Mean, Median, Mode: Age Distribution Of 30 People

by Kenji Nakamura 50 views

Hey guys! Let's dive into some interesting data analysis. We've got a group of 30 people and their ages: 24, 3, 29, 6, 5, 17, 25, 24, 36, 42, 30, 16, 14, 12, 8, 4, 8, 37, 32, 40, 37, 26, 28, 15, 17, 41, 20, 18, 27, 42. Our mission today is to figure out the mean, median, and mode of this age data. These three measures give us different insights into the central tendency and distribution of the data. So, let’s put on our math hats and get started!

What are Mean, Median, and Mode?

Before we jump into the calculations, let's make sure we're all on the same page about what mean, median, and mode actually mean. These are fundamental concepts in statistics, and understanding them is crucial for interpreting data effectively.

Mean: The Average Age

The mean, often called the average, is calculated by adding up all the values in a dataset and then dividing by the number of values. It gives us a sense of the typical value in the dataset. In our case, it's the average age of the group. The mean is sensitive to outliers, meaning extreme values can significantly affect it. Imagine if we had a 100-year-old person in our group; the mean age would likely increase quite a bit. To find the mean age in our group, we will sum all the ages together. This means we will add 24 + 3 + 29 + 6 + 5 + 17 + 25 + 24 + 36 + 42 + 30 + 16 + 14 + 12 + 8 + 4 + 8 + 37 + 32 + 40 + 37 + 26 + 28 + 15 + 17 + 41 + 20 + 18 + 27 + 42. After calculating this sum, we get a total of 746. To compute the mean, we will divide this total by the number of individuals in the group, which is 30. Therefore, the mean age is 746 divided by 30. So, 746 / 30 gives us a mean age of approximately 24.87 years. This means that, on average, the individuals in this group are around 24.87 years old. Understanding the mean helps us get a general sense of the central tendency of the age distribution within the group.

Median: The Middle Ground

The median is the middle value in a dataset when the values are arranged in ascending order. It's less sensitive to outliers than the mean because it only considers the central position of the data. If we have an odd number of values, the median is simply the middle value. If we have an even number of values, like in our case, the median is the average of the two middle values. To find the median age, we first need to arrange the ages in ascending order. This helps us identify the middle values easily. So, let’s sort the data: 3, 4, 5, 6, 8, 8, 12, 14, 15, 16, 17, 17, 18, 20, 24, 24, 25, 26, 27, 28, 29, 30, 32, 36, 37, 37, 40, 41, 42, 42. Since we have 30 ages, which is an even number, the median will be the average of the 15th and 16th values. Looking at our sorted list, the 15th value is 24, and the 16th value is also 24. Thus, to find the median, we calculate the average of these two values. The average of 24 and 24 is simply (24 + 24) / 2 = 48 / 2 = 24. Therefore, the median age of this group is 24 years. This means that half of the individuals are younger than 24 years, and half are older than 24 years. The median provides a robust measure of central tendency, particularly useful when the dataset may contain outliers or extreme values that could skew the mean.

Mode: The Most Frequent Age

The mode is the value that appears most frequently in a dataset. A dataset can have one mode (unimodal), more than one mode (multimodal), or no mode if no value is repeated. The mode is useful for identifying the most common occurrence in the data. To determine the mode, we need to count how many times each age appears in the dataset. Let's take a look at the ages again: 24, 3, 29, 6, 5, 17, 25, 24, 36, 42, 30, 16, 14, 12, 8, 4, 8, 37, 32, 40, 37, 26, 28, 15, 17, 41, 20, 18, 27, 42. Now, let’s count the occurrences of each age. By counting, we find that the ages 8, 17, 24, 37, and 42 each appear twice, which is more frequent than any other age in the dataset. Therefore, there are multiple modes in this dataset, making it a multimodal dataset. The modes are 8, 17, 24, 37, and 42 years. This indicates that these ages are the most common within the group. Understanding the mode can help in identifying trends or common characteristics within the data. For instance, in this group, we see a few distinct age clusters that are more prevalent than others.

