Calculating Percentage Of Runners Below A Certain Time Using Normal Distribution

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In the realm of mathematics and statistics, understanding data distribution is crucial for making informed decisions and predictions. One common type of distribution is the normal distribution, often referred to as the bell curve. This distribution is characterized by its symmetrical shape, with the majority of data points clustered around the mean. In this article, we will delve into a scenario involving the running times of 15-year-old athletes, which are approximately normally distributed. Our primary goal is to determine the percentage of runners who achieve times less than a specific threshold. This analysis will involve leveraging the properties of the normal distribution, including the mean and standard deviation, to calculate probabilities and understand the spread of data.

Let's consider a scenario where the running times of all 15-year-old runners in a particular race are approximately normally distributed. The mean time, denoted by $\mu$, is 18 seconds, and the standard deviation, denoted by $\sigma$, is 1.2 seconds. Our objective is to calculate the percentage of runners who have times less than 14.4 seconds. This problem requires us to apply our understanding of normal distributions and z-scores to determine the probability of a runner achieving a time within the specified range. To effectively address this problem, we will explore the concepts of z-scores, cumulative distribution functions, and how to interpret these values in the context of the given scenario. This problem provides a practical application of statistical concepts in a real-world setting, highlighting the importance of understanding data distributions in various fields.

Before we proceed with the calculations, let's review some key concepts related to normal distributions:

  • Normal Distribution: A normal distribution, often called a Gaussian distribution, is a continuous probability distribution that is symmetrical around its mean. It is characterized by its bell-shaped curve, where the majority of data points cluster around the mean. The normal distribution is widely used in statistics to model various phenomena, such as heights, weights, and test scores.

  • Mean (\μ\mu\\): The mean is the average value of a dataset. In a normal distribution, the mean represents the center of the distribution. It is calculated by summing all the data points and dividing by the total number of data points.

  • Standard Deviation (\σ\sigma\\): The standard deviation measures the spread or dispersion of data points around the mean. A smaller standard deviation indicates that the data points are clustered closer to the mean, while a larger standard deviation indicates a wider spread. It is calculated as the square root of the variance.

  • Z-Score: A z-score, also known as a standard score, represents the number of standard deviations a data point is away from the mean. It is calculated by subtracting the mean from the data point and dividing the result by the standard deviation. Z-scores allow us to standardize data from different normal distributions, making it easier to compare and analyze them. The formula for calculating the z-score is:

    z=fracx−musigmaz = \\frac{x - \\mu}{\\sigma}

    where:

    • x is the data point
    • \μ\mu\\ is the mean
    • \σ\sigma\\ is the standard deviation

  • Cumulative Distribution Function (CDF): The cumulative distribution function (CDF) of a normal distribution gives the probability that a random variable is less than or equal to a certain value. In other words, it calculates the area under the normal curve to the left of a given point. The CDF is often denoted by Φ(z), where z is the z-score. CDF values range from 0 to 1, representing probabilities from 0% to 100%.

To determine the percentage of runners with times less than 14.4 seconds, we need to follow these steps:

  1. Calculate the z-score:

    Using the formula for z-score, we have:

    z=frac14.4−181.2=frac−3.61.2=−3z = \\frac{14.4 - 18}{1.2} = \\frac{-3.6}{1.2} = -3

    This means that 14.4 seconds is 3 standard deviations below the mean.

  2. Find the probability using the CDF:

    We need to find the probability that a runner's time is less than 14.4 seconds, which corresponds to finding the area under the normal curve to the left of z = -3. We can use a z-table or a statistical calculator to find the CDF value for z = -3.

    Looking up the z-score of -3 in a standard normal distribution table (or using a calculator), we find that the corresponding probability is approximately 0.0013.

    This value represents the probability of a runner having a time less than 14.4 seconds.

  3. Convert the probability to a percentage:

    To express the probability as a percentage, we multiply it by 100:

    0.0013times100=0.13%0. 0013 \\times 100 = 0.13\%

The core of this problem lies in understanding how to translate a specific value (14.4 seconds) within a normal distribution into a probability. The z-score serves as the bridge between the raw data value and the standard normal distribution, which is a normal distribution with a mean of 0 and a standard deviation of 1. By calculating the z-score, we essentially standardize the value, allowing us to compare it to the standard normal distribution and determine its relative position.

The z-score of -3 tells us that 14.4 seconds is significantly below the average time of 18 seconds. Specifically, it is 3 standard deviations below the mean. This is a substantial deviation, suggesting that very few runners would achieve a time this low.

The cumulative distribution function (CDF) is the key to finding the probability. The CDF for a given z-score gives the area under the standard normal curve to the left of that z-score. This area represents the probability of observing a value less than the corresponding value in the original distribution. In our case, we are interested in the probability of a runner's time being less than 14.4 seconds, which corresponds to the area under the standard normal curve to the left of z = -3.

Looking up the z-score of -3 in a z-table or using a statistical calculator provides the CDF value, which is approximately 0.0013. This value signifies that there is a 0.0013 probability (or 0.13%) of a runner achieving a time less than 14.4 seconds. This is a very small probability, which aligns with our understanding that 14.4 seconds is a significantly fast time compared to the average time.

This analysis has practical implications in various fields, particularly in sports and athletics. Coaches and trainers can use this type of statistical analysis to assess the performance of athletes, identify outliers, and set realistic goals. For instance, knowing the distribution of running times can help a coach determine the likelihood of an athlete achieving a particular time and tailor training programs accordingly.

Furthermore, this analysis can be extended to compare the performance of different groups of athletes. By comparing the distributions of running times for different age groups, genders, or training programs, we can gain insights into the factors that influence athletic performance. This can be valuable for developing effective training strategies and identifying talent.

However, it is important to consider the limitations of this analysis. The assumption of a normal distribution may not always hold true in real-world scenarios. Factors such as athlete fatigue, weather conditions, and course difficulty can influence running times and deviate from the normal distribution. Therefore, it is crucial to carefully evaluate the data and consider potential confounding factors when interpreting the results.

In conclusion, by calculating the z-score and using the cumulative distribution function, we determined that approximately 0.13% of the 15-year-old runners have times less than 14.4 seconds. This example demonstrates the power of statistical analysis in understanding data distributions and making predictions. By applying the concepts of normal distribution, z-scores, and CDF, we can gain valuable insights into various phenomena and make informed decisions in a wide range of fields. This analysis underscores the importance of a strong foundation in statistical concepts for anyone working with data, whether it be in sports, science, business, or any other domain.

Calculating Percentage of Runners Below a Certain Time Using Normal Distribution