Why do the mean and standard deviation have no predictive power for a uniform distribution?

Conditions

• The distribution is uniform.
• All outcomes within the support interval are equally likely.

Reasoning, step by step

1. Recall the definition of a uniform distribution, where every event is equally likely.
2. Analyze the role of the mean and standard deviation as summary statistics.
3. Conclude that because probability is spread evenly, these statistics cannot predict which specific outcome will occur.

Example

For the bus waiting time on [0, 60], the mean is 30 minutes and the standard deviation is 17.321 minutes. However, waiting exactly 30 minutes is no more likely than waiting 5 minutes or 55 minutes, because all waiting times are equally likely.

Common misconceptions

• Assuming that the mean represents the most likely outcome in any distribution.
• Believing that a low standard deviation implies outcomes are tightly clustered around the mean in a uniform distribution.

Watch the explanation

Explore next

• How does the video distinguish a uniform distribution from a normal distribution?
• What is the formula for the mean of a continuous uniform distribution?
• What is the formula for the standard deviation of a continuous uniform distribution?