Calculating Mean, Median, and Mode for Our Data

Now that we know what these measures are, let’s calculate them for our dataset. This will give us a clear picture of the age distribution in our group of 30 people.

Step-by-Step Calculation

  1. Mean: To calculate the mean, we add up all the ages and divide by the total number of people (30).

    • Sum of ages: 24 + 3 + 29 + 6 + 5 + 17 + 25 + 24 + 36 + 42 + 30 + 16 + 14 + 12 + 8 + 4 + 8 + 37 + 32 + 40 + 37 + 26 + 28 + 15 + 17 + 41 + 20 + 18 + 27 + 42 = 746
    • Mean = 746 / 30 = 24.87

  2. Median: To find the median, we first need to arrange the ages in ascending order. Then, we identify the middle value (or the average of the two middle values).

    • Sorted ages: 3, 4, 5, 6, 8, 8, 12, 14, 15, 16, 17, 17, 18, 20, 24, 24, 25, 26, 27, 28, 29, 30, 32, 36, 37, 37, 40, 41, 42, 42
    • Since we have 30 values (an even number), the median is the average of the 15th and 16th values, which are both 24.
    • Median = (24 + 24) / 2 = 24

  3. Mode: To determine the mode, we look for the age(s) that appear most frequently.

    • By counting the occurrences of each age, we find that 8, 17, 24, 37, and 42 each appear twice, which is more frequent than any other age.
    • Mode = 8, 17, 24, 37, 42

Interpreting the Results

Okay, now we've got the mean, median, and mode. But what do these numbers actually tell us about the age distribution in our group? Let's break it down.

  • Mean (24.87): The average age of the group is approximately 24.87 years. This gives us a general idea of the central age, but it's important to remember that this is just an average and doesn't tell us about the spread of ages.
  • Median (24): The median age is 24 years. This means that half of the people in the group are younger than 24, and half are older. The median is a more robust measure of central tendency than the mean, especially when there are outliers in the data.
  • Mode (8, 17, 24, 37, 42): We have multiple modes, which indicates that there are several ages that are more common than others. This suggests that there might be different subgroups within the larger group, each with its own common age.

What Does It All Mean?

When we compare the mean and median, we see they are quite close (24.87 and 24, respectively). This suggests that the age distribution is fairly symmetrical, meaning it's not heavily skewed by extremely high or low ages. If the mean were significantly higher than the median, it would indicate that there are some older individuals pulling the average up. The fact that we have multiple modes also gives us some interesting insights. It tells us that there isn't one single age that dominates the group. Instead, there are several ages that appear frequently, which could indicate different clusters or generations within the group. For example, we see modes at younger ages (8 and 17), mid-range ages (24), and older ages (37 and 42). This could suggest a diverse group with members from various life stages.

Why This Matters

Understanding the mean, median, and mode isn't just about crunching numbers; it's about gaining a deeper understanding of the data. In this case, we've learned about the age distribution of a group of people. This kind of analysis can be useful in many different fields, from marketing to public health to social research. For example, if you were planning an event for this group, knowing the age distribution could help you tailor the activities and offerings to suit the attendees. If you were conducting a health study, understanding the age demographics could help you identify potential health trends or risk factors. So, the next time you encounter a dataset, remember the power of mean, median, and mode. They're your tools for unlocking the story hidden within the numbers!

Conclusion

So, there you have it! We've successfully calculated the mean, median, and mode for the given age data. The mean age is approximately 24.87 years, the median age is 24 years, and the modes are 8, 17, 24, 37, and 42 years. By understanding these measures, we've gained a valuable insight into the age distribution of this group. Keep these concepts in mind, and you'll be well-equipped to analyze and interpret data in all sorts of situations. Keep exploring and stay curious, guys!