that the average height of a group of boys in a class is around 4.5 feet. It gives you not just an idea but also a detailed understanding of their heights compared to each other.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Mode: The most frequently occurring value in a data set.

  • Frequency Table: A structured way to present how often different values occur.

  • Modal Class: The specific category with the highest frequency in a frequency table.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • In a survey of 20 families, the frequency of family sizes was recorded, showing specific patterns indicating population characteristics.

  • If the mode is calculated to be 3.286, it signifies that family sizes mostly hover around 3-5 members.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 Rhymes Time

  • When you want to know the most, just look where the numbers boast.

📖 Fascinating Stories

  • Imagine a family reunion where the most common family size is celebrated, representing the majority—this is similar to the mode in a data set!

🧠 Other Memory Gems

  • FAMP - Frequency, Amount, Modal class, and Practical use.

🎯 Super Acronyms

MFM - Most Frequent Member defines the Mode.

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Mode

    Definition:

    The value that appears most frequently in a data set.

  • Term: Frequency Table

    Definition:

    A table that displays the number of occurrences of each value in a data set.

  • Term: Modal Class

    Definition:

    The class interval in a frequency distribution with the highest frequency.

  • Term: Class Size

    Definition:

    The difference between the upper and lower boundaries of a class interval.