In a recent LinkedIn poll aimed at CFA candidates, we posed a question about the probability calculation under a standard normal distribution. The results, intriguingly, highlighted a common area of confusion among finance professionals. Here’s a closer look at the question, the correct answer, and why other options might have seemed plausible to many.

The Question and Its Outcome

The question asked was: “If a random variable Z follows a standard normal distribution, what is the probability that Z is less than 2?” Here’s a breakdown of the poll responses:

• Approximately 0.9772: 52% (Correct Answer)
• Approximately 0.5: 7%
• Approximately 0.0228: 13%
• Approximately 0.95: 28%

Correct Answer: Approximately 0.9772

The correct answer to this question is approximately 0.9772. This figure represents the cumulative probability that a standard normally distributed variable (Z) will have a value less than 2. In a standard normal distribution, the mean is 0 and the standard deviation is 1. The Z-score of 2 means that the value is 2 standard deviations above the mean. Using the Z-table, or the cumulative distribution function for the standard normal distribution, the probability that Z is less than 2 is approximately 0.9772.

Analyzing Incorrect Responses

• Approximately 0.5: This answer may have been chosen by those who mistakenly thought of the median or the middle value of the distribution. Since the mean and median of a normal distribution are the same (zero for the standard normal distribution), it could seem intuitive to select 0.5, but this ignores the question’s specific reference to a Z-score of 2.
• Approximately 0.0228: This represents the probability of Z being more than 2 standard deviations away from the mean on the upper side (or less than -2 on the lower side). It’s the complement of the correct answer (1 – 0.9772). Candidates may have chosen this by confusing “less than 2” with “more than 2” or by mistakenly considering the upper tail area beyond 2 standard deviations.
• Approximately 0.95: This answer was surprisingly popular, with 28% selecting it. This likely stems from a common rounding of probabilities associated with certain confidence intervals in a normal distribution. For instance, approximately 95% of data falls within two standard deviations from the mean (from -2 to 2, symmetrically). Therefore, those who chose this option may have incorrectly recalled the probability associated with being within two standard deviations of the mean, rather than specifically less than 2.

Educational Takeaway

This question underscores the importance of thoroughly understanding the properties and calculations associated with the normal distribution, a foundational concept in statistics and financial analysis. The common selection of option D suggests a potential mix-up between different statistical rules and emphasizes the need for clear understanding when applying these concepts in real-world scenarios.

In conclusion, the majority of candidates were able to identify the correct answer, demonstrating good comprehension of normal distributions. However, the significant minority opting for other answers highlights an area where further review could be beneficial. Our discussion aims to clarify these misunderstandings and strengthen the grasp of such crucial concepts among finance professionals.