Draft:Rule of Five (statistics)

In statistics the rule of five is a statistical rule of thumb used to quickly estimate the median of a population. It states that there is a 93.75% chance that the true median of a population lies between the smallest and largest values in any random sample of five taken from that population.

The rule of five offers a way to reduce uncertainty and make faster business decisions without extensive data collection. Instead of surveying an entire population, the rule of five involves selecting a random sample of five members to represent the population. This statistical tool is used across disciplines like business research, software engineering, statistical computing, data analytics, and social research.

How it Works

The rule is based on the probability of randomly selecting values above or below the median of a population. There's an equal chance (50%) of picking a random value above or below the median, similar to a coin flip. The probability of selecting five values that are all above the median (akin to flipping five heads in a row) is $(\frac{1}{2}{)}^5=3.125\mathrm{\%}$. The same probability applies to selecting five values all below the median.

Therefore, the probability of all five values being either above or below the median is $3.125\mathrm{\%}+3.125\mathrm{\%}=6.25\mathrm{\%}$. Consequently, the probability that at least one value is above the median and at least one is below (meaning the median falls within the range of the sample) is $100\mathrm{\%}-6.25\mathrm{\%}=93.75\mathrm{\%}$.

Usefulness

The rule of five is valuable when:

• A quick estimate of the median is needed.
• Resources or time are limited, making extensive data collection impractical.
• An acceptable level of accuracy is sufficient for decision-making or trend prediction.
• It can be particularly useful when there are high levels of uncertainty to begin with.

Limitations

• Randomness is critical: The selected sample of five must be truly random to avoid bias and ensure the rule's validity.
• The rule provides an estimate of the median and may not replace the need for thorough data analysis in all situations.
• The range might be very wide with a sample size of only five.

History and Origin

The rule of five was conceived by Douglas W. Hubbard, an expert in risk management, metrics, and decision analysis. He introduced it in his book "How to Measure Anything: Finding the Value of Intangibles in Business". Hubbard chose the number five because it’s memorable and the smallest sample size that provides a probability greater than 90%.