Statistical Tables for Testing Data Closer to Expectation than Compatible with Random Sampling

Abstract

A p-value <0.05 is generally used as a cut-off level to indicate a significant difference from what we expect. A p-value of >0.05, then, indicates no significant difference. The larger the p-value the smaller the chance of a difference. A p-value of 1.00 means 0% chance of a difference, while a p-value of 0.95 means a chance of difference close to 0. A p-value of >0.95 literally means that we have >95% chance of finding a result less close to expectation, which means a chance of <(1 − 0.95), i.e., <0.05 of finding a result this close or closer. Using the traditional 5% decision level, this would mean, that we have a strong argument that such data are not completely random. The example from the previous chapter is used once more. In a Mendelian experiment the expected ratio of yellow-peas/-green peas is 1/1. A highly representative random sample of n = 100 might consist of 50 yellow and 50 green peas. However, the larger the sample the smaller the chance of finding exactly fifty/fifty. The chance of exactly 5,000 yellow/5,000 green peas or even the chance of a result very close to this result is, due to large variability in biological processes, almost certainly zero. In a sample of 10,000 peas, you might find 4,997 yellow and 5,003 green peas. What is the chance of finding a result this close to expectation? A chi-square test produces here a p > 0.95 of finding a result less close, and consequently, <0.05 of finding a result this close or closer. Using the 5% decision level, this would mean, that we have a strong argument that these data are not completely random. The example is actually based on some true historic facts, Mendel improved his data (Cleophas and Cleophas 2001).