What Are The Mean, Median, And Mode Of The Following Set Of Runner Weights In Kilograms 90.7, 89.5, 95.4, 92.1, 82.6, 92.5, 94.4, 89.5, 86.7, 90.4, 94.4, 97.1, 89.5?

Jun 19, 2025 by ADMIN 166 views

Analyzing Runner Weights Unveiling Mean, Median, and Mode in District Marathon

In this article, we delve into the weights of the runners participating in the annual district level marathon. We will analyze the provided dataset to calculate the mean, median, and mode, providing a comprehensive understanding of the weight distribution among the runners. Understanding these statistical measures can offer valuable insights into the physical characteristics of the athletes and potentially inform training strategies.

The dataset consists of the following weights (in kilograms): 90.7, 89.5, 95.4, 92.1, 82.6, 92.5, 94.4, 89.5, 86.7, 90.4, 94.4, 97.1, 89.5. Our goal is to compute the mean, median, and mode for this dataset. These statistical measures will help us understand the central tendency and distribution of the runners' weights. The mean represents the average weight, the median indicates the middle value when the weights are arranged in ascending order, and the mode signifies the most frequently occurring weight. By calculating these measures, we can gain a clearer picture of the weight profile of the marathon participants.

Calculating the Mean

The mean, often referred to as the average, is a fundamental statistical measure that provides a central value for a dataset. To calculate the mean, we sum all the values in the dataset and divide by the number of values. In this case, we sum the weights of all the runners and divide by the total number of runners, which is 13. The formula for the mean ( ${\bar{x}}$ ) is:

${\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}}$

Where:

${\bar{x}}$ is the mean,
${\sum_{i=1}^{n} x_i}$ is the sum of all the values,
${n}$ is the number of values.

Applying this formula to our dataset, we first sum the weights:

7 + 89.5 + 95.4 + 92.1 + 82.6 + 92.5 + 94.4 + 89.5 + 86.7 + 90.4 + 94.4 + 97.1 + 89.5 = 1174.4

Next, we divide this sum by the number of runners (13):

${\frac{1174.4}{13} \approx 90.34}$

Therefore, the mean weight of the runners is approximately 90.34 kilograms. This value gives us an idea of the average weight among the marathon participants. Understanding the mean weight can be crucial for coaches and trainers in designing training programs tailored to the average physical attributes of the runners.

Determining the Median

The median is another essential measure of central tendency, representing the middle value in a dataset when the values are arranged in ascending or descending order. Unlike the mean, the median is not affected by extreme values or outliers, making it a robust measure for datasets with skewed distributions. To find the median, we first need to sort the weights in ascending order:

6, 86.7, 89.5, 89.5, 89.5, 90.4, 90.7, 92.1, 92.5, 94.4, 94.4, 95.4, 97.1

Since there are 13 values in the dataset (an odd number), the median is the middle value, which is the 7th value in the sorted list. In this case, the median is 90.7 kilograms. If the dataset had an even number of values, the median would be the average of the two middle values.

The median weight of 90.7 kilograms indicates that half of the runners weigh less than or equal to 90.7 kilograms, and half weigh more. Comparing the median to the mean (90.34 kilograms) provides insight into the distribution's symmetry. In this case, the mean and median are quite close, suggesting a relatively symmetrical distribution of weights. The median is particularly useful in datasets where extreme values might skew the mean, providing a more representative measure of central tendency.

Identifying the Mode

The mode is the value that appears most frequently in a dataset. It is a simple yet informative measure, particularly useful for categorical or discrete data. In our dataset of runner weights, we look for the weight that occurs most often. Examining the list of weights:

7, 89.5, 95.4, 92.1, 82.6, 92.5, 94.4, 89.5, 86.7, 90.4, 94.4, 97.1, 89.5

We can see that the weight 89.5 kilograms appears three times, which is more frequent than any other weight in the dataset. The weight 94.4 kilograms appears twice. Therefore, the mode is 89.5 kilograms.

The mode of 89.5 kilograms indicates that this is the most common weight among the runners in the marathon. The mode can be particularly useful in understanding the most typical or prevalent value in a dataset. In this context, it suggests that a significant number of runners weigh around 89.5 kilograms. Unlike the mean and median, the mode can be non-unique (i.e., there can be more than one mode) or may not exist if all values occur with the same frequency. Understanding the mode provides valuable information about the most common characteristics within a group.

