The Number Of Plastic Bottles Collected By Students During A Weekend Cleanup Drive Was: 15, 20, 18, 22, And 19. What Is The Median Number Of Bottles Collected?

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During a weekend cleanup drive, students collected a varying number of plastic bottles, and determining the median number of bottles collected is a fundamental statistical task. The median represents the middle value in a dataset when the values are arranged in ascending order. Understanding how to calculate the median is crucial in various fields, including environmental studies, data analysis, and general statistics.

To find the median, we must first arrange the given data in ascending order. The dataset representing the number of plastic bottles collected by students is: 15, 20, 18, 22, and 19. Arranging these numbers in ascending order gives us: 15, 18, 19, 20, 22. The median is the middle value in this ordered list.

Steps to Find the Median

  1. Arrange the Data: The first step in finding the median is to arrange the dataset in ascending order. This means ordering the numbers from the smallest to the largest. In our case, the ordered list is: 15, 18, 19, 20, 22.

  2. Identify the Middle Value: Once the data is ordered, the next step is to identify the middle value. The method for finding the middle value differs slightly depending on whether the dataset has an odd or even number of data points.

    • Odd Number of Data Points: If the dataset has an odd number of data points, the median is simply the middle number. For example, in our ordered list (15, 18, 19, 20, 22), there are five data points, which is an odd number. The middle number is 19, as there are two numbers to its left and two numbers to its right.
    • Even Number of Data Points: If the dataset has an even number of data points, the median is the average of the two middle numbers. For instance, if we had an additional data point, say 25, the ordered list would be 15, 18, 19, 20, 22, 25. In this case, there are six data points, which is an even number. The two middle numbers are 19 and 20. To find the median, we would calculate the average of these two numbers: (19 + 20) / 2 = 19.5. Therefore, the median would be 19.5.

  3. Determine the Median: In our original dataset (15, 18, 19, 20, 22), there are five data points, which is an odd number. The middle value is 19. Therefore, the median number of plastic bottles collected is 19.

Importance of the Median

The median is a crucial statistical measure because it represents the central tendency of a dataset. Unlike the mean (average), the median is not affected by extreme values or outliers. This makes the median a more robust measure of central tendency in situations where the data may contain unusually high or low values.

Robustness to Outliers

Outliers are data points that significantly differ from other values in the dataset. These extreme values can skew the mean, making it a less representative measure of central tendency. However, the median remains unaffected by outliers because it only considers the middle value(s) of the ordered data.

For example, consider the dataset 15, 18, 19, 20, 100. The mean of this dataset is (15 + 18 + 19 + 20 + 100) / 5 = 34.4. The median, however, is 19. In this case, the outlier (100) significantly inflates the mean, while the median remains a more accurate representation of the central tendency of the data.

Applications in Environmental Studies

In environmental studies, the median is often used to analyze data related to pollution levels, waste collection, and other environmental metrics. For instance, when assessing the effectiveness of a cleanup drive, the median number of items collected can provide a more accurate picture of the typical collection effort than the mean, especially if there are significant variations in the amount of waste collected by different individuals or groups.

Alternative Measures of Central Tendency

While the median is a robust measure of central tendency, it is essential to understand other measures, such as the mean and mode, to gain a comprehensive understanding of the data.

Mean

The mean, also known as the average, is calculated by summing all the values in the dataset and dividing by the number of values. The mean is straightforward to calculate and is widely used in various statistical analyses. However, as mentioned earlier, the mean is sensitive to outliers.

In our example dataset (15, 18, 19, 20, 22), the mean is (15 + 18 + 19 + 20 + 22) / 5 = 18.8. The mean provides a general sense of the average number of bottles collected, but it can be skewed if there are extreme values.

Mode

The mode is the value that appears most frequently in the dataset. A dataset can have one mode (unimodal), multiple modes (multimodal), or no mode if all values appear only once. The mode is useful for identifying the most common value in the dataset.

In our example dataset (15, 18, 19, 20, 22), each value appears only once, so there is no mode. If the dataset were 15, 18, 19, 19, 20, 22, the mode would be 19 because it appears twice.

Choosing the Right Measure

The choice between the mean, median, and mode depends on the nature of the data and the purpose of the analysis. The mean is suitable for datasets with a normal distribution and no significant outliers. The median is preferable when dealing with skewed data or datasets with outliers. The mode is useful for identifying the most common value in a dataset.

Practical Examples

  1. Income Analysis: When analyzing income data, the median income is often used instead of the mean income because income distributions are typically skewed. A few high earners can significantly inflate the mean income, making the median a more representative measure of the income level of the average person.
  2. Test Scores: In education, the mean test score is often used to evaluate the performance of students. However, if there are students who score significantly higher or lower than the rest, the median can provide a more accurate picture of the typical student's performance.
  3. Sales Data: In business, the median sales figure can be used to understand the central sales trend. This is particularly useful in situations where a few large sales can skew the average sales figure.

Conclusion

In summary, the median number of plastic bottles collected during the weekend cleanup drive is 19. The median is a robust measure of central tendency that is not affected by outliers, making it a valuable tool for data analysis in various fields. Understanding how to calculate and interpret the median is essential for anyone working with data, whether in environmental studies, statistics, or other disciplines. By arranging the data in ascending order and identifying the middle value, we can gain a clear understanding of the central tendency of the dataset. While the mean and mode also provide valuable insights, the median's resilience to extreme values makes it a particularly useful measure in many real-world scenarios.

Practice Questions

To further enhance your understanding of calculating the median, consider the following practice questions:

  1. What is the median of the dataset: 25, 30, 18, 35, 40?
  2. Find the median of the dataset: 10, 12, 15, 18, 20, 22.
  3. Calculate the median for the dataset: 5, 8, 10, 5, 12.

By working through these examples, you will solidify your ability to calculate and interpret the median in different contexts. Remember, the median is a powerful tool for understanding the central tendency of a dataset, especially when dealing with skewed data or outliers.

The median number of bottles collected is 19, which corresponds to option B.