Question

NFL data from the 2006 football season reported the number of yards gained by each of the league's 167 wide receivers: a) According to the Normal model, what percent of receivers would you expect to gain fewer yards than 2 standard deviations below the mean number of yards? b) For these data, what does that mean? c) Explain the problem in using a Normal model here.

Short answer

a) About 2.5% gain fewer yards; b) implies lower performance; c) data might not follow a normal distribution.

Step-by-step solution

Understanding the Normal Distribution

To solve the problem, we need to know that a normal distribution is a probability distribution that is symmetric about the mean. In a standard normal distribution, the mean is 0, and the standard deviation is 1. We also know that about 95% of data within a normal distribution would lie within two standard deviations (both positive and negative) from the mean.

Find the Percent of Receivers Gaining Fewer Yards

The percentage of receivers that gain fewer yards than 2 standard deviations below the mean corresponds to the cumulative probability for the z-score -2. In a standard normal distribution, about 2.5% of the data lies below 2 standard deviations from the mean. Thus, we expect approximately 2.5% of receivers to fall below this threshold.

Interpret the Results for the Data

If 2.5% of the receivers gained fewer yards than 2 standard deviations below the mean, it implies that these receivers are outliers in terms of performance, gaining significantly fewer yards compared to an average receiver of the dataset.

Discuss the Problem with Using the Normal Model

The issue with applying a normal model here is that real-world data, like the number of yards, might not perfectly follow a normal distribution. Factors such as team strategy, player injuries, or playtime can cause skewness, suggesting the actual data may not be symmetric or might have different peaks and tail behaviors compared to a standard normal distribution.

Key concepts

Standard Deviation

The concept of standard deviation is essential for understanding how data is distributed in a set. In any dataset, the standard deviation measures how much the data points deviate from the mean, or average, of the dataset. It's a crucial indicator of data spread. A smaller standard deviation means that the data points are closer to the mean, while a larger standard deviation signifies that the data points are spread out over a wider range.
When analyzing NFL receiver yards, the standard deviation helps determine how consistent the receivers are in gaining yards. A low standard deviation would imply most receivers gain a similar number of yards, whereas a high standard deviation would suggest a wide variance in receiver performance.

Z-score

A z-score is a statistical measurement that describes a value's position relative to the mean of a group of values. It is expressed in terms of standard deviations from the mean. Therefore, a z-score allows us to understand how far any given data point lies from the average.
For example, if an NFL receiver's yards has a z-score of -2, it means this receiver's yards are 2 standard deviations below the mean yards gained by all receivers. The z-score is vital for determining outliers or significant deviations from the average.

Cumulative Probability

Cumulative probability refers to the probability that a random variable is less than or equal to a particular value. In the context of normal distribution, cumulative probability helps identify what proportion of data falls below a certain z-score, which we use to make probabilistic predictions about the dataset.
For instance, in the NFL data example, the cumulative probability of a z-score of -2 indicates that about 2.5% of data is expected to fall below this point. Thus, it tells us directly the likelihood of receivers gaining less than a specific yard number. This information can be crucial for determining how unusual a particular performance might be.

Data Skewness

Data skewness refers to the asymmetry in a statistical distribution. In perfectly normal distributions, the data should be symmetric, meaning it has no skewness. However, in real-world scenarios, data often shows some degree of skewness either to the right or left.
When applying a normal model to data, noticing skewness is important to ensure accuracy. For the NFL data, external factors like player injuries and playing time may create skewness, making the symmetrical assumptions of a normal model less applicable. Real-world datasets often need adjustments for skewness before making precise predictions or analyses based on normal distribution.