6.5: Unusual Results

Unusual results are those that have a very low chance of occurring. Unusual results can be identified using probabilities and the range rule of thumb. In problems involving probability, unusual results can be observed in 2 instances – an unusually high number of successes or an unusually low number of successes.

According to the range rule of thumb, any value above or below two standard deviations, 2σ from the mean, μ is considered unusual.

Maximum unusual value = μ + 2σ

Minimum unusual value = μ - 2σ

Unusual results are events that have very low chances of occurrence. They can be identified either with the range rule of thumb or the probability values.

Consider the probability distribution of seat occupancy in a carpool with a mean of 3.5 and a standard deviation of 1.2.

According to the range rule of thumb, the majority of the random variable values must lie within two standard deviations of the mean.

All the remaining data values that fall outside this range are unusual values.

To identify unusual results from probability values, consider the probability distribution of the number of heads in a coin tossed five times.

Since the probability of zero or fewer heads is less than 0.05, those results can be labeled unusual.

Similarly, if the probability of five or more heads is less than 0.05, those results are also unusual.

The cut-off value of two sigmas for random variables and 0.05 for probability is not rigid. It can be decided based on the context of the problem.