Summary of Statistical Measures

To summarize, we have calculated the three primary measures of central tendency for the runners' weights in the annual district level marathon:

Mean: 90.34 kilograms
Median: 90.7 kilograms
Mode: 89.5 kilograms

The mean weight provides an average value, the median represents the middle value, and the mode identifies the most frequent weight. These measures offer a comprehensive understanding of the weight distribution among the marathon runners. The mean and median are relatively close, indicating a fairly symmetrical distribution of weights. The mode highlights the most common weight among the runners.

These statistical measures can be valuable for various purposes. For instance, coaches can use this information to tailor training programs based on the average weight and weight distribution of the runners. Medical staff can use this data to understand the general health profile of the participants. Furthermore, these statistics can be compared with data from previous years to identify any trends or changes in the physical characteristics of the runners.

Implications and Applications

The analysis of mean, median, and mode provides valuable insights into the dataset, but it's equally important to understand the implications and applications of these measures in real-world scenarios. In the context of the marathon runners' weights, these statistics can be used in several ways.

Training Program Design

The mean weight of 90.34 kilograms can be used as a benchmark for designing training programs. Coaches can consider this average weight when planning workouts and nutritional strategies. For example, understanding the average weight can help in determining appropriate training loads and dietary recommendations. If the runners are significantly above or below this average, individualized plans can be developed to cater to their specific needs.

Health and Nutritional Assessments

The median weight (90.7 kilograms) provides a more robust measure of central tendency, less influenced by extreme values. This can be particularly useful in health assessments. If there are runners with significantly higher or lower weights, the median gives a more representative central value. Health professionals can use this information to identify potential health risks associated with being underweight or overweight. Nutritional strategies can then be tailored to ensure that runners maintain a healthy weight, optimizing their performance and overall well-being.

Equipment and Gear Selection

The mode (89.5 kilograms) highlights the most common weight among the runners. This information can be valuable for equipment manufacturers and retailers. For instance, understanding the most common weight range can help in designing and stocking appropriate sizes of running gear, such as shoes and apparel. Ensuring that equipment is suitable for the majority of participants can enhance their comfort and performance during the marathon.

Comparative Analysis

The statistical measures calculated for this group of runners can be compared with those from other marathons or previous years. This comparative analysis can reveal trends and changes in the physical characteristics of marathon participants over time. For example, if the mean weight has increased over the years, it might indicate changes in training habits, dietary practices, or other factors. Such comparisons can inform future training strategies and health interventions.

Risk Assessment and Injury Prevention

Understanding the distribution of weights can also aid in risk assessment and injury prevention. Runners who are significantly above or below the average weight may be at a higher risk of certain types of injuries. For instance, heavier runners may be more prone to joint stress, while underweight runners may face issues related to fatigue and energy depletion. By identifying these potential risks, coaches and medical staff can implement preventive measures to ensure the safety and well-being of the runners.

Conclusion

In conclusion, analyzing the weights of the runners in the annual district level marathon using measures such as mean, median, and mode provides valuable insights into the physical characteristics of the participants. The mean weight was calculated to be 90.34 kilograms, the median weight was 90.7 kilograms, and the mode was 89.5 kilograms. These statistical measures offer a comprehensive view of the central tendency and distribution of weights within the group.

These insights have numerous practical applications, including designing training programs, conducting health and nutritional assessments, selecting appropriate equipment, performing comparative analyses, and implementing risk assessment and injury prevention strategies. By understanding the weight profile of the runners, coaches, medical staff, and other stakeholders can make informed decisions to optimize performance, ensure runner safety, and promote overall well-being. The use of statistical analysis in this context exemplifies how data-driven insights can enhance various aspects of sports and athletics.

By examining the mean, median, and mode, we gain a deeper understanding of the runners' physical attributes, which can inform training and health management strategies. This analysis underscores the importance of statistical measures in providing meaningful insights from data.

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Find the mean, median, and mode of the following runner weights (in kilograms): 90.7, 89.5, 95.4, 92.1, 82.6, 92.5, 94.4, 89.5, 86.7, 90.4, 94.4, 97.1, 89.5.