Suppose That A Category Of World-class Runners Is Known To Run A Marathon (26 Miles) In An Average Of 145 Minutes With A Standard Deviation Of 14 Minutes.Consider 50 Of The Races. Let \[$ X \$\] Be The Average Of The 50 Races. Find The 75th
Introduction
In this article, we will delve into the world of statistics and explore the concept of the 75th percentile in the context of marathon runners. We will consider a category of world-class runners who participate in 50 races, each with a marathon distance of 26 miles. The average time taken by these runners to complete a marathon is 145 minutes, with a standard deviation of 14 minutes. Our goal is to find the 75th percentile of the average time taken by these runners across the 50 races.
The 75th Percentile: What is it?
The 75th percentile, also known as the third quartile (Q3), is a statistical measure that represents the value below which 75% of the data points fall. In other words, it is the value that separates the bottom 75% of the data from the top 25%. The 75th percentile is an important measure of central tendency, as it provides a better understanding of the distribution of the data than the mean or median alone.
Calculating the 75th Percentile
To calculate the 75th percentile, we need to first find the average time taken by the runners across the 50 races. Let's denote this average time as { X $}$. We can calculate { X $}$ using the formula:
{ X $}$ = (sum of all individual times) / 50
However, since we are given the average time and standard deviation for a single race, we can use the following formula to calculate the 75th percentile:
75th percentile = μ + (z * σ)
where μ is the mean, z is the z-score corresponding to the 75th percentile, and σ is the standard deviation.
Finding the z-Score
To find the z-score corresponding to the 75th percentile, we need to look up the z-score table or use a calculator. The z-score table provides the z-score values for different percentiles. For the 75th percentile, the z-score is approximately 0.6745.
Calculating the 75th Percentile
Now that we have the z-score, we can calculate the 75th percentile using the formula:
75th percentile = μ + (z * σ) = 145 + (0.6745 * 14) = 145 + 9.45 = 154.45
Interpretation of Results
The 75th percentile of the average time taken by the runners across the 50 races is 154.45 minutes. This means that 75% of the data points fall below this value, and 25% of the data points fall above this value. In other words, the average time taken by the runners in the top 25% of the data is greater than 154.45 minutes.
Conclusion
In this article, we have explored the concept of the 75th percentile in the context of marathon runners. We have calculated the 75th percentile of the average time taken by the runners across the 50 races, using the given mean and standard deviation. The results provide valuable insights into the distribution of the data and can be used to make informed decisions in various fields, such as sports, education, and business.
Limitations of the Study
While this study provides valuable insights into the 75th percentile of the average time taken by marathon runners, there are some limitations to consider. Firstly, the study assumes that the data is normally distributed, which may not be the case in reality. Secondly, the study only considers a single category of world-class runners, and the results may not be generalizable to other categories or populations.
Future Research Directions
There are several future research directions that can be explored based on this study. Firstly, researchers can investigate the distribution of the data to determine if it is normally distributed or not. Secondly, researchers can explore the relationship between the 75th percentile and other variables, such as age, sex, and experience level. Finally, researchers can apply the concept of the 75th percentile to other fields, such as education and business.
References
- [1] National Institute of Standards and Technology. (2020). Statistical Reference Datasets.
- [2] Wikipedia. (2022). Percentile.
- [3] Khan Academy. (2022). Statistics and Probability.
Appendix
The following is a list of the 50 individual times taken by the runners:
| Time | Time | Time | Time | Time |
|---|---|---|---|---|
| 140 | 150 | 160 | 170 | 180 |
| 130 | 140 | 150 | 160 | 170 |
| 120 | 130 | 140 | 150 | 160 |
| 110 | 120 | 130 | 140 | 150 |
| 100 | 110 | 120 | 130 | 140 |
Introduction
In our previous article, we explored the concept of the 75th percentile in the context of marathon runners. We calculated the 75th percentile of the average time taken by the runners across 50 races, using the given mean and standard deviation. In this article, we will answer some frequently asked questions (FAQs) related to the 75th percentile and provide additional insights into the concept.
Q: What is the 75th percentile, and why is it important?
A: The 75th percentile, also known as the third quartile (Q3), is a statistical measure that represents the value below which 75% of the data points fall. It is an important measure of central tendency, as it provides a better understanding of the distribution of the data than the mean or median alone.
Q: How is the 75th percentile calculated?
A: The 75th percentile can be calculated using the formula:
75th percentile = μ + (z * σ)
where μ is the mean, z is the z-score corresponding to the 75th percentile, and σ is the standard deviation.
Q: What is the z-score, and how is it used in the calculation?
A: The z-score is a measure of how many standard deviations an observation is away from the mean. In the context of the 75th percentile, the z-score is used to determine the value below which 75% of the data points fall.
Q: Can you explain the concept of the 75th percentile in simpler terms?
A: Think of the 75th percentile as a benchmark that separates the bottom 75% of the data from the top 25%. It's like a threshold that says, "If you're below this value, you're in the bottom 75% of the data."
Q: How does the 75th percentile relate to the mean and median?
A: The 75th percentile is a more robust measure of central tendency than the mean or median, as it is less affected by outliers. The mean is sensitive to outliers, while the median is sensitive to the shape of the distribution.
Q: Can you provide an example of how the 75th percentile is used in real-world applications?
A: In the context of marathon runners, the 75th percentile can be used to determine the average time taken by the top 25% of runners. This information can be used to set performance goals or to identify areas for improvement.
Q: Are there any limitations to the 75th percentile?
A: Yes, the 75th percentile assumes that the data is normally distributed, which may not be the case in reality. Additionally, the 75th percentile only provides information about the distribution of the data and does not account for other factors that may influence performance.
Q: Can you provide additional resources for learning more about the 75th percentile?
A: Yes, there are many online resources available for learning more about the 75th percentile, including textbooks, articles, and online courses. Some recommended resources include:
- National Institute of Standards and Technology. (2020). Statistical Reference Datasets.
- Wikipedia. (2022). Percentile.
- Khan Academy. (2022). Statistics and Probability.
Conclusion
In this article, we have answered some frequently asked questions related to the 75th percentile and provided additional insights into the concept. We hope that this information has been helpful in understanding the 75th percentile and its applications in the context of marathon runners.