What is Z score? What is Z score used for?
Z score is based on normal distribution, which is the most commonly used distribution for a continuous variable. In the famous '6-sigma' concept, 6 is the Z score and sigma refers to the standard deviation. Essentially, Z score is about the extent of deviation from the mean. It is generally acceptable to assume that the height distribution is normally distributed. The tallest NBA player (Manute Bol) is more than 3-sigma taller than the mean, which can be written as Z>3. On the other hand, the shortest man of the world (Chandra Bahadur Dangi) is more than 3-sigma shorter than the mean, which can be written as Z<-3. The majority of the population are within 2-sigma of the mean, with Z between -2 and 2. In mass production of commercial goods such as cars, the '6-sigma' concept was adopted to reduce the defect rate and ensure good quality of car production. '6-sigma' is a very high standard, as the probability of Z>6 is close to 0. Even when the 1.5 sigma mean shift is considered, the defect rate standard is still highly stringent at under 3.4 per million.
What is p value from Z score?
For a one-sided Z-test, if |z|=1.282 or more, p<0.10; if |z|=1.645 or more, p<0.05; if |z|=2.327 or more, p<0.01; if |z|=3.091 or more, p<0.001.
For a two-sided Z-test, if |z|=1.645 or more, p<0.10; if |z|=1.960 or more, p<0.05; if |z|=2.576 or more, p<0.01; if |z|=3.291 or more, p<0.001.
In the famous six-sigma test, try z=6, you get P(z>6)=9.8659e-10. If there is a 1.5 sigma shift with z=4.5, you get P(z>4.5)=0.0000034, or about 3.4 per million.