Tukey's method

Used in ANOVA to create confidence intervals for all pairwise differences between factor level means while controlling the family error rate to a level you specify. It is important to consider the family error rate when making multiple comparisons because your chances of making a type I error for a series of comparisons is greater than the error rate for any one comparison alone. To counter this higher error rate, Tukey's method adjusts the confidence level for each individual interval so that the resulting simultaneous confidence level is equal to the value you specify.

For example, you are measuring the response times for memory chips. You sampled 25 chips from five different manufacturers. The ANOVA resulted in a p-value of 0.01, leading you to conclude that at least one of the manufacturer means is different from the others.  

You decide to look at all 10 comparisons between the five plants to determine specifically which means are different.  Using Tukey's method, you specify that the entire set of comparisons should have a family error rate of 0.05 (equivalent to a 95% joint confidence level).  Minitab calculates that the 10 individual confidence levels need to be 99.35% in order to obtain the 95% joint confidence level. These wider Tukey confidence intervals provide less precise estimates of the population parameter but limit the probability that one or more of the confidence intervals does not contain the true difference to a maximum of 5%. Understanding this context, you can then look at the confidence intervals to see if any do not include zero, suggesting a significant difference.