13.3.2. Example 1

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:

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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

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.

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.

Therefore, the mode of the data above is 3.286.

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.

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.